# How do you solve x - 2y = 5 and 2x - 3y = 10 using matrices?

Apr 5, 2016

Given
$\textcolor{w h i t e}{\text{XXX}} x - 2 y = 5$
$\textcolor{w h i t e}{\text{XXX}} 2 x - 3 y = 10$

Re-writing as augmented matrix:
$\textcolor{w h i t e}{\text{XXX") [(1,-2,5),(2,-3,10)]color(white)("XXX}} \left.\begin{matrix}\left[1\right] \\ \left[2\right]\end{matrix}\right.$

Subtracting $2$ times the first row from the second row:
$\textcolor{w h i t e}{\text{XXX") [(1,-2,5),(0,1,0)]color(white)("XXX}} \left.\begin{matrix}\left[1\right] \\ \left[3\right] = \left[2\right] - 2 \times \left[1\right]\end{matrix}\right.$

Adding $2$ times row [3] to the first row:
$\textcolor{w h i t e}{\text{XXX") [(1,0,5),(0,1,0)]color(white)("XXX}} \left.\begin{matrix}\left[4\right] \\ \left[3\right] = \left[1\right] + 2 \times \left[3\right]\end{matrix}\right.$

Re-writing with variables:
$\textcolor{w h i t e}{\text{XXX}} x = 5$
$\textcolor{w h i t e}{\text{XXX}} y = 0$