# How do you solve 5x + 3y = -8 and 4x + 6y = 8 using matrices?

Jun 27, 2018

The solution is $\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}- 4 \\ 4\end{matrix}\right)$

#### Explanation:

The equation in matrix form is

$A X = B$

where

The matrix $A = \left(\begin{matrix}5 & 3 \\ 4 & 6\end{matrix}\right)$

$X = \left(\begin{matrix}x \\ y\end{matrix}\right)$

and

$B = \left(\begin{matrix}- 8 \\ 8\end{matrix}\right)$

The solution is

$X = {A}^{-} 1 B$

The inverse of $A$ is

${A}^{-} 1 = \frac{1}{|} \left(5 , 3\right) , \left(4 , 6\right) | \left(\begin{matrix}6 & - 3 \\ - 4 & 5\end{matrix}\right)$

$= \frac{1}{18} \left(\begin{matrix}6 & - 3 \\ - 4 & 5\end{matrix}\right)$

Therefore,

$X = \frac{1}{18} \left(\begin{matrix}6 & - 3 \\ - 4 & 5\end{matrix}\right) \left(\begin{matrix}- 8 \\ 8\end{matrix}\right)$

$= \frac{1}{18} \left(\begin{matrix}- 72 \\ 72\end{matrix}\right)$

$= \left(\begin{matrix}- 4 \\ 4\end{matrix}\right)$

The solution is

$\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}- 4 \\ 4\end{matrix}\right)$