Solve system of linear equations, using matrix method.
2x-y=-2
3x+4y=3
2x-y=-2
3x+4y=3
Solution and Explanation
The given system of equations can be written in the form of AX = B, where
A=\(\begin{vmatrix} 2 &-1 \\ 3& 4 \end{vmatrix}\), X=\(\begin{vmatrix} x \\ y \end{vmatrix}\)and B=\(\begin{vmatrix} -2 \\ 3 \end{vmatrix}\)
\(Now |A|=8+3=11≠0\)
Thus, A is non-singular. Therefore, its inverse exists.
Now,
A-1=\(\frac{1}{|A|}\)adjA=\(\frac{1}{11}\)\(\begin{vmatrix} 4 &1 \\ -3& 2 \end{vmatrix}\)
so X=A-1B=\(\frac{1}{11}\)\(\begin{vmatrix} 4 &1 \\ -3& 2 \end{vmatrix}\)\(\begin{vmatrix} -2 \\ 3 \end{vmatrix}\)
\(\Rightarrow \begin{vmatrix} x \\ y \end{vmatrix}=\frac{1}{11}\begin{vmatrix} -8+3 \\ 6+6 \end{vmatrix}=\frac{1}{11}\begin{vmatrix} -5 \\ 12 \end{vmatrix}\)=\(\begin{vmatrix} -\frac{5}{11} \\ \frac{12}{11} \end{vmatrix}\)
Hence x=\(-\frac{5}{11}\) and y=\(\frac{12}{11}\)
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