How do you find the solution set of a matrix? Thank you!

Question

1 0 0 2 3 | 4

0 1 0 5 6 | 9

0 0 1 1 1 | 7

I'd like to see how to do this big matrix

Answer 1

If we denote the variables by x₁ through x₅, then we have:

x₁ + 2x₄ + 3x₅ = 4

x₂ + 5x₄ + 6x₅ = 9

x₃ + x₄ + x₅ = 7

Let x₄ = s and x₅ = t

Then x₁ = 4 - 2s - 3t, x₂ = 9 - 5s - 6t, x₃ = 7 - s - t (s and t are parameters, i.e, any real numbers)

Solution set (x₁, ..., x₅) = (4 - 2s - 3t, 9 - 5s - 6t, 7 - s - t, s, t) =

s(-2, -5, -1, 1, 0) + t(-3, -6, -1, 0, 1) + (4, 9, 7, 0, 0)