Solve system of linear equations, using matrix method. 5x + 2y = … - Vidyadip

Question

Solve system of linear equations, using matrix method.

5x + 2y = 3

3x + 2y = 5

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Answer

Matrix form of given equations is AX = B $\Rightarrow\ \begin{bmatrix}5&2\\3&2\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}3\\5\end{bmatrix}$
$\text{Here}\ \text{A}= \begin{bmatrix}5&2\\3&2\end{bmatrix},\ \text{X}=\begin{bmatrix}x\\y\end{bmatrix}\text{and B}=\begin{bmatrix}3\\5\end{bmatrix}$
$\therefore\ \text{|A|}=\begin{vmatrix}5&2\\3&2\end{vmatrix}=10-6=4\neq0$
Therefore, solution is unique and $\text{X=A}^{-1}\text{B}=\frac{1}{\text{|A|}}\text{(adj. A)B}$
$\Rightarrow\ \begin{bmatrix}x\\y\end{bmatrix}=\frac{1}{4}\begin{bmatrix}2&-2\\-3&5\end{bmatrix}\begin{bmatrix}3\\5\end{bmatrix}$
$=\frac{1}{4}\begin{bmatrix}6-10\\-9+25\end{bmatrix}=\begin{bmatrix}1\\4\end{bmatrix}\begin{bmatrix}-4\\16\end{bmatrix}=\begin{bmatrix}-1\\4\end{bmatrix}$
Therefore, x = -1 and y = 4

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