# How do you solve 2x + 6y = 2 and x + 5y = 3 using matrices?

Oct 23, 2016

$\left(x , y\right) = \left(- 2 , 1\right)$
(see below for solution using matrices).

#### Explanation:

Write the given equations
$\textcolor{w h i t e}{\text{XXX}} 2 x + 6 y = 2$
$\textcolor{w h i t e}{\text{XXX}} x + 5 y = 3$
in augmented matrix form
$\textcolor{w h i t e}{\text{XXX}} \left(\begin{matrix}2 & 6 & 2 \\ 1 & 5 & 3\end{matrix}\right)$

Divide the first row by $2$
$\textcolor{w h i t e}{\text{XXX}} \left(\begin{matrix}1 & 3 & 1 \\ 1 & 5 & 3\end{matrix}\right)$

Subtract the (new) first row from the second row
$\textcolor{w h i t e}{\text{XXX}} \left(\begin{matrix}1 & 3 & 1 \\ 0 & 2 & 2\end{matrix}\right)$

Divide the (new) second row by $2$
$\textcolor{w h i t e}{\text{XXX}} \left(\begin{matrix}1 & 3 & 1 \\ 0 & 1 & 1\end{matrix}\right)$

Subtract $3$ times this second row from this first row
$\textcolor{w h i t e}{\text{XXX}} \left(\begin{matrix}1 & 0 & - 2 \\ 0 & 1 & 1\end{matrix}\right)$

Note that this is equivalent to the expressions
color(white)("XXX"){: {:(1x,+0y,=-2),(0x,+1y,=1):}} rArr {:(x=-2),(y=1):}