Matrices
Given: $\;$ $\begin{bmatrix} 3 & -2 \\ -4 & 4 \end{bmatrix} \begin{bmatrix} 2x \\ 1 \end{bmatrix} + 2 \begin{bmatrix} -4 \\ 5 \end{bmatrix} = \begin{bmatrix} 8 \\ 4y \end{bmatrix}$, find the value of $x$ and $y$.
$\begin{bmatrix}
3 & -2 \\
-4 & 4
\end{bmatrix} \begin{bmatrix}
2x \\
1
\end{bmatrix} + 2 \begin{bmatrix}
-4 \\
5
\end{bmatrix} = \begin{bmatrix}
8 \\
4y
\end{bmatrix}$
i.e. $\;$ $\begin{bmatrix}
6x - 2 \\
-8x + 4
\end{bmatrix} + \begin{bmatrix}
-8 \\
10
\end{bmatrix} = \begin{bmatrix}
8 \\
4y
\end{bmatrix}$
i.e. $\;$ $\begin{bmatrix}
6x - 10 \\
-8x + 14
\end{bmatrix} = \begin{bmatrix}
8 \\
4y
\end{bmatrix}$
When two matrices are equal, their corresponding elements are equal.
i.e. $\;$ $6x - 10 = 8$ $\implies$ $6x = 18$ $\implies$ $x = 3$
and, $\;$ $-8x + 14 = 4y$
i.e. $\;$ $-8 \times 3 + 14 = 4y$ $\implies$ $-10 = 4y$ $\implies$ $y = - \dfrac{5}{2}$
$\therefore \;$ $x = 3$, $\;$ $y = - \dfrac{5}{2}$