Finding variables in a 2x2 matrix multiplication

Question

How do I find $b$ and $d$ in the equation: $\begin{bmatrix}6 & 25\\12 & 15\end{bmatrix} \times \begin{bmatrix}2 & b\\5 & d\end{bmatrix}$ = $\begin{bmatrix}22 & 17\\10 & 22\end{bmatrix}$

Answer 1

Note that the first column of the resultant matrix is $[137,99]^T$.

On the one hand

$$\begin{bmatrix}6 & 25\\12 & 15\end{bmatrix} \begin{bmatrix}2 & b\\5 & d\end{bmatrix} = \begin{bmatrix}137 & 17\\99 & 22\end{bmatrix}$$

and on the other hand

$$\begin{bmatrix}6 & 25\\12 & 15\end{bmatrix} \begin{bmatrix}2 & b\\5 & d\end{bmatrix} = \begin{bmatrix}137 & 6b+25d\\99 & 12b+25d\end{bmatrix}.$$

We, then, have to solve the following system of linear equations:

$$6b+25d=17$$ $$12b+25d=22.$$

This should not cause any problem:

The solutions are $b=\frac56$ and $d=\frac{12}{25}$. These numbers give the second column of the given resultant matrix.