Solve the system of linear equation, using matrix method 5x + 2y … - Vidyadip

Question

Solve the system of linear equation, using matrix method $5x + 2y = 3; 3x + 2y = 5$

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Answer

Matrix form of given equations is $ AX = B$
$\Rightarrow \left[ {\begin{array}{*{20}{c}} 5\ \ 2 \\ 3\ \ 2 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 3 \\ 5 \end{array}} \right]$
Here $A = \left[\begin{array}{ll} 5\ \ 2 \\ 3\ \ 2 \end{array}\right], X = \left[\begin{array}{l} x \\ y \end{array}\right]$ and $B = \left[\begin{array}{l} 3 \\ 5 \end{array}\right]$
$\therefore |A| = \left|\begin{array}{ll} 5\ \ 2 \\ 3\ \ 2 \end{array}\right| = 10 - 6 = 4 \ne 0$
Therefore, solution is unique and $X = A^{-1}B = \frac{1}{|A|} (\text{adj}\ A B)$
$ = \left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] = \frac{1}{4}\left[ {\begin{array}{*{20}{c}} 2&-2 \\ { - 3}5 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 3 \\ 5 \end{array}} \right]$
$= \frac{1}{4}\left[ {\begin{array}{*{20}{c}} {6 - 10} \\ { - 9 + 25} \end{array}} \right] = \frac{1}{4}\left[ {\begin{array}{*{20}{c}} { - 4} \\ {16} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 1} \\ 4 \end{array}} \right]$
Therefore, $x = -1$ and $y = 4$

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