# Use an inverse matrix to solve system of linear equations.x+2y=-1x-2y=3

Use an inverse matrix to solve system of linear equations.
$x+2y=-1$
$x-2y=3$

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Step 1
In matrix form, given system of equations cab be written as
$x+2y=-1$
$x-2y=3$
$AX=B$
$A=\left[\begin{array}{cc}1& 2\\ 1& -2\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right],B=\left[\begin{array}{c}-1\\ 3\end{array}\right]$
We know that
${\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$
Step 2
Therefore,
${A}^{-1}=\frac{1}{1\left(-2\right)-2\left(1\right)}\left[\begin{array}{cc}-2& -2\\ -1& 1\end{array}\right]$
${A}^{-1}=-\frac{1}{4}\left[\begin{array}{cc}-2& -2\\ -1& 1\end{array}\right]$
Therefore, solution of system of equations is
$X={A}^{-1}B$
$X=-\frac{1}{4}\left[\begin{array}{cc}-2& -2\\ -1& 1\end{array}\right]\left[\begin{array}{c}-1\\ 3\end{array}\right]$
$X=-\frac{1}{4}\left[\begin{array}{cc}2& -6\\ 1& 3\end{array}\right]\left[\begin{array}{c}1\\ -1\end{array}\right]$
$x=1,y=-1$
Result:$x=1,y=-1$