Given: $\;$ $A = \begin{bmatrix} 2 & -6 \\ 2 & 0 \end{bmatrix}$, $\;$ $B = \begin{bmatrix} -3 & 2 \\ 4 & 0 \end{bmatrix}$ $\;$ and $\;$ $C = \begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix}$. Find matrix $X$ such that $A + 2 X = 2B + C$.
Let $X = \begin{bmatrix}
p & q \\
r & s
\end{bmatrix}$
Given: $\;$ $A + 2 X = 2 B + C$
i.e. $\;$ $\begin{bmatrix}
2 & -6 \\
2 & 0
\end{bmatrix} + 2 \begin{bmatrix}
p & q \\
r & s
\end{bmatrix} = 2 \begin{bmatrix}
-3 & 2 \\
4 & 0
\end{bmatrix} + \begin{bmatrix}
4 & 0 \\
0 & 2
\end{bmatrix}$
i.e. $\;$ $\begin{bmatrix}
2 + 2p & -6 + 2q \\
2 + 2r & 0 + 2s
\end{bmatrix} = \begin{bmatrix}
-6 + 4 & 4 + 0 \\
8 + 0 & 0 + 2
\end{bmatrix}$
i.e. $\;$ $\begin{bmatrix}
2 + 2p & -6 + 2q \\
2 + 2r & 2s
\end{bmatrix} = \begin{bmatrix}
-2 & 4 \\
8 & 2
\end{bmatrix}$
Two matrices are equal when all their corresponding elements are equal.
i.e. $\;$ $2 + 2p = -2$ $\implies$ $2 p = -4$ $\implies$ $p = -2$
$-6 + 2q = 4$ $\implies$ $2 q = 10$ $\implies$ $q = 5$
$2 + 2 r = 8$ $\implies$ $2 r = 6$ $\implies$ $r = 3$
$2 s = 2$ $\implies$ $s = 1$
$\therefore \;$ $X = \begin{bmatrix}
-2 & 5 \\
3 & 1
\end{bmatrix}$