Question

Solve the following simultaneous equations using Cramer's rule.
$3x-2y=3$;
$2x+y=16$.

Answer 1

Writing the system of equations in matrix form $AX=B$

where $A$ is the $2\times 2$ coefficient matrix, $X=\left(\begin{array}{c}x\\ y\end{array}\right)$ and $B$ is the $2\times 1$ matrix of constants.

Here, $A=\left(\begin{array}{cc}3& -2\\ 2& 1\end{array}\right)$ and $B=\left(\begin{array}{c}3\\ 16\end{array}\right)$

We can write, $X=A^{-1}B$, where

$A^{-1}=\frac{1}{|A|}adj\left(A\right)$

$A^{-1}=\frac{1}{7}\times \left(\begin{array}{cc}1& 2\\ -2& 3\end{array}\right)$

Substituting the values we get,

$\left(\begin{array}{c}x\\ y\end{array}\right)=$ $\frac{1}{7}\times \left(\begin{array}{cc}1& 2\\ -2& 3\end{array}\right)$ $\times \left(\begin{array}{c}3\\ 16\end{array}\right)$

$\left(\begin{array}{c}x\\ y\end{array}\right)=$ $\frac{1}{7}\times \left(\begin{array}{c}3\times 1+2\times 16\\ (-2)\times 3+3\times 16\end{array}\right)$

$\left(\begin{array}{c}x\\ y\end{array}\right)=$ $\left(\begin{array}{c}5\\ 6\end{array}\right)$.

Hence, $x=5$ and $y=6$.