Question

Use the Gaussian elimination to find the complete solution of equation or show that none exists.
5x+12y+z=10
2x+5y+2z=-1
x+2y-3z=5

Answer 1

There's no solution.

Multiply the second row by two and add it to the third row. Then this combination gives us
5x + 12 y + z = 3.

But the first equation is 5x + 12y + z = 10.

Since 3 can never equal 10, no solution can exist.