Question

Find system by matrix 4) 4x – 2y = 14
x - 6y=-13

Answer 1

$\huge{x=5,y=3}$

Step-by-step explanation:

    Given system of equations, $4x\textrm{ }-\textrm{ }2y\textrm{ }=\textrm{ }14\textrm{ , }x\textrm{ }-\textrm{ }6y\textrm{ }=\textrm{ }-13$

    The system can be equivalenty represented as a matrix product as $\left[\begin{array}{cc}4&-2\\1&-6\end{array}\right]\left[\begin{array}{c}x\\y\end{array}\right]\textrm{ = }\left[\begin{array}{c}14\\-13\end{array}\right]$

   This can be simplifies,  $Ax=B$⇒$x=A^{-1} B$

For a matrix $A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$ of size $2$ × $2$, $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d&-b\\-c&a\end{array}\right]$.

So $\left[\begin{array}{cc}4&-2\\1&-6\end{array}\right]^{-1}=\frac{1}{-22}\left[\begin{array}{cc}-6&2\\-1&4\end{array}\right]$

$\left[\begin{array}{c}x\\y\end{array}\right]=\frac{1}{-22}\left[\begin{array}{cc}-6&2\\-1&4\end{array}\right]\left[\begin{array}{c}14\\-13\end{array}\right]=\left[\begin{array}{c}5\\3\end{array}\right]$

∴$x=5,y=3$