Question

Solve the following system of equation using inverse matrix method.
5x + 2y=4
7x + 3y=5​

Answer 1

`5x+2y = 4~~~~~~~~~~...(i)\\\\7x +3y = 5~~~~~~~~~~...(ii)\\\\\text{Write in AX = B form.}\\\\~~~~~~\begin{bmatrix}5&2\\7&3 \end{bmatrix} \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix} 4\\5 \end{bmatrix}\\\\\\\implies \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix} 4\\5 \end{bmatrix}\begin{bmatrix}5&2\\7&3 \end{bmatrix}^(-1)\\\\\\`

`\\\implies \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix} 4\\5 \end{bmatrix}\cdot\frac{\begin{bmatrix} 3&-7\\-2&5\end{bmatrix}^(T)}{\begin{vmatrix}5&2\\ 7&3 \end{vmatrix}}\\\\\\\implies \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix} 4\\5 \end{bmatrix}\cdot\frac{\begin{bmatrix} 3&-2\\-7&5\end{bmatrix}}{5(3) -2(7)}\\\\\\`

`\\\implies \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix} 4\\5 \end{bmatrix}\cdot\begin{bmatrix} 3&-2\\-7&5\end{bmatrix}\\\\\\ \implies \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix}12-10\\-28+25 \end{bmatrix}\\\\\\ \implies \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix}2\\-3\end{bmatrix}\\\\\\\text{Hence,}~ (x,y) = (2,-3)`