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

User Roger Ray
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1 Answer

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\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.

User Shammoo
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