# How can i use an inverse matrix to solve this equation? -x + y= 4, -2x + y = 0

Jul 22, 2017

$x , y \in \left\{4 , 8\right\}$

#### Explanation:

$- x + y = 4 , - 2 x + y = 0$

Rewrite these equations using matrices

$\left(\begin{matrix}- 1 & 1 \\ - 2 & 1\end{matrix}\right) \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}4 \\ 0\end{matrix}\right)$

Let $A = \left(\begin{matrix}- 1 & 1 \\ - 2 & 1\end{matrix}\right)$

Then $A \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}4 \\ 0\end{matrix}\right)$

And $\left(\begin{matrix}x \\ y\end{matrix}\right) = {A}^{-} 1 \left(\begin{matrix}4 \\ 0\end{matrix}\right)$

$\det \left(A\right) = - 1 \left(1\right) - \left(- 2\right) 1 = 1 \Rightarrow {A}^{-} 1 = \left(\begin{matrix}1 & - 4 \\ 2 & - 1\end{matrix}\right)$

$\left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}1 & - 4 \\ 2 & - 1\end{matrix}\right) \left(\begin{matrix}4 \\ 0\end{matrix}\right) = \left(\begin{matrix}4 \\ 8\end{matrix}\right)$

$\therefore x = 4 , y = 8$

You may have noticed there's another way to do it. From the second equation, we get $y = 2 x$. Subbing this into the first equation allows us to solve for $x$ and $y$