# How do you solve x-2y=2 and 3x-5y=9 using matrices?

Feb 3, 2017

$x = 8 , y = 3$

#### Explanation:

Put the eqns in matrix form

$x - 2 y = 2$
$3 x - 5 y = 9$

become

$\left(\begin{matrix}1 & - 2 \\ 3 & - 5\end{matrix}\right)$$\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}2 \\ 9\end{matrix}\right)$

we therefore have

$M \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}2 \\ 9\end{matrix}\right)$

by premultiplying both sides by ${M}^{- 1}$

${M}^{- 1} M \left(\begin{matrix}x \\ y\end{matrix}\right) = {M}^{- 1} \left(\begin{matrix}2 \\ 9\end{matrix}\right)$

$I \left(\begin{matrix}x \\ y\end{matrix}\right) = {M}^{- 1} \left(\begin{matrix}2 \\ 9\end{matrix}\right)$

where $I = \left(\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right)$

$\therefore \left(\begin{matrix}x \\ y\end{matrix}\right) = {M}^{- 1} \left(\begin{matrix}2 \\ 9\end{matrix}\right)$

so we need to find the inverse of the matrix.

$M = \left(\begin{matrix}1 & - 2 \\ 3 & - 5\end{matrix}\right)$

for any $2 \times 2$ mx

$M = \left(\begin{matrix}a & b \\ c & d\end{matrix}\right)$

${M}^{- 1} = \frac{1}{\Delta} \left(\begin{matrix}d & - b \\ - c & a\end{matrix}\right) \text{ providing } \Delta M \ne 0$

find its determinant

$\Delta M = | \left(1 , - 2\right) , \left(3 , - 5\right) |$

$\Delta M = 1 \times - 5 - \left(3 \times - 2\right) = - 5 - - 6 = 1$

$\because \Delta M \ne 0 \text{ the inverse exists}$

$\therefore {M}^{- 1} = \frac{1}{1} \left(\begin{matrix}- 5 & 2 \\ - 3 & 1\end{matrix}\right) = \left(\begin{matrix}- 5 & 2 \\ - 3 & 1\end{matrix}\right)$

so$\left(\begin{matrix}x \\ y\end{matrix}\right) = {M}^{- 1} \left(\begin{matrix}2 \\ 9\end{matrix}\right) = \left(\begin{matrix}- 5 & 2 \\ - 3 & 1\end{matrix}\right) \left(\begin{matrix}2 \\ 9\end{matrix}\right)$

$\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}- 5 \times 2 + 2 \times 9 \\ - 3 \times 2 + 1 \times 9\end{matrix}\right)$

$\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}- 10 + 18 \\ - 6 + 9\end{matrix}\right)$

$\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}- 10 + 18 \\ - 6 + 9\end{matrix}\right)$

$\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}8 \\ 3\end{matrix}\right)$