Solving a system of equation with matrices

[Mr Davis 97]

I have the following system of equations: $2t-4s=-2;~-t+2s=-1;~3t-5s=3$. With them, I form the matrix
\begin{bmatrix}
2 & -4 & -2 \\
-1 & 2 & -1 \\
3 & -5 & 3
\end{bmatrix}
Which turns out to be row equivalent to
\begin{bmatrix}
1 & 0 & 11 \\
0 & 1 & 6 \\
0 & 0 & 0
\end{bmatrix}
so $s=11,~t=6$. However, this satisfies only the first and third equation and not the second. Shouldn't it satisfy all of the equations, since I got a valid result from doing row reduction?

 

[fresh_42]

Have you worked with $A_{23}=-1$ as in your second equation, or with $A_{23}=-2$ in your matrix?
And can the first two equations hold true simultaneously at all?

 

[Mr Davis 97]

Have you worked with $A_{23}=-1$ as in your second equation, or with $A_{23}=-2$ in your matrix?
And can the first two equations hold true simultaneously at all?

I fixed it. Am I doing the row reduction wrong? Should I be getting an inconsistent system?

 

[fresh_42]

Multiply the second equation by $-2$ and compare it with the first.

 

[Mr Davis 97]

Multiply the second equation by $-2$ and compare it with the first.

2t - 4s = -2 and 2t - 4s = 2, which cannot be possible. So I guess I just did the row reduction wrong.