I just want to check that I did a certain problem correctly. This is it: $$a+b=3 \pmod{26}\\2a+b=7 \pmod{26}$$ Solve for $a$ and $b$
Now I setup the augmented matrix: $$\left[ \begin{array}{ccc} 1 & 1 & 3 \\ 2 & 1 & 7 \end{array} \right]$$
After doing elementary row operations I get: $$\left[ \begin{array}{ccc} 1 & 0 & 4 \\ 0 & 1 & -1 \end{array} \right]$$
That yields: $$4-1=3\pmod{26}\\ 8-1=7\pmod{26}$$
Did I do that correctly?
The matrix manipulations are correct, but the interpretation of the answer isn't. After doing row operations, your matrix is the augmented matrix of the system \begin{align*} a \qquad & \equiv \phantom{-}4 \pmod{26} \\ \qquad b & \equiv -1 \pmod{26}. \end{align*} So that's your solution: $a\equiv4\pmod{26}$, $b\equiv-1\equiv25\pmod{26}$. You can check that this solution satisfies both original congruences (and also check that the solution you originally obtained does not).