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

Question

Solve system of linear equations, using matrix method.

2x - y = -2

3x + 4y = 3

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Answer

Matrix form of given equations is AX = B $\Rightarrow\ \begin{bmatrix}2&-1\\3&4\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-2\\3\end{bmatrix}$

$\text{Here}\ \text{A}=\begin{bmatrix}2&-1\\3&4\end{bmatrix},\ \text{X}=\begin{bmatrix}x\\y\end{bmatrix}\text{and B}=\begin{bmatrix}-2\\3\end{bmatrix}$

$\therefore\ \text{|A|}=\begin{vmatrix}2&-1\\3&4\end{vmatrix}=8-(-3)=8+3=11\neq0$

Therefore, solution is Unique and X = $\text{A}^{-1}\text{B}=\frac{1}{\text{|A|}}(\text{adj. A)B}$

$\Rightarrow\ \begin{bmatrix}x\\y\end{bmatrix}=\frac{1}{11}\begin{bmatrix}4&1\\-3&2\end{bmatrix}\begin{bmatrix}-2\\3\end{bmatrix}=\frac{1}{11}\begin{bmatrix}-8+3\\6+6\end{bmatrix}=\frac{1}{11}\begin{bmatrix}-5\\12\end{bmatrix}=\begin{bmatrix}\frac{-5}{11}\\\frac{12}{11}\end{bmatrix}$

Therefore, $x=\frac{-5}{11}\text{and}\ y=\frac{12}{11}$

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