Question: How would I find distinct solutions in a matrix equation?

How would I find and check 10 distinct solutions to the matrix equation Matrix(2, 4, {(1, 1) = 1, (1, 2) = 2, (1, 3) = 3, (1, 4) = 4, (2, 1) = 5, (2, 2) = 6, (2, 3) = 7, (2, 4) = 8}) Vector(4, {(1) = x[1], (2) = x[2], (3) = x[3], (4) = x[4]}) = Vector(2, {(1) = 10, (2) = 10}) ? I know that I need to use LinearSolve, but I'm not sure how to do it exactly...
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