If $2\begin{bmatrix$}[r] 3 & 4 \ 5 & x \end{bmatrix} + \begin{bmatrix}[r] 1 & y \ 0 & 1 \end{bmatrix} = \begin{bmatrix}[r] z & 0 \ 10 & 5 \end{bmatrix}, find the values of x, y and z.

Given,

$\Rightarrow 2\begin{bmatrix$}[r] 3 & 4 \ 5 & x \end{bmatrix} + \begin{bmatrix}[r] 1 & y \ 0 & 1 \end{bmatrix} = \begin{bmatrix}[r] z & 0 \ 10 & 5 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 6 & 8 \ 10 & 2x \end{bmatrix} + \begin{bmatrix}[r] 1 & y \ 0 & 1 \end{bmatrix} = \begin{bmatrix}[r] z & 0 \ 10 & 5 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 6 + 1 & 8 + y \ 10 + 0 & 2x + 1 \end{bmatrix} = \begin{bmatrix}[r] z & 0 \ 10 & 5 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 7 & 8 + y \ 10 & 2x + 1 \end{bmatrix} = \begin{bmatrix}[r] z & 0 \ 10 & 5 \end{bmatrix} \\[1em] \Rightarrow 7 = z, 8 + y = 0 \text{ and } 2x + 1 = 5 \\[0.5em] \Rightarrow z = 7, y = -8 \text{ and } 2x = 4 \\[0.5em] \Rightarrow z = 7, y = -8 \text{ and } x = 2.

∴ x = 2, y = -8 and z = 7.

Hence, the values are x = 2, y = -8 and z = 7.