Find the values of x and y if $\begin{bmatrix$}[r] x + y & y \ 2x & x - y \end{bmatrix} \begin{bmatrix}[r] 2 \ -1 \end{bmatrix} = \begin{bmatrix}[r] 3 \ 2 \end{bmatrix}.

Given,

$\begin{bmatrix$}[r] x + y & y \ 2x & x - y \end{bmatrix} \begin{bmatrix}[r] 2 \ -1 \end{bmatrix} = \begin{bmatrix}[r] 3 \ 2 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] (x + y) \times 2 + y \times (-1) \ 2x \times 2 + (x - y) \times (-1) \end{bmatrix} = \begin{bmatrix}[r] 3 \ 2 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 2x + 2y - y \ 4x - x + y \end{bmatrix} = \begin{bmatrix}[r] 3 \ 2 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 2x + y \ 3x + y \end{bmatrix} = \begin{bmatrix}[r] 3 \ 2 \end{bmatrix} \\[1em]

By definition of equality of matrices we have,

⇒ 2x + y = 3 or y = 3 - 2x     (…Eq 1)
⇒ 3x + y = 2                         (…Eq 2)

Putting value of y from equation 1 in equation 2,

⇒ 3x + 3 - 2x = 2
⇒ x + 3 = 2
⇒ x = -1.

Now finding value of y,

⇒ y = 3 - 2x
⇒ y = 3 - 2(-1)
⇒ y = 3 + 2
⇒ y = 5.

Hence, the value of x = -1 and y = 5.