Solve the system of equations &2 x+5 y=1 &3 x+2 y=7 - Vidyadip

Question

Solve the system of equations $\begin{aligned} &2 x+5 y=1\\ &3 x+2 y=7 \end{aligned}$

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Answer

The system of equations can be written in the form $AX = B,$ where
$A = \left[\begin{array}{ll} {2} & {5} \\ {3} & {2} \end{array}\right], X=\left[\begin{array}{l} {x} \\ {y} \end{array}\right]$ and $B = \left[\begin{array}{l} {1} \\ {7} \end{array}\right]$
Now, $A = –11 \neq 0,$
Hence $,A$ is nonsingular matrix and so has a unique solution.
Note that $A^{-1}=-\frac{1}{11}\left[\begin{array}{cc} {2} & {-5} \\ {-3} & {2} \end{array}\right]$
Therefore $X = A^{–1}B = –\frac{1}{11}\left[\begin{array}{cc} {2} & {-5} \\ {-3} & {2} \end{array}\right]\left[\begin{array}{l} {1} \\ {7} \end{array}\right]$
i.e., $\left[\begin{array}{l} {x} \\ {y} \end{array}\right]=-\frac{1}{11}\left[\begin{array}{c} {-33} \\ {11} \end{array}\right]=\left[\begin{array}{c} {3} \\ {-1} \end{array}\right]$
Hence, $x = 3, y = – 1$

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