Question

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

Answer 1

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

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)=(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)=(1)/(-22)\left[\begin{array}{cc}-6&2\\-1&4\end{array}\right]`

`\left[\begin{array}{c}x\\y\end{array}\right]=(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`