# Question:Example for finding all solutions to a system of equations

## Question:Example for finding all solutions to a system of equations

This is a training example for finding all solutions to a system of equations. If you look at the graph, you can count 36 solutions, but I managed to find 20 relatively good approximations. No attempts to get more solutions, which are also visible as intersections of graphs, did not lead to success. Therefore, there is a suspicion that there are only 20 solutions.
Is it so?

``` restart: with(plots):
a:=8.:
f1 := x1^4-1.999*x1^2*x2^2+x2^4-1;
f2 := tan(x1+x2)-x2*sin(x1);
implicitplot([f1, f2], x1 = -a .. a, x2 = -a .. a, numpoints = 25000, scaling = constrained,  color = [red, blue], thickness = 1);```
```   1, (3.192246883291975), (-3.0395187374365404)
2, (3.0952031367176476), (-3.2447717313041897)
3, (0.5881900748267959), (-1.160066226905079)
4, (-0.936866718243322), (-1.3700058362814254)
5, (-2.555853694651265), (-2.7399958564861953)
6, (-3.2556241416421168), (-3.3964651254113774)
7, (-3.583319843955091), (-3.7077839724189228)
8, (-5.364827188794712), (-5.401998918608201)
9, (-5.398295356665546), (-5.360818510223991)
10, (-3.769206506106412), (-3.6477855329362683)
11,(-1.3978806247566642), (-0.9772190664843745)
12, (-1.192159295544978),(0.6492335177657542)
13, (-3.0867255059416623),(2.927375855548188)
14, (-3.18519036357835), (3.329801919022179)
15, (2.0108268901120754), (2.243492422396739)
16, (3.1133812329649766), (3.261937603184373)
17, (4.0265558604742715), (4.130826167761226)
18, (4.124539552922121), (4.019977762680433)
19, (3.172338340844501), (3.018365965761908)
20, (2.1945695320368097), (1.9558412553082192)
```

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