Question

Solve the following equations using Cramer's Rule
$2x-y+3z=9,x+y+z=6,x-y+z=2$

Answer 1

$\Delta =\left|\begin{array}{ccc}2& -1& 3\\ 1& 1& 1\\ 1& -1& 1\end{array}\right|=2(1+1)-(-1)(1-1)+3(-2)=4-6=-2$

Now, ${\Delta }_x=\left|\begin{array}{ccc}9& -1& 3\\ 6& 1& 1\\ 2& -1& 1\end{array}\right|=14{\Delta }_y=\left|\begin{array}{ccc}2& 9& 3\\ 1& 6& 1\\ 1& 2& 1\end{array}\right|=-2{\Delta }_z=\left|\begin{array}{ccc}2& -1& 9\\ 1& 1& 6\\ 1& -1& 2\end{array}\right|=-6$

So,According to Cramer's Rule $x=\frac{{\Delta }_x}{\Delta }=\frac{14}{-2}=-7y=\frac{{\Delta }_y}{\Delta }=\frac{-2}{-2}=1z=\frac{{\Delta }_z}{\Delta }=\frac{-6}{-2}=3$