Question

Find the solution of the following.

$2x-y=-2$ and $3x+4y=3$.

Answer 1

The given system of equations can be written in the form of $AX=B$, where

$A=\left[\begin{array}{cc}2& -1\\ 3& 4\end{array}\right],X=\left[\begin{array}{c}x\\ y\end{array}\right],B=\left[\begin{array}{c}-2\\ 3\end{array}\right]$

Now, $|A|=8+3=11\ne 0$

Thus, A is non-singular. Therefore, its inverse exists.

Now,

$A^{-1}=\frac{1}{|A|}adjA$

$A^{-1}=\frac{1}{11}\left[\begin{array}{cc}4& 1\\ -3& 2\end{array}\right]$

Therefore,

$X=A^{-1}B=\frac{1}{11}\left[\begin{array}{cc}4& 1\\ -3& 2\end{array}\right]\left[\begin{array}{c}-2\\ 3\end{array}\right]$

$\left[\begin{array}{c}x\\ y\end{array}\right]=\frac{1}{11}\left[\begin{array}{c}-5\\ 12\end{array}\right]=\left[\begin{array}{c}-\frac{5}{11}\\ \frac{12}{11}\end{array}\right]$

Hence, $x=-\frac{5}{11}$ and $y=\frac{12}{11}$.