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

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–1B = –\(\frac{1}{11}\left[\begin{array}{cc} {2} & {-5} \\ {-3} & {2} \end{array}\right]\left[\begin{array}{l} {1} \\ {7} \end{array}\right]\)
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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