qu.1.topic=Solving Systems of Linear Equations@

qu.1.1.mode=Inline@
qu.1.1.name=System of Equations into a Matrix@
qu.1.1.comment=<p>Make sure you enter the coefficients in the order <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>z</mi></mrow></mstyle></math>&nbsp;in each row of the matrix.&nbsp; Don't forget the sign.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=range(-10,10);
$b=range(-10,10);
$c=range(-10,10);
$d=range(-10,10);
$e=range(-10,10);
$f=range(-10,10);
$g=range(-10,10);
$h=range(-10,10);
$i=range(-10,10);
$v=maple("
v1:=Matrix([[$a,$b,$c],[$d,$e,$f],[$g,$h,$i]]):
v2:=MathML[ExportPresentation](v1):
v3:=MathML[ExportPresentation]($a*x+($b)*y+($c)*z=0):
v4:=MathML[ExportPresentation]($d*x+($e)*y+($f)*z=0):
v5:=MathML[ExportPresentation]($g*x+($h)*y+($i)*z=0):
convert(v1,string),convert(v2,string),v3,v4,v5
");
$ans=switch(0,$v);
$anspretty=switch(1,$v);
$eq1=switch(2,$v);
$eq2=switch(3,$v);
$eq3=switch(4,$v);@
qu.1.1.uid=93916ecd-05f2-4334-95b1-2e564475fe83@
qu.1.1.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=System Of Equations Into A Matrix;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.1.1.weighting=1@
qu.1.1.numbering=alpha@
qu.1.1.part.1.name=sro_id_1@
qu.1.1.part.1.maple_answer=printf("$anspretty");@
qu.1.1.part.1.editing=useHTML@
qu.1.1.part.1.question=(Unset)@
qu.1.1.part.1.libname=@
qu.1.1.part.1.mode=Maple@
qu.1.1.part.1.allow2d=2@
qu.1.1.part.1.plot=@
qu.1.1.part.1.maple=ans:=$ans:
grade:=0:
for i from 1 to 3 do
for j from 1 to 3 do
if ans[i,j] = $RESPONSE[i,j]
then grade:=grade+0.11112:
end if;
end;
end;
grade;@
qu.1.1.part.1.type=maple@
qu.1.1.question=<p>Translate the following system of equations into a Matrix <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mi>x</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>y</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>z</mi></mrow></mtd></mtr></mtable></mfenced></mrow></mrow></mstyle></math>=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>0</mn></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math> equals the system of equations.</p><p>$eq1</p><p>$eq2</p><p>$eq3</p><p>&nbsp;</p><p>(To input your answer, right-click on the box below to bring up the symbols bar, select the button with a square made out of 9 smaller squares and select the appropriate size of matrix.)</p><p align="left"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>A</mi></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.2.mode=Multiple Choice@
qu.1.2.name=systems of equations -- unique solution@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=range(2,8,1);
$b=range(2,8,1);
$c=range(5,8,1);
$eq1=mathml($a*x+($b+1)*y);
$eq2=mathml($b*$c*y+$a*$c*x);
$d=range(2,4,1);
$e=$c*$d;@
qu.1.2.uid=6e1d8349-2c98-48b2-a262-9ef37f0a72c6@
qu.1.2.info=  Course=Introductory Mathematical Economics;
  Topic=System Of Equations;
  Sub-Topic=Number Of Solutions;
  Author=Asha Sadanand;
  Difficulty=Medium;
@
qu.1.2.question=<p>The following system of equations</p>
<p>$eq1=$d</p>
<p>$eq2 = $e</p>@
qu.1.2.answer=2@
qu.1.2.choice.1=is underdetermined@
qu.1.2.choice.2=has a unique solution@
qu.1.2.choice.3=has no solution@
qu.1.2.choice.4=has an infinite number of solutions@
qu.1.2.fixed=@

qu.1.3.question=<p>Give the coefficient matrix of the following system. Use the order <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Z</mi></mrow></mstyle></math>.</p>
<p>$Eq1<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$c1</mi></mrow></mstyle></math></p>
<p>$Eq2<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$c2</mi></mrow></mstyle></math></p>
<p>$Eq3<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$c3</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<blockquote></blockquote>@
qu.1.3.maple=LinearAlgebra[Equal]($RESPONSE,$ans);@
qu.1.3.allow2d=2@
qu.1.3.maple_answer=printf(MathML[ExportPresentation]($ans));@
qu.1.3.type=maple@
qu.1.3.mode=Maple@
qu.1.3.name=coefficient matrix - random 3x3@
qu.1.3.comment=<p>Make sure you enter the coefficients in the order&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>Y</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>Z</mi></mrow></mstyle></math> in each row of the matrix. Don't forget the sign.</p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$v=maple("
randomize():
m:=LinearAlgebra[RandomMatrix](3,generator=rand(-9..9),attributes=[nonsingular]):
m1:=MathML[ExportPresentation](m):
eq1:=MathML[ExportPresentation](m[1,3]*Z+m[1,1]*X+m[1,2]*Y):
eq2:=MathML[ExportPresentation]($m[2,1]*X+$m[2,3]*Z+$m[2,2]*Y):
eq3:=MathML[ExportPresentation]($m[3,2]*Y+$m[3,1]*X+$m[3,3]*Z):
convert(m,string), m1, eq1, eq2, eq3
");
$ans=switch(0,$v);
$anspretty=switch(1,$v);
$Eq1=switch(2,$v);
$Eq2=switch(3,$v);
$Eq3=switch(4,$v);
$c1=range(-8,8,1);
$c2=range(-8,8,1);
$c3=range(-8,8,1);@
qu.1.3.uid=7f3c488e-4b6b-46fc-949f-71221b58b05b@
qu.1.3.info=  Course=Introductory Mathematical Economics;
  Topic=Systems Of Equations;
  Sub-Topic=Matrix Representation;
  Author=Asha Sadanand;
  Difficulty=Medium;
@

qu.1.4.mode=Inline@
qu.1.4.name=Jumbled Equation@
qu.1.4.comment=<p>The rearranged system looks like this:</p>
<p>$ansApretty</p>
<p>$ansBpretty</p>
<p>$ansCpretty</p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$ka=range(1,9);
$kb=range(1,9);
$kc=range(1,9);
$a1=range(0,10);
$a2=range(-10,10);
$a3=range(-10,10);
$a4=range(-10,10);
$b1=range(0,10);
$b2=range(-10,10);
$b3=range(-10,10);
$b4=range(-10,10);
$c1=range(0,10);
$c2=range(-10,10);
$c3=range(-10,10);
$c4=range(-10,10);
$v=maple("
  eqaAns:=MathML[ExportPresentation](sort(($a1)*x+($a2)*y+($a3)*z=($a4),order=plex(x,y,z))):
if $ka=1 then
  eqa:=MathML[ExportPresentation](sort(($a1)*x+($a2)*y+($a3)*z=($a4),order=plex(x,y,z))):
elif $ka=2 then
  eqa:=MathML[ExportPresentation](sort(($a1)*x+($a2)*y+($a3)*z=($a4),order=plex(y,x,z))):
elif $ka=3 then
  eqa:=MathML[ExportPresentation](sort(($a1)*x+($a2)*y+($a3)*z=($a4),order=plex(z,x,y))):
elif $ka=4 then
  eqa:=MathML[ExportPresentation](sort(($a1)*x+($a2)*y+($a3)*z=($a4),order=plex(z,y,x))):
elif $ka=5 then
  eqa:=MathML[ExportPresentation](sort(($a1)*x+($a2)*y+($a3)*z=($a4),order=plex(x,z,y))):
elif $ka=6 then
  eqa:=MathML[ExportPresentation](-($a4)+($a2)*y+($a3)*z=-($a1)*x):
elif $ka=7 then
  eqa:=MathML[ExportPresentation](-($a4)+($a3)*z=-($a1)*x-($a2)*y):
elif $ka=8 then
  eqa:=MathML[ExportPresentation](-($a4)+($a2)*y+($a1)*x=-($a3)*z):
elif $ka=9 then
  eqa:=MathML[ExportPresentation](($a2)*y+($a3)*z=-($a1)*x+($a4)):
end if:

eqbAns:=MathML[ExportPresentation](sort(($b1)*x+($b2)*y+($b3)*z=($b4),order=plex(x,y,z))):

if $kb=1 then
  eqb:=MathML[ExportPresentation](sort(($b1)*x+($b2)*y+($b3)*z=($b4),order=plex(x,y,z))):
elif $kb=2 then
  eqb:=MathML[ExportPresentation](sort(($b1)*x+($b2)*y+($b3)*z=($b4),order=plex(y,x,z))):
elif $kb=3 then
  eqb:=MathML[ExportPresentation](sort(($b1)*x+($b2)*y+($b3)*z=($b4),order=plex(z,x,y))):
elif $kb=4 then
  eqb:=MathML[ExportPresentation](sort(($b1)*x+($b2)*y+($b3)*z=($b4),order=plex(z,y,x))):
elif $kb=5 then
  eqb:=MathML[ExportPresentation](sort(($b1)*x+($b2)*y+($b3)*z=($b4),order=plex(x,z,y))):
elif $kb=6 then
  eqb:=MathML[ExportPresentation](-($b4)+($b2)*y+($b3)*z=-($b1)*x):
elif $kb=7 then
  eqb:=MathML[ExportPresentation](-($b4)+($b3)*z=-($b1)*x-($b2)*y):
elif $kb=8 then
  eqb:=MathML[ExportPresentation](-($b4)+($b2)*y+($b1)*x=-($b3)*z):
elif $kb=9 then
  eqb:=MathML[ExportPresentation](($b2)*y+($b3)*z=-($b1)*x+($b4)):
end if:

eqcAns:=MathML[ExportPresentation](sort(($c1)*x+($c2)*y+($c3)*z=($c4),order=plex(x,y,z))):

if $kc=1 then
  eqc:=MathML[ExportPresentation](sort(($c1)*x+($c2)*y+($c3)*z=($c4),order=plex(x,y,z))):
elif $kc=2 then
eqc:=MathML[ExportPresentation](sort(($c1)*x+($c2)*y+($c3)*z=($c4),order=plex(y,x,z))):
elif $kc=3 then
eqc:=MathML[ExportPresentation](sort(($c1)*x+($c2)*y+($c3)*z=($c4),order=plex(z,x,y))):
elif $kc=4 then
eqc:=MathML[ExportPresentation](sort(($c1)*x+($c2)*y+($c3)*z=($c4),order=plex(z,y,x))):
elif $kc=5 then
eqc:=MathML[ExportPresentation](sort(($c1)*x+($c2)*y+($c3)*z=($c4),order=plex(x,z,y))):
elif $kc=6 then
eqc:=MathML[ExportPresentation](-($c4)+($c2)*y+($c3)*z=-($c1)*x):
elif $kc=7 then
eqc:=MathML[ExportPresentation](-($c4)+($c3)*z=-($c1)*x-($c2)*y):
elif $kc=8 then
eqc:=MathML[ExportPresentation](-($c4)+($c2)*y+($c1)*x=-($c3)*z):
elif $kc=9 then
eqc:=MathML[ExportPresentation](($c2)*y+($c3)*z=-($c1)*x+($c4)):
end if:

convert(eqaAns,string),convert(eqa,string),convert(eqbAns,string),convert(eqb,string),convert(eqcAns,string),convert(eqc,string)");
$equationA=switch(1,$v);
$equationB=switch(3,$v);
$equationC=switch(5,$v);
$ansApretty=switch(0,$v);
$ansBpretty=switch(2,$v);
$ansCpretty=switch(4,$v);@
qu.1.4.uid=22d40be9-8639-4d96-b771-0d3954f627ad@
qu.1.4.info=  Course=Introductory Mathematical Economics;
  Topic=Algebra;
  Sub-Topic=Rearranging Equations;
  Author=Katherine Dare;
  Difficulty=Easy;
@
qu.1.4.weighting=1,1,1,1,1,1,1,1,1,1,1,1@
qu.1.4.numbering=alpha@
qu.1.4.part.1.name=sro_id_1@
qu.1.4.part.1.answer.units=@
qu.1.4.part.1.numStyle=   @
qu.1.4.part.1.editing=useHTML@
qu.1.4.part.1.showUnits=false@
qu.1.4.part.1.question=(Unset)@
qu.1.4.part.1.mode=Numeric@
qu.1.4.part.1.grading=exact_value@
qu.1.4.part.1.negStyle=both@
qu.1.4.part.1.answer.num=$a1@
qu.1.4.part.2.name=sro_id_2@
qu.1.4.part.2.answer.units=@
qu.1.4.part.2.numStyle=   @
qu.1.4.part.2.editing=useHTML@
qu.1.4.part.2.showUnits=false@
qu.1.4.part.2.question=(Unset)@
qu.1.4.part.2.mode=Numeric@
qu.1.4.part.2.grading=exact_value@
qu.1.4.part.2.negStyle=both@
qu.1.4.part.2.answer.num=$a2@
qu.1.4.part.3.name=sro_id_3@
qu.1.4.part.3.answer.units=@
qu.1.4.part.3.numStyle=   @
qu.1.4.part.3.editing=useHTML@
qu.1.4.part.3.showUnits=false@
qu.1.4.part.3.question=(Unset)@
qu.1.4.part.3.mode=Numeric@
qu.1.4.part.3.grading=exact_value@
qu.1.4.part.3.negStyle=both@
qu.1.4.part.3.answer.num=$a3@
qu.1.4.part.4.name=sro_id_4@
qu.1.4.part.4.answer.units=@
qu.1.4.part.4.numStyle=   @
qu.1.4.part.4.editing=useHTML@
qu.1.4.part.4.showUnits=false@
qu.1.4.part.4.question=(Unset)@
qu.1.4.part.4.mode=Numeric@
qu.1.4.part.4.grading=exact_value@
qu.1.4.part.4.negStyle=both@
qu.1.4.part.4.answer.num=$a4@
qu.1.4.part.5.name=sro_id_5@
qu.1.4.part.5.answer.units=@
qu.1.4.part.5.numStyle=   @
qu.1.4.part.5.editing=useHTML@
qu.1.4.part.5.showUnits=false@
qu.1.4.part.5.question=(Unset)@
qu.1.4.part.5.mode=Numeric@
qu.1.4.part.5.grading=exact_value@
qu.1.4.part.5.negStyle=both@
qu.1.4.part.5.answer.num=$b1@
qu.1.4.part.6.name=sro_id_6@
qu.1.4.part.6.answer.units=@
qu.1.4.part.6.numStyle=   @
qu.1.4.part.6.editing=useHTML@
qu.1.4.part.6.showUnits=false@
qu.1.4.part.6.question=(Unset)@
qu.1.4.part.6.mode=Numeric@
qu.1.4.part.6.grading=exact_value@
qu.1.4.part.6.negStyle=both@
qu.1.4.part.6.answer.num=$b2@
qu.1.4.part.7.name=sro_id_7@
qu.1.4.part.7.answer.units=@
qu.1.4.part.7.numStyle=   @
qu.1.4.part.7.editing=useHTML@
qu.1.4.part.7.showUnits=false@
qu.1.4.part.7.question=(Unset)@
qu.1.4.part.7.mode=Numeric@
qu.1.4.part.7.grading=exact_value@
qu.1.4.part.7.negStyle=both@
qu.1.4.part.7.answer.num=$b3@
qu.1.4.part.8.name=sro_id_8@
qu.1.4.part.8.answer.units=@
qu.1.4.part.8.numStyle=   @
qu.1.4.part.8.editing=useHTML@
qu.1.4.part.8.showUnits=false@
qu.1.4.part.8.question=(Unset)@
qu.1.4.part.8.mode=Numeric@
qu.1.4.part.8.grading=exact_value@
qu.1.4.part.8.negStyle=both@
qu.1.4.part.8.answer.num=$b4@
qu.1.4.part.9.name=sro_id_9@
qu.1.4.part.9.answer.units=@
qu.1.4.part.9.numStyle=   @
qu.1.4.part.9.editing=useHTML@
qu.1.4.part.9.showUnits=false@
qu.1.4.part.9.question=(Unset)@
qu.1.4.part.9.mode=Numeric@
qu.1.4.part.9.grading=exact_value@
qu.1.4.part.9.negStyle=both@
qu.1.4.part.9.answer.num=$c1@
qu.1.4.part.10.name=sro_id_10@
qu.1.4.part.10.answer.units=@
qu.1.4.part.10.numStyle=   @
qu.1.4.part.10.editing=useHTML@
qu.1.4.part.10.showUnits=false@
qu.1.4.part.10.question=(Unset)@
qu.1.4.part.10.mode=Numeric@
qu.1.4.part.10.grading=exact_value@
qu.1.4.part.10.negStyle=both@
qu.1.4.part.10.answer.num=$c2@
qu.1.4.part.11.name=sro_id_11@
qu.1.4.part.11.answer.units=@
qu.1.4.part.11.numStyle=   @
qu.1.4.part.11.editing=useHTML@
qu.1.4.part.11.showUnits=false@
qu.1.4.part.11.question=(Unset)@
qu.1.4.part.11.mode=Numeric@
qu.1.4.part.11.grading=exact_value@
qu.1.4.part.11.negStyle=both@
qu.1.4.part.11.answer.num=$c3@
qu.1.4.part.12.name=sro_id_12@
qu.1.4.part.12.answer.units=@
qu.1.4.part.12.numStyle=   @
qu.1.4.part.12.editing=useHTML@
qu.1.4.part.12.showUnits=false@
qu.1.4.part.12.question=(Unset)@
qu.1.4.part.12.mode=Numeric@
qu.1.4.part.12.grading=exact_value@
qu.1.4.part.12.negStyle=both@
qu.1.4.part.12.answer.num=$c4@
qu.1.4.question=<p>Rearrange the following system of equations to be in the form <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>c</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>z</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>constant</mi></mrow></mstyle></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&ge;</mo><mn>0</mn></mrow></mstyle></math>.</p><p>$equationA</p><p>$equationB</p><p>$equationC</p><p>&nbsp;</p><p>&nbsp;(If the coefficient is zero, enter 0. If the coefficient is one, enter 1. Enter all other numbers normally.)</p><p><span>&nbsp;</span><1><span> </span>x+<span>&nbsp;</span><2><span> </span>y+<span>&nbsp;</span><3><span> </span>z=<span>&nbsp;</span><4><span>&nbsp;</span></p><p><span>&nbsp;</span><5><span> </span>x+<span>&nbsp;</span><6><span> </span>y+<span>&nbsp;</span><7><span> </span>z=<span>&nbsp;</span><8><span>&nbsp;</span></p><p><span>&nbsp;</span><9><span> </span>x+<span>&nbsp;</span><10><span> </span>y+<span>&nbsp;</span><11><span> </span>z=&nbsp;<span> </span><12><span>&nbsp;</span></p>@

qu.1.5.mode=Inline@
qu.1.5.name=Solving a 4x4 system of equations@
qu.1.5.comment=<p>x=$anspretty</p>@
qu.1.5.editing=useHTML@
qu.1.5.hint.1=Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>and pre-multiply both sides of the equation by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.@
qu.1.5.solution=@
qu.1.5.algorithm=$v=maple("
randomize():
with(LinearAlgebra):
v1:=RandomMatrix(4,4,generator=(-3..3)):
while LinearAlgebra[Determinant](v1) = 0 do v1:=RandomMatrix(4,4,generator=(-3..3)) end do:
v2:=RandomMatrix(4,1,generator=(-5..5)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string)
");
$A=switch(3,$v);
$ans=switch(1,$v);
$b=switch(5,$v);
$anspretty=switch(4,$v);@
qu.1.5.uid=0fe8b0c7-6e0c-495f-a12e-067faf6f01d5@
qu.1.5.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Solving A System;
  Author=Katherine Dare;
  Difficulty=Hard;
  Feature=Students Use Equation Editor;
@
qu.1.5.weighting=1@
qu.1.5.numbering=alpha@
qu.1.5.part.1.name=sro_id_1@
qu.1.5.part.1.maple_answer=printf("$anspretty");@
qu.1.5.part.1.editing=useHTML@
qu.1.5.part.1.question=(Unset)@
qu.1.5.part.1.libname=@
qu.1.5.part.1.mode=Maple@
qu.1.5.part.1.allow2d=2@
qu.1.5.part.1.plot=@
qu.1.5.part.1.maple=ans:=$ans:
grade:=0:
for i from 1 to 4 do
if $ans[i,1] = convert($RESPONSE,Matrix)[i,1]
then grade:=grade+0.25:
end if;
end;
grade;@
qu.1.5.part.1.type=maple@
qu.1.5.question=<p>Solve for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>4</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math>:</p><p>$A&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>4</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math>&nbsp; =$b</p><p>&nbsp;</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>4</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.6.mode=Inline@
qu.1.6.name=Solving a 3x3 system of equations - market clearing problem - Cramer's rule@
qu.1.6.comment=<p>Note: the method below solves for all prices. If you only need to find one, consider using Cramer's rule instead.</p>
<p>Equate each firm's demand to its supply and rearrange and the resulting system of equations can be written as the following matrix problem:</p>
<p>$Amatrix<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$constants</p>
<p>The inverse of $Amatrix is:</p>
<p>$inv</p>
<p>&nbsp;</p>
<p>Pre-multiplying both sides of the equation by the inverse gives:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$inv<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow></mstyle></math>$constants=$prices</p>@
qu.1.6.editing=useHTML@
qu.1.6.hint.1=Equate each firm's supply and demand.@
qu.1.6.hint.2=Rearrange so that all the variables are on the left hand side and all the constants are on the right hand side.@
qu.1.6.hint.3=Translate system into matrix form.@
qu.1.6.hint.4=Use Cramer's rule to solve.@
qu.1.6.solution=@
qu.1.6.algorithm=$k=range(1,3);
$a=range(1,2);
$b=range(1,2);
$c=range(1,2);
$v=maple("
randomize():
v1:=LinearAlgebra[RandomMatrix](3,generator=rand(3..6),attributes=[nonsingular]):
v1[1..1,1..1]:=(-1)*v1[1,1]:
v1[2..2,2..2]:=(-1)*v1[2,2]:
v1[3..3,3..3]:=(-1)*v1[3,3]:
while LinearAlgebra[Determinant](v1) = 0 do v1 := LinearAlgebra[RandomMatrix](3, generator = rand(1 .. 5), attributes = [nonsingular]): 
v1[1..1,1..1]:=(-1)*v1[1,1]:
v1[2..2,2..2]:=(-1)*v1[2,2]:
v1[3..3,3..3]:=(-1)*v1[3,3]:
end do:
v2:=LinearAlgebra[RandomMatrix](3,1,generator=rand(5..15)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v6:=MathML[ExportPresentation](v1.v2):
q1d:=MathML[ExportPresentation](-(v3[1,1])+(v1[1,1]+$a)*P1+v1[1,2]*P2+v1[1,3]*P3):
q2d:=MathML[ExportPresentation](-(v3[2,1])+v1[2,1]*P1+(v1[2,2]+$b)*P2+v1[2,3]*P3):
q3d:=MathML[ExportPresentation](-(v3[3,1])+v1[3,1]*P1+v1[3,2]*P2+(v1[3,3]+$c)*P3):
q1s:=MathML[ExportPresentation]($a*P1):
q2s:=MathML[ExportPresentation]($b*P2):
q3s:=MathML[ExportPresentation]($c*P3):
P1:=v2[1,1]:
P2:=v2[2,1]:
P3:=v2[3,1];
inv:=LinearAlgebra[MatrixInverse](v1):
invpretty:=MathML[ExportPresentation](inv):
prices:=MathML[ExportPresentation](v2):
constants:=MathML[ExportPresentation](v3):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(q1d,string),convert(q2d,string),convert(q3d,string),convert(q1s,string),convert(q2s,string),convert(q3s,string),P1,P2,P3,invpretty,prices,constants
");
$q1d=switch(4,$v);
$q2d=switch(5,$v);
$q3d=switch(6,$v);
$q1s=switch(7,$v);
$q2s=switch(8,$v);
$q3s=switch(9,$v);
$P1=switch(10,$v);
$P2=switch(11,$v);
$P3=switch(12,$v);
$inv=switch(13,$v);
$prices=switch(14,$v);
$Amatrix=switch(3,$v);
$constants=switch(15,$v);
$ans=switch($k-1,$P1,$P2,$P3);@
qu.1.6.uid=43118d26-8db8-4650-8e17-7c59a90c8083@
qu.1.6.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Solving A System Of Equations;
  Author=Katherine Dare;
  Difficulty=Hard;
@
qu.1.6.weighting=1@
qu.1.6.numbering=alpha@
qu.1.6.part.1.name=sro_id_1@
qu.1.6.part.1.answer.units=@
qu.1.6.part.1.numStyle=   @
qu.1.6.part.1.editing=useHTML@
qu.1.6.part.1.showUnits=false@
qu.1.6.part.1.question=(Unset)@
qu.1.6.part.1.mode=Numeric@
qu.1.6.part.1.grading=exact_value@
qu.1.6.part.1.negStyle=both@
qu.1.6.part.1.answer.num=$ans@
qu.1.6.question=<p>Three firms sell related products in a market. Their supply functions are:</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q1s</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q2s</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>3</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q3s</p><p>&nbsp;</p><p>The firms face the following demands:</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>1</mn></mrow><mrow><mi>d</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q1d</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q2d</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>3</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q3d</p><p>&nbsp;</p><p>&nbsp;</p><p><span>Given the above supply and demand functions, what price must firm $k charge to equate supply and demand? Note that you do not need to solve for all equilibrium prices. <br /></span></p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mi mathvariant='normal'>$k</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.1.7.mode=Multiple Choice@
qu.1.7.name=systems of equations -- infinity of solutions@
qu.1.7.comment=<p>In general, you need to have as many <em>unique</em> equations as unknowns for a fully determined system.</p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$a=range(2,8,1);
$b=range(2,8,1);
$c=range(5,8,1);
$eq1=mathml($a*x+$b*y);
$eq2=mathml($b*$c*y+$a*$c*x);
$d=range(2,4,1);
$e=$c*$d;@
qu.1.7.uid=3d0786f8-5795-49d3-bc41-d3933736a4f5@
qu.1.7.info=  Course=Introductory Mathematical Economics;
  Topic=System Of Equations;
  Sub-Topic=Number Of Solutions;
  Author=Asha Sadanand;
  Difficulty=Medium;
@
qu.1.7.question=<p>The following system of equations</p>
<p>$eq1<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$d</mi></mrow></mstyle></math></p>
<p>$eq2 <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>$e</mi></mrow></mstyle></math></p>@
qu.1.7.answer=4@
qu.1.7.choice.1=is overdetermined@
qu.1.7.choice.2=has a unique solution@
qu.1.7.choice.3=has no solution@
qu.1.7.choice.4=has an infinite number of solutions@
qu.1.7.fixed=@

qu.1.8.mode=Inline@
qu.1.8.name=Solving a 3x3 system of equations - market clearing problem@
qu.1.8.comment=<p>Equate each firm's demand to its supply and rearrange and the resulting system of equations can be written as the following matrix problem:</p>
<p>$Amatrix<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$constants</p>
<p>The inverse of $Amatrix is:</p>
<p>$inv</p>
<p>&nbsp;</p>
<p>Pre-multiplying both sides of the equation by the inverse gives:</p>
<p>$inv<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow></mstyle></math>$constants=$prices</p>@
qu.1.8.editing=useHTML@
qu.1.8.hint.1=Equate each firm's supply and demand.@
qu.1.8.hint.2=Rearrange so that all the variables are on the left hand side and all the constants are on the right hand side.@
qu.1.8.hint.3=Translate system into matrix form.@
qu.1.8.hint.4=Invert and pre-multiply to solve for vector of prices.@
qu.1.8.solution=@
qu.1.8.algorithm=$a=range(1,2);
$b=range(1,2);
$c=range(1,2);
$v=maple("
randomize():
v1:=LinearAlgebra[RandomMatrix](3,generator=rand(3..6),attributes=[nonsingular]):
v1[1..1,1..1]:=(-1)*v1[1,1]:
v1[2..2,2..2]:=(-1)*v1[2,2]:
v1[3..3,3..3]:=(-1)*v1[3,3]:
while LinearAlgebra[Determinant](v1) = 0 do v1 := LinearAlgebra[RandomMatrix](3, generator = rand(1 .. 5), attributes = [nonsingular]): 
v1[1..1,1..1]:=(-1)*v1[1,1]:
v1[2..2,2..2]:=(-1)*v1[2,2]:
v1[3..3,3..3]:=(-1)*v1[3,3]:
end do:
v2:=LinearAlgebra[RandomMatrix](3,1,generator=rand(5..15)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v6:=MathML[ExportPresentation](v1.v2):
q1d:=MathML[ExportPresentation](-(v3[1,1])+(v1[1,1]+$a)*P1+v1[1,2]*P2+v1[1,3]*P3):
q2d:=MathML[ExportPresentation](-(v3[2,1])+v1[2,1]*P1+(v1[2,2]+$b)*P2+v1[2,3]*P3):
q3d:=MathML[ExportPresentation](-(v3[3,1])+v1[3,1]*P1+v1[3,2]*P2+(v1[3,3]+$c)*P3):
q1s:=MathML[ExportPresentation]($a*P1):
q2s:=MathML[ExportPresentation]($b*P2):
q3s:=MathML[ExportPresentation]($c*P3):
P1:=v2[1,1]:
P2:=v2[2,1]:
P3:=v2[3,1];
inv:=LinearAlgebra[MatrixInverse](v1):
invpretty:=MathML[ExportPresentation](inv):
prices:=MathML[ExportPresentation](v2):
constants:=MathML[ExportPresentation](v3):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(q1d,string),convert(q2d,string),convert(q3d,string),convert(q1s,string),convert(q2s,string),convert(q3s,string),P1,P2,P3,invpretty,prices,constants
");
$q1d=switch(4,$v);
$q2d=switch(5,$v);
$q3d=switch(6,$v);
$q1s=switch(7,$v);
$q2s=switch(8,$v);
$q3s=switch(9,$v);
$P1=switch(10,$v);
$P2=switch(11,$v);
$P3=switch(12,$v);
$inv=switch(13,$v);
$prices=switch(14,$v);
$Amatrix=switch(3,$v);
$constants=switch(15,$v);@
qu.1.8.uid=1c2c8860-58a9-4ff7-9885-1dc0dab3a82c@
qu.1.8.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Solving A System Of Equations;
  Author=Katherine Dare;
  Difficulty=Very Hard;
@
qu.1.8.weighting=1,1,1@
qu.1.8.numbering=alpha@
qu.1.8.part.1.name=sro_id_1@
qu.1.8.part.1.answer.units=@
qu.1.8.part.1.numStyle=   @
qu.1.8.part.1.editing=useHTML@
qu.1.8.part.1.showUnits=false@
qu.1.8.part.1.question=(Unset)@
qu.1.8.part.1.mode=Numeric@
qu.1.8.part.1.grading=exact_value@
qu.1.8.part.1.negStyle=both@
qu.1.8.part.1.answer.num=$P1@
qu.1.8.part.2.name=sro_id_2@
qu.1.8.part.2.answer.units=@
qu.1.8.part.2.numStyle=   @
qu.1.8.part.2.editing=useHTML@
qu.1.8.part.2.showUnits=false@
qu.1.8.part.2.question=(Unset)@
qu.1.8.part.2.mode=Numeric@
qu.1.8.part.2.grading=exact_value@
qu.1.8.part.2.negStyle=both@
qu.1.8.part.2.answer.num=$P2@
qu.1.8.part.3.name=sro_id_3@
qu.1.8.part.3.answer.units=@
qu.1.8.part.3.numStyle=   @
qu.1.8.part.3.editing=useHTML@
qu.1.8.part.3.showUnits=false@
qu.1.8.part.3.question=(Unset)@
qu.1.8.part.3.mode=Numeric@
qu.1.8.part.3.grading=exact_value@
qu.1.8.part.3.negStyle=both@
qu.1.8.part.3.answer.num=$P3@
qu.1.8.question=<p>Three firms sell related products in a market. Their supply functions are:</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q1s</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q2s</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>3</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q3s</p><p>&nbsp;</p><p>The firms face the following demands:</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>1</mn></mrow><mrow><mi>d</mi></mrow></msubsup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q1d</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q2d</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msubsup><mi>q</mi><mrow><mn>3</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$q3d</p><p>&nbsp;</p><p>&nbsp;</p><p><span>Given the above supply and demand functions, what price must each firm charge to equate supply and demand?</span></p><p>&nbsp;</p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><1><span>&nbsp;</span></span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span>&nbsp;</span></span></p><p><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>P</mi><mrow><mn>3</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span>&nbsp;</span><br /></span></p><p>&nbsp;</p>@

qu.1.9.mode=Inline@
qu.1.9.name=systems of equations - one parameter, many solutions@
qu.1.9.comment=<p>This system has multiple solutions with one parameter. This can be seen using a variety of methods. One method is to form the 3x4 matrix of coefficients.</p>
<p>$A1</p>
<p>Then apply elementary row operation until the row echelon form is achieved.&nbsp;</p>
<p>$A2</p>
<p>Now the answers can be simply read from the matrix.</p>
<p>&nbsp;</p>@
qu.1.9.editing=useHTML@
qu.1.9.solution=@
qu.1.9.algorithm=$m=maple("
randomize():
m0:=LinearAlgebra[RandomMatrix](2,generator=rand(-3 .. 3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m0) = 0 do m0:= LinearAlgebra[RandomMatrix](2, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
M0:=MathML[ExportPresentation](m0):
m1:=Matrix([[1,0,m0[1,1]],[0,1,m0[2,1]],[0,0,0]]):
M1:=MathML[ExportPresentation](m1):
m2:=Matrix([[m0[1,2]],[m0[2,2]],[0]]):
M2:=MathML[ExportPresentation](m2):
m3:=LinearAlgebra[RandomMatrix](3,generator=rand(-3 .. 3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m3) = 0 do m3:= LinearAlgebra[RandomMatrix](3, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
M3:=MathML[ExportPresentation](m3):
n0:=m3.m1:
N0:=MathML[ExportPresentation](n0):
n1:=m3.m2:
N1:=MathML[ExportPresentation](n1):
N3:=MathML[ExportPresentation](Matrix([[n0[1,1],n0[1,2],n0[1,3],n1[1,1]],[n0[2,1],n0[2,2],n0[2,3],n1[2,1]],[n0[3,1],n0[3,2],n0[3,3],n1[3,1]]])):
n4:=MathML[ExportPresentation]((LinearAlgebra[ReducedRowEchelonForm](<n0|n1>))):
e1:=MathML[ExportPresentation](n0[1,1]*X+n0[1,2]*Y+n0[1,3]*Z-n1[1,1]=0):
e2:=MathML[ExportPresentation](n0[2,1]*X+n0[2,2]*Y+n0[2,3]*Z-n1[2,1]=0):
e3:=MathML[ExportPresentation](n0[3,1]*X+n0[3,2]*Y+n0[3,3]*Z-n1[3,1]=0):
e1,e2,e3,m0[1,1],m0[1,2],m0[2,1],m0[2,2],M0,M1,M2,M3,N0,N1,N3,n4
");
$E1=switch(0,$m);
$E2=switch(1,$m);
$E3=switch(2,$m);
$a11=switch(3,$m);
$a12=switch(4,$m);
$a21=switch(5,$m);
$a22=switch(6,$m);
$A1=switch(13,$m);
$A2=switch(14,$m);@
qu.1.9.uid=022bf50a-65e9-4c21-8f98-3b265ce15d88@
qu.1.9.info=  Course=Introductory Mathematical Economics;
  Topic=Systems Of Equations;
  Sub-Topic=Infinite Solutions -- One Parameter;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.1.9.weighting=1,1,1,1@
qu.1.9.numbering=alpha@
qu.1.9.part.1.grader=exact@
qu.1.9.part.1.name=sro_id_1@
qu.1.9.part.1.editing=useHTML@
qu.1.9.part.1.display.permute=true@
qu.1.9.part.1.answer.3=has a unique solution@
qu.1.9.part.1.question=(Unset)@
qu.1.9.part.1.answer.2=has an infinite number of solutions.@
qu.1.9.part.1.answer.1=has no solutions.@
qu.1.9.part.1.mode=List@
qu.1.9.part.1.display=menu@
qu.1.9.part.1.credit.3=0.0@
qu.1.9.part.1.credit.2=1.0@
qu.1.9.part.1.credit.1=0.0@
qu.1.9.part.2.name=sro_id_2@
qu.1.9.part.2.maple_answer=$a12-Z*($a11)@
qu.1.9.part.2.editing=useHTML@
qu.1.9.part.2.question=(Unset)@
qu.1.9.part.2.libname=@
qu.1.9.part.2.mode=Maple@
qu.1.9.part.2.allow2d=1@
qu.1.9.part.2.plot=@
qu.1.9.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.9.part.2.type=formula@
qu.1.9.part.3.name=sro_id_3@
qu.1.9.part.3.maple_answer=$a22-Z*($a21)@
qu.1.9.part.3.editing=useHTML@
qu.1.9.part.3.question=(Unset)@
qu.1.9.part.3.libname=@
qu.1.9.part.3.mode=Maple@
qu.1.9.part.3.allow2d=1@
qu.1.9.part.3.plot=@
qu.1.9.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.9.part.3.type=formula@
qu.1.9.part.4.name=sro_id_4@
qu.1.9.part.4.maple_answer=Z@
qu.1.9.part.4.editing=useHTML@
qu.1.9.part.4.question=(Unset)@
qu.1.9.part.4.libname=@
qu.1.9.part.4.mode=Maple@
qu.1.9.part.4.allow2d=1@
qu.1.9.part.4.plot=@
qu.1.9.part.4.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.9.part.4.type=formula@
qu.1.9.question=<p>Consider the following system of equations:</p><p>$E1</p><p>$E2</p><p>$E3</p><p>Choose the answer that correctly describes the system</p><p>The system<span>&nbsp;</span><1><span> </span>.</p><p>&nbsp;</p><p>Give the solutions for the system, if they exist.</p><p>If there is no solution type "no solution" (without quotes) in each case.</p><p>If there are many solutions, for a two parameter solution type&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> in the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>box and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math>in the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>box, and then give <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math>;</p><p>For a one parameter solution type <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math> in the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>box and then give <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math>.</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span> </span>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span> </span>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><4><span>&nbsp;</span></p>@

qu.1.10.mode=Multiple Choice@
qu.1.10.name=systems of equations -- no solution@
qu.1.10.comment=<p>This system of equations has no solutions because looking at the left hand sides of the two equations the second is two times the first; however, looking at the right hand sides the second equation has $d which is certainly not two times $c, the right hand side of the first equation.</p>@
qu.1.10.editing=useHTML@
qu.1.10.solution=@
qu.1.10.algorithm=$a=range(2,8,1);
$b=range(2,8,1);
$eq1=mathml($a*x+$b*y);
$eq2=mathml($b*2*y+$a*2*x);
$c=range(5,8,1);
$d=range(2,4,1);@
qu.1.10.uid=c61bcb41-4ac7-46a4-a71e-89319869b7bb@
qu.1.10.info=  Course=Introductory Mathematical Economics;
  Topic=System Of Equations;
  Sub-Topic=Two Equations, No Solution;
  Difficulty=Easy;
  Author=Asha Sadanand;
@
qu.1.10.question=<p>The following system of equations</p>
<p>$eq1 =$c</p>
<p>$eq2 = $d</p>@
qu.1.10.answer=3@
qu.1.10.choice.1=is underdetermined@
qu.1.10.choice.2=has a unique solution@
qu.1.10.choice.3=has no solution@
qu.1.10.choice.4=has an infinite number of solutions@
qu.1.10.fixed=@

qu.1.11.mode=Inline@
qu.1.11.name=systems of equations - no solution@
qu.1.11.comment=<p>This system of equations has no solution because it is inconsistent. This can be seen by applying elementary row operations to the matrix for the equations.</p>
<p>The matrix of coefficients is $M5, and the column of constant terms is $M6.</p>
<p>If we combine these to form a 3x4 matrix and then use elementary row operations to reduce it to the Row Echelon Form we find $M8.</p>
<p>We can see that there is no solution by looking at the last row which is essentially saying that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>X</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>Y</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>Z</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>.&nbsp; There is no solution to this equation.</p>@
qu.1.11.editing=useHTML@
qu.1.11.solution=@
qu.1.11.algorithm=$a=range(1,9,1);
$b=range(1,9,1);
$c=range(1,9,1);
$d=range(1,3,1);
$e=range(1,3,1);
$m=maple("
randomize():
m0:=LinearAlgebra[RandomMatrix](3,generator=rand(2..5),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m0) = 0 do m0 := LinearAlgebra[RandomMatrix](3, generator = rand(2 .. 5), attributes = [nonsingular]) end do:
m4:=MathML[ExportPresentation](m0):
m1:=MathML[ExportPresentation](m0[1,1]*(X-$a)+m0[1,2]*(Y-$b)+m0[1,3]*(Z-$c)=0):
m2:=MathML[ExportPresentation](m0[2,1]*(X-$a)+m0[2,2]*(Y-$b)+m0[2,3]*(Z-$c)=0):
m3:=MathML[ExportPresentation]($d*(m0[1,1]*(X-$a)+m0[1,2]*(Y-$b)+m0[1,3]*(Z-$c))-$e*(m0[2,1]*(X-$a)+m0[2,2]*(Y-$b)+m0[2,3]*(Z-$c))+1=0
):
A:=Matrix([[m0[1,1],m0[1,2],m0[1,3]],[m0[2,1],m0[2,2],m0[2,3]],[($d*m0[1,1]-$e*m0[2,1]),($d*m0[1,2]-$e*m0[2,2]),($d*m0[1,3]-$e*m0[2,3])]]):
B:=Matrix([[($a*m0[1,1]+$b*m0[1,2]+$c*m0[1,3])],[($a*m0[2,1]+$b*m0[2,2]+$c*m0[2,3])],[($d*($a*m0[1,1]+$b*m0[1,2]+$c*m0[1,3])-$e*($a*m0[2,1]+$b*m0[2,2]+$c*m0[2,3])-1)]]):
m4:=MathML[ExportPresentation](m0):
m5:=MathML[ExportPresentation](A):
m6:=MathML[ExportPresentation](B):
m7:=(LinearAlgebra[ReducedRowEchelonForm](<A|B>)):
m8:=MathML[ExportPresentation](m7):
m1,m2,m3,m4,m5,m6,convert(m7,string),m8
");
$M1=switch(0,$m);
$M2=switch(1,$m);
$M3=switch(2,$m);
$M5=switch(4,$m);
$M6=switch(5,$m);
$M8=switch(7,$m);@
qu.1.11.uid=9b5446ec-7bdb-4d5a-973b-102db2cf9085@
qu.1.11.info=  Course=Introductory Mathematical Economics;
  Topic=System Of Equations;
  Sub-Topic=No Solutions;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.1.11.weighting=1,1,1,1@
qu.1.11.numbering=alpha@
qu.1.11.part.1.grader=exact@
qu.1.11.part.1.name=sro_id_1@
qu.1.11.part.1.editing=useHTML@
qu.1.11.part.1.display.permute=true@
qu.1.11.part.1.answer.3=has a unique solution@
qu.1.11.part.1.question=(Unset)@
qu.1.11.part.1.answer.2=has an infinite number of solutions.@
qu.1.11.part.1.answer.1=has no solutions.@
qu.1.11.part.1.mode=List@
qu.1.11.part.1.display=menu@
qu.1.11.part.1.credit.3=0.0@
qu.1.11.part.1.credit.2=0.0@
qu.1.11.part.1.credit.1=1.0@
qu.1.11.part.2.name=sro_id_2@
qu.1.11.part.2.maple_answer="no solution"@
qu.1.11.part.2.editing=useHTML@
qu.1.11.part.2.question=(Unset)@
qu.1.11.part.2.libname=@
qu.1.11.part.2.mode=Maple@
qu.1.11.part.2.allow2d=0@
qu.1.11.part.2.plot=@
qu.1.11.part.2.maple=evalb($ANSWER="$RESPONSE");@
qu.1.11.part.2.type=maple@
qu.1.11.part.3.name=sro_id_3@
qu.1.11.part.3.maple_answer="no solution"@
qu.1.11.part.3.editing=useHTML@
qu.1.11.part.3.question=(Unset)@
qu.1.11.part.3.libname=@
qu.1.11.part.3.mode=Maple@
qu.1.11.part.3.allow2d=0@
qu.1.11.part.3.plot=@
qu.1.11.part.3.maple=evalb($ANSWER="$RESPONSE");@
qu.1.11.part.3.type=maple@
qu.1.11.part.4.name=sro_id_4@
qu.1.11.part.4.maple_answer="no solution"@
qu.1.11.part.4.editing=useHTML@
qu.1.11.part.4.question=(Unset)@
qu.1.11.part.4.libname=@
qu.1.11.part.4.mode=Maple@
qu.1.11.part.4.allow2d=0@
qu.1.11.part.4.plot=@
qu.1.11.part.4.maple=evalb($ANSWER="$RESPONSE");@
qu.1.11.part.4.type=maple@
qu.1.11.question=<p>Consider the following system of equations:</p><p>&nbsp;</p><p>$M1</p><p>$M2</p><p>$M3</p><p>&nbsp;</p><p>Choose the answer that correctly describes the system</p><p>The system<span>&nbsp;</span><1><span> </span>.</p><p>&nbsp;</p><p>Give the solutions for the system, if they exist.</p><p>If there is no solution type "no solution" (without quotes) in each case.</p><p>If there are many solutions,&nbsp; for a two parameter solution type <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> in the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>box and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math> in the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>box and then give <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math>.</p><p>For a one parameter solutions type <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math> in the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>box and then give <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span> </span>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span> </span>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><4><span>&nbsp;</span></p>@

qu.1.12.question=<p>A system of equations can be written in the matrix form <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Ax</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>b</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>Give the coefficient matrix of the following system. Use the order <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub></mrow></mstyle></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>7</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>4</mn></mrow></mstyle></math></p>@
qu.1.12.maple=LinearAlgebra[Equal]($RESPONSE,$ans);@
qu.1.12.allow2d=2@
qu.1.12.maple_answer=printf(MathML[ExportPresentation]($ans));@
qu.1.12.type=maple@
qu.1.12.mode=Maple@
qu.1.12.name=coefficient matrix fixed matrix@
qu.1.12.comment=@
qu.1.12.editing=useHTML@
qu.1.12.solution=@
qu.1.12.algorithm=$ans=maple("
Matrix(2,3,[[1,2,-3],[7,-1,2]])
");@
qu.1.12.uid=a2673e73-204e-4a77-bf09-af6ad843d9b5@
qu.1.12.info=  Course=Introductory Mathematical Economics;
  Topic=Coefficient Matrix Fixed Coefficients;
  Sub-Topic=;
  Author=Asha Sadanand;
  Difficulty=Easy;
@

qu.1.13.mode=Inline@
qu.1.13.name=systems of equations --unique solution@
qu.1.13.comment=<p>There are various ways to solve this question.</p>
<p>One way is to take the matrix of coefficients $M4 and combine it with the column of constants $M6, to form a 3x4 matrix. Then use elementary row operations to reduce the matrix to its Row Echelon Form $M8. Now we can just read the solutions from the matrix: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$a, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$b, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$c.</p>@
qu.1.13.editing=useHTML@
qu.1.13.solution=@
qu.1.13.algorithm=$a=range(1,9,1);
$b=range(1,9,1);
$c=range(1,9,1);
$m=maple("
randomize():
m0:=LinearAlgebra[RandomMatrix](3,generator=rand(-3..3),attributes=[nonsingular]):
while LinearAlgebra[Determinant](m0) = 0 do m0 := LinearAlgebra[RandomMatrix](3, generator = rand(-3 .. 3), attributes = [nonsingular]) end do:
m1:=MathML[ExportPresentation](m0[1,1]*(X-$a)+m0[1,2]*(Y-$b)+m0[1,3]*(Z-$c)=0):
m2:=MathML[ExportPresentation](m0[2,1]*(X-$a)+m0[2,2]*(Y-$b)+m0[2,3]*(Z-$c)=0):
m3:=MathML[ExportPresentation](m0[3,1]*(X-$a)+m0[3,2]*(Y-$b)+m0[3,3]*(Z-$c)=0):
m4:=MathML[ExportPresentation](m0):
m5:=Matrix([[($a*m0[1,1]+$b*m0[1,2]+$c*m0[1,3])],[($a*m0[2,1]+$b*m0[2,2]+$c*m0[2,3])],[($a*m0[3,1]+$b*m0[3,2]+$c*m0[3,3])]]):
m6:=MathML[ExportPresentation](m5):
m7:=(LinearAlgebra[ReducedRowEchelonForm](<m0|m5>)):
m8:=MathML[ExportPresentation](m7):
m1,m2,m3,convert(m0,string),m4,m6,m8
");
$M0=switch(3,$m);
$M1=switch(0,$m);
$M2=switch(1,$m);
$M3=switch(2,$m);
$M4=switch(4,$m);
$M6=switch(5,$m);
$M8=switch(6,$m);@
qu.1.13.uid=6446633e-ca44-43de-ac1c-e4f7720256f5@
qu.1.13.info=  Course=Introductory Mathematical Economics;
  Topic=System Of Equations;
  Sub-Topic=Unique Solution;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.1.13.weighting=1,1,1,1@
qu.1.13.numbering=alpha@
qu.1.13.part.1.grader=exact@
qu.1.13.part.1.name=sro_id_1@
qu.1.13.part.1.editing=useHTML@
qu.1.13.part.1.display.permute=true@
qu.1.13.part.1.answer.3=has a unique solution@
qu.1.13.part.1.question=(Unset)@
qu.1.13.part.1.answer.2=has an infinite number of solutions.@
qu.1.13.part.1.answer.1=has no solutions.@
qu.1.13.part.1.mode=List@
qu.1.13.part.1.display=menu@
qu.1.13.part.1.credit.3=1.0@
qu.1.13.part.1.credit.2=0.0@
qu.1.13.part.1.credit.1=0.0@
qu.1.13.part.2.name=sro_id_2@
qu.1.13.part.2.maple_answer=$a@
qu.1.13.part.2.editing=useHTML@
qu.1.13.part.2.question=(Unset)@
qu.1.13.part.2.libname=@
qu.1.13.part.2.mode=Maple@
qu.1.13.part.2.allow2d=1@
qu.1.13.part.2.plot=@
qu.1.13.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.13.part.2.type=formula@
qu.1.13.part.3.name=sro_id_3@
qu.1.13.part.3.maple_answer=$b@
qu.1.13.part.3.editing=useHTML@
qu.1.13.part.3.question=(Unset)@
qu.1.13.part.3.libname=@
qu.1.13.part.3.mode=Maple@
qu.1.13.part.3.allow2d=1@
qu.1.13.part.3.plot=@
qu.1.13.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.13.part.3.type=formula@
qu.1.13.part.4.name=sro_id_4@
qu.1.13.part.4.maple_answer=$c@
qu.1.13.part.4.editing=useHTML@
qu.1.13.part.4.question=(Unset)@
qu.1.13.part.4.libname=@
qu.1.13.part.4.mode=Maple@
qu.1.13.part.4.allow2d=1@
qu.1.13.part.4.plot=@
qu.1.13.part.4.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.13.part.4.type=formula@
qu.1.13.question=<p>Consider the following system of equations:</p><p>$M1</p><p>$M2</p><p>$M3</p><p>Choose the answer that correctly describes the system</p><p>The system<span>&nbsp;</span><1><span> </span>.</p><p>&nbsp;</p><p>Give the solutions for the system, if they exist.</p><p>If there is no solution type "no solution" (without quotes) in each case.</p><p>If there are many solutions, for a two parameter solution type&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>box and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math> in the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>box, and then give <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math>.</p><p>For one parameter solutions type <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math> in the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>box and then give <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math>.</p><p><br />&nbsp;<br /><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span> </span>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span> </span>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><4><span>&nbsp;</span></p>@

qu.1.14.mode=Multiple Choice@
qu.1.14.name=systems of equations -- no solution@
qu.1.14.comment=<p>In general, if you encounter a contradiction (eg. <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow></mstyle></math>) then there is no solution.</p>@
qu.1.14.editing=useHTML@
qu.1.14.solution=@
qu.1.14.algorithm=$a=range(2,8,1);
$b=range(2,8,1);
$eq1=mathml($a*x+$b*y);
$eq2=mathml($b*2*y+$a*2*x);
$c=range(5,8,1);
$d=range(2,4,1);@
qu.1.14.uid=b6b83558-2edf-483f-a1de-0090214b9715@
qu.1.14.info=  Course=Introductory Mathematical Economics;
  Topic=Systems Of Equations;
  Sub-Topic=Simple Two Equations;
  Author=Asha Sadanand;
  Difficulty=Easy;
@
qu.1.14.question=<p>The following system of equations</p>
<p>$eq1 =$c</p>
<p>$eq2 = $d</p>@
qu.1.14.answer=3@
qu.1.14.choice.1=is underdetermined@
qu.1.14.choice.2=has a unique solution@
qu.1.14.choice.3=has no solution@
qu.1.14.choice.4=has an infinite number of solutions@
qu.1.14.fixed=@

qu.1.15.mode=Inline@
qu.1.15.name=Solving a 2x2 system of equations@
qu.1.15.comment=<p>x=$anspretty</p>@
qu.1.15.editing=useHTML@
qu.1.15.hint.1=Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>and pre-multiply both sides of the equation by <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>A</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup></mrow></mstyle></math>.@
qu.1.15.solution=@
qu.1.15.algorithm=$v=maple("
randomize():
with(LinearAlgebra):
v1:=RandomMatrix(2,2,generator=(-10..10)):
while LinearAlgebra[Determinant](v1) = 0 do v1:=RandomMatrix(2,2,generator=(-10..10)) end do:
v2:=RandomMatrix(2,1,generator=(-10..10)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string)
");
$A=switch(3,$v);
$ans=switch(1,$v);
$b=switch(5,$v);
$anspretty=switch(4,$v);@
qu.1.15.uid=367ca141-1c4a-47e4-a44a-54c93b9197e3@
qu.1.15.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Solving A System;
  Author=Katherine Dare;
  Difficulty=Easy;
  Feature=Students Use Equation Editor;
@
qu.1.15.weighting=1@
qu.1.15.numbering=alpha@
qu.1.15.part.1.name=sro_id_1@
qu.1.15.part.1.maple_answer=printf("$anspretty");@
qu.1.15.part.1.editing=useHTML@
qu.1.15.part.1.question=(Unset)@
qu.1.15.part.1.libname=@
qu.1.15.part.1.mode=Maple@
qu.1.15.part.1.allow2d=2@
qu.1.15.part.1.plot=@
qu.1.15.part.1.maple=ans:=$ans:
grade:=0:
for i from 1 to 2 do
if convert(ans,Matrix)[i,1] = convert($RESPONSE,Matrix)[i,1]
then grade:=grade+0.5:
end if;
end;
grade;@
qu.1.15.part.1.type=maple@
qu.1.15.question=<p>Solve for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math>:</p><p>$A&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math>&nbsp; =$b</p><p>&nbsp;</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.16.mode=Inline@
qu.1.16.name=Solving a 3x3 system of equations@
qu.1.16.comment=<p>x=$anspretty</p>@
qu.1.16.editing=useHTML@
qu.1.16.solution=@
qu.1.16.algorithm=$v=maple("
randomize():
with(LinearAlgebra):
v1:=RandomMatrix(3,3,generator=(-3..3)):
while LinearAlgebra[Determinant](v1) = 0 do v1:=RandomMatrix(3,3,generator=(-3..3)) end do:
v2:=RandomMatrix(3,1,generator=(-9..9)):
v3:=v1.v2:
v4:=MathML[ExportPresentation](v1):
v5:=MathML[ExportPresentation](v2):
v6:=MathML[ExportPresentation](v3):
convert(v1,string),convert(v2,string),convert(v3,string),convert(v4,string),convert(v5,string),convert(v6,string)
");
$A=switch(3,$v);
$ans=switch(1,$v);
$b=switch(5,$v);
$anspretty=switch(4,$v);@
qu.1.16.uid=f6c54f46-4c0b-4a03-9896-6d666e533645@
qu.1.16.info=  Course=Introductory Mathematical Economics;
  Topic=Matrix Algebra;
  Sub-Topic=Solving A System;
  Author=Katherine Dare;
  Difficulty=Medium;
  Feature=Students Use Equation Editor;
@
qu.1.16.weighting=1@
qu.1.16.numbering=alpha@
qu.1.16.part.1.name=sro_id_1@
qu.1.16.part.1.maple_answer=printf("$anspretty");@
qu.1.16.part.1.editing=useHTML@
qu.1.16.part.1.question=(Unset)@
qu.1.16.part.1.libname=@
qu.1.16.part.1.mode=Maple@
qu.1.16.part.1.allow2d=2@
qu.1.16.part.1.plot=@
qu.1.16.part.1.maple=ans:=$ans:
grade:=0:
for i from 1 to 3 do
if $ans[i,1] = convert($RESPONSE,Matrix)[i,1]
then grade:=grade+0.33334:
end if;
end;
grade;@
qu.1.16.part.1.type=maple@
qu.1.16.question=<p>Solve for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math>:</p><p>$A&nbsp; <font size="2"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math></font>&nbsp; =$b</p><p>&nbsp;</p><p>&nbsp;</p><p><font size="2"><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='[' close=']' separators=','><mtable rowalign='baseline' columnalign='center' groupalign='{left}' rowspacing='1.0ex'><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>x</mi><mrow><mn>3</mn></mrow></msub></mrow></mtd></mtr></mtable></mfenced></mrow></mstyle></math></font>=<span>&nbsp;</span><1><span>&nbsp;</span></p>@

qu.1.17.mode=Inline@
qu.1.17.name=systems of equations - two parameter, many solutions@
qu.1.17.comment=<p>This is a system with multiple solutions with two parameters. This can be seen from the fact that all three equations are just multiples of one another. The second equation is $f times the first equation, and the third equation is $e times the first equation. To find the solutions we let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math>be the parameters, and just solve the first equation $M1 for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math>. The solution is <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$Ans.</p>@
qu.1.17.editing=useHTML@
qu.1.17.solution=@
qu.1.17.algorithm=$a=range(1,9,1);
$b=range(1,9,1);
$c=range(1,9,1);
$d=range(2,9,1);
$e=range(2,9,1);
$f=range(1,9,1);
$g=range(1,9,1);
$h=range(1,9,1);
$A=maple("
m:=Matrix([[$f,$g,$h],[$f*$d,$g*$d,$h*$d],[$f*$e,$g*$e,$h*$e]]):
m4:=MathML[ExportPresentation](m):
m1:=MathML[ExportPresentation](m[1,1]*(X-$a)+m[1,2]*(Y-$b)+m[1,3]*(Z-$c)=0):
m2:=MathML[ExportPresentation](m[2,1]*(X-$a)+m[2,2]*(Y-$b)+m[2,3]*(Z-$c)=0):
m3:=MathML[ExportPresentation](m[3,1]*(X-$a)+m[3,2]*(Y-$b)+m[3,3]*(Z-$c)=0):
q:=MathML[ExportPresentation]($a +((($g*$b)+($h*$c))/$f)-($g/$f)*Y-($h/$f)*Z):
m4,m1,m2,m3,convert(q,string)
");
$M1=switch(1,$A);
$M2=switch(2,$A);
$M3=switch(3,$A);
$ans=$a +frac((($g*$b)+($h*$c)),$f)-frac($g,$f)*Y-frac($h,$f)*Z;
$Ans=switch(4,$A);@
qu.1.17.uid=6be7a3dc-8e16-4e1e-9c5f-4b880ab6d384@
qu.1.17.info=  Course=Introductory Mathematical Economics;
  Topic=Systems Of Equations;
  Sub-Topic=Infinite Solutions -- Two Parameter;
  Author=Asha Sadanand;
  Difficulty=Hard;
@
qu.1.17.weighting=1,1,1,1@
qu.1.17.numbering=alpha@
qu.1.17.part.1.grader=exact@
qu.1.17.part.1.name=sro_id_1@
qu.1.17.part.1.editing=useHTML@
qu.1.17.part.1.display.permute=true@
qu.1.17.part.1.answer.3=has a unique solution@
qu.1.17.part.1.question=(Unset)@
qu.1.17.part.1.answer.2=has an infinite number of solutions.@
qu.1.17.part.1.answer.1=has no solutions.@
qu.1.17.part.1.mode=List@
qu.1.17.part.1.display=menu@
qu.1.17.part.1.credit.3=0.0@
qu.1.17.part.1.credit.2=1.0@
qu.1.17.part.1.credit.1=0.0@
qu.1.17.part.2.name=sro_id_2@
qu.1.17.part.2.maple_answer=$ans@
qu.1.17.part.2.editing=useHTML@
qu.1.17.part.2.question=(Unset)@
qu.1.17.part.2.libname=@
qu.1.17.part.2.mode=Maple@
qu.1.17.part.2.allow2d=1@
qu.1.17.part.2.plot=@
qu.1.17.part.2.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.17.part.2.type=formula@
qu.1.17.part.3.name=sro_id_3@
qu.1.17.part.3.maple_answer=Y@
qu.1.17.part.3.editing=useHTML@
qu.1.17.part.3.question=(Unset)@
qu.1.17.part.3.libname=@
qu.1.17.part.3.mode=Maple@
qu.1.17.part.3.allow2d=1@
qu.1.17.part.3.plot=@
qu.1.17.part.3.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.17.part.3.type=formula@
qu.1.17.part.4.name=sro_id_4@
qu.1.17.part.4.maple_answer=Z@
qu.1.17.part.4.editing=useHTML@
qu.1.17.part.4.question=(Unset)@
qu.1.17.part.4.libname=@
qu.1.17.part.4.mode=Maple@
qu.1.17.part.4.allow2d=1@
qu.1.17.part.4.plot=@
qu.1.17.part.4.maple=resp:=subs({x=X,y=Y,z=Z},$RESPONSE);
evalb(($ANSWER)=(resp));@
qu.1.17.part.4.type=formula@
qu.1.17.question=<p>Consider the following system of equations:</p><p>&nbsp;</p><p>&nbsp;$M1</p><p>$M2</p><p>$M3</p><p>&nbsp;</p><p>Choose the answer that correctly describes the system</p><p>The system<span>&nbsp;</span><1><span> </span>.</p><p>&nbsp;</p><p>Give the solutions for the system, if they exist.</p><p>If there is no solution type "no solution" (without quotes) in each case.</p><p>If there are many solutions, for a two parameter solution type&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math> box and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math> in the&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>box, and give <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math>;</p><p>For a one parameter solutions type <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math> in the <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>box and then give <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi></mrow></mstyle></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi></mrow></mstyle></math> in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi></mrow></mstyle></math>.</p><p>&nbsp;</p><p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><2><span> </span>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><3><span> </span>, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>Z</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math><span>&nbsp;</span><4><span>&nbsp;</span></p>@

