[1 1 -1 : 5]

[1 2 -3 : 9]

[1 -1 3 : 3]

The left side of the : is the coefficients of x,y,z in our system and the numbers on the right side are our values. We need to be this is row echloen form.

First replace row 2 by (-row 2 + row 1). Then

[1 1 -1 : 5]

[0 -1 2 : -4]

[1 -1 3 : 3]

Now, replace row 3 by (-row 3 + row 1). Then

[1 1 -1 : 5]

[0 -1 2 : -4]

[0 2 -4 : -2]

Now multiply row 2 by -1. Then

[1 1 -1 : 5]

[0 1 -2 : 4]

[0 2 -4 : -2]

Now, replace row 3 by (-1/2 row 3 + row 2). Then

[1 1 -1 : 5]

[0 1 -2 : 4]

[0 0 0 : 5]

This third row implies that 0x + 0y + 0z = 5 i.e. 0 = 5 which is impossible. No solution!