Question

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20

4 -1 3 12
1 4 6 -32
5 3 9 20

I believe that the solution does not exist and how can I prove that?

Answer 1

never mind the matrix stuff.

Suppose we subtract #3 - #1
I get x + 4y + 6z = 8

but #2 says:
x + 4y + 6z = -32
which contradicts the above, so ....

no solution