Solve by the matrix method: $2x + y = 5$ and $3x + 2y = 7$. What is $x$?

Explanation

1. 01

   Write it as $AX = B$ with $A = \begin{bmatrix}2 & 1 \\ 3 & 2\end{bmatrix}$ and $B = \begin{bmatrix}5 \\ 7\end{bmatrix}$; $|A| = 1$, so $X = A^{-1}B$.

2. 02

   $A^{-1} = \frac{1}{1}\begin{bmatrix}2 & -1 \\ -3 & 2\end{bmatrix}$, and $A^{-1}B = \begin{bmatrix}3 \\ -1\end{bmatrix}$.

3. 03

   So $x = 3$ and $y = -1$.