Question
Solve system of linear equations, using matrix method.
4x - 3y = 3
3x - 5y = 7
✓
Answer
Matrix form of given equations AX = B $\Rightarrow\ \begin{bmatrix}4&-3\\3&-5\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}3\\7\end{bmatrix}$ $\text{Here}\ \text{A}=\begin{bmatrix}4&-3\\3&-5\end{bmatrix},\ \text{X}=\begin{bmatrix}x\\y\end{bmatrix}\text{and B}=\begin{bmatrix}3\\7\end{bmatrix}$ $\therefore\ \text{|A|}=\begin{vmatrix}4&-3\\3&-5\end{vmatrix}=-20-(-9)=-20+9=-11\neq0$ Therefore, solution is unique and $\text{X=A}^{-1}\text{B}=\frac{1}{\text{|A|}}\text{(adj. A)B}$ $\Rightarrow\ \begin{bmatrix}x\\y\end{bmatrix}=\frac{1}{-11}\begin{bmatrix}-5&3\\-3&4\end{bmatrix}\begin{bmatrix}3\\7\end{bmatrix}$ $=\frac{1}{-11}\begin{bmatrix}-15+21\\-9+28\end{bmatrix}=\frac{1}{-11}\begin{bmatrix}6\\19\end{bmatrix}=\begin{bmatrix}\frac{-6}{11}\\\frac{-19}{11}\end{bmatrix}$ Therefore, $x=\frac{-6}{11}\text{and}\ y=\frac{-19}{11}$
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