Question

Determine the set of solutions of the system of linear equations below by using the matrix:
A. ax + by = p and cx + dy = q
B. 4x + 2y = 9 and 2x + 5y = 9

Answer 1

$\displaystyle \left[\begin{array}{ccc}a&b\\c&d\end{array}\right]\left[\begin{array}{ccc}x\\y\end{array}\right]=\left[\begin{array}{ccc}p\\q\end{array}\right]\\\left[\begin{array}{ccc}x\\y\end{array}\right]=\frac{1}{ad-bc}\left[\begin{array}{ccc}d&-b\\-c&a\end{array}\right]\left[\begin{array}{ccc}p\\q\end{array}\right]\\\left[\begin{array}{ccc}x\\y\end{array}\right]=\frac{1}{ad-bc}\left[\begin{array}{ccc}dp-bq\\-cp+aq\end{array}\right]\displaystyle \left[\begin{array}{ccc}x\\y\end{array}\right]=\left[\begin{array}{ccc}\frac{dp-bq}{ad-bc}\\\frac{aq-cp}{ad-bc}\end{array}\right]\\\\\\\boxed{\boxed{\text{HP}=\left\{\frac{dp-bq}{ad-bc};\frac{aq-cp}{ad-bc}\right\}}}$

$\displaystyle \left[\begin{array}{ccc}4&2\\2&5\end{array}\right]\left[\begin{array}{ccc}x\\y\end{array}\right]=\left[\begin{array}{ccc}9\\9\end{array}\right]\\\left[\begin{array}{ccc}x\\y\end{array}\right]=\frac{1}{4\cdot5-2\cdot2}\left[\begin{array}{ccc}5&-2\\-2&4\end{array}\right]\left[\begin{array}{ccc}9\\9\end{array}\right]\\\left[\begin{array}{ccc}x\\y\end{array}\right]=\frac{1}{20-4}\left[\begin{array}{ccc}45-18\\-18+36\end{array}\right]\displaystyle \left[\begin{array}{ccc}x\\y\end{array}\right]=\frac{1}{16}\left[\begin{array}{ccc}27\\18\end{array}\right]\\\left[\begin{array}{ccc}x\\y\end{array}\right]=\left[\begin{array}{ccc}\frac{27}{16}\\\frac{18}{16}\end{array}\right]\\\\\boxed{\boxed{\text{HP}=\left\{\frac{27}{16};\frac{9}{8}\right\}}}$