Question

-x=-y+2 and 3y+4=2x solve using matrices

Answer 1

`\begin{cases}-x=-y+2\\3y+4=2x\end{cases}\implies\begin{cases}x-y=-2\\2x-3y=4\end{cases}`

In matrix form, this is

`\begin{bmatrix}1&-1\\2&-3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-2\\4\end{bmatrix}`

The coefficient matrix has determinant

`\begin{vmatrix}1&-1\\2&-3\end{vmatrix}=-3+2=-1\\eq0`

so it has an inverse, which is

`\begin{bmatrix}1&-1\\2&-3\end{bmatrix}^(-1)=\begin{bmatrix}3&-1\\2&-1\end{bmatrix}`

Multiply both sides by the inverse matrix:

`\begin{bmatrix}3&-1\\2&-1\end{bmatrix}\begin{bmatrix}1&-1\\2&-3\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}3&-1\\2&-1\end{bmatrix}\begin{bmatrix}-2\\4\end{bmatrix}`

`\implies\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-10\\-8\end{bmatrix}`

so that
`x=-10` and
`y=-8`.