# How do you solve 2x+4y=10, 6x+2y=10 using Cramer's rule?

Mar 2, 2017

$\left(x , y\right) = \left(1 , 2\right)$
$\textcolor{w h i t e}{\text{XXX}}$see below for determination using Cramer's Rule.

#### Explanation:

Re-writing the given equations as an augmented matrix:
color(white)("XXX")( (2,4,"|",10),(6,2,"|",10))

and using the standard derived square matrices:
$M = \left(\begin{matrix}2 & 4 \\ 6 & 2\end{matrix}\right) , {M}_{x} = \left(\begin{matrix}10 & 4 \\ 10 & 2\end{matrix}\right) , {M}_{y} = \left(\begin{matrix}2 & 10 \\ 6 & 10\end{matrix}\right)$

We can calculate the Determinants:
$\textcolor{w h i t e}{\text{XXX}} {D}_{M} = 2 \times 2 - 6 \times 4 = - 20$
$\textcolor{w h i t e}{\text{XXX}} {D}_{{M}_{x}} = 10 \times 2 - 10 \times 4 = - 20$
$\textcolor{w h i t e}{\text{XXX}} {D}_{{M}_{y}} = 2 \times 10 - 6 \times 10 = - 40$

Cramer's Rule Tells us:
$\textcolor{w h i t e}{\text{XXX}} x = \frac{{D}_{{M}_{x}}}{{D}_{M}} = \frac{- 20}{- 20} = 1$
and
$\textcolor{w h i t e}{\text{XXX}} y = \frac{{D}_{{M}_{y}}}{{D}_{M}} = \frac{- 40}{- 20} = 2$