Question

Using determinants to solve the equations
$x+2y+3z=6,2x+4y+z=17,3x+2y+9z=2$.
Find $x+y+z$?

Answer 1

Given system of eqautions are $x+2y+3z=6,2x+4y+z=17,3x+2y+9z=2$

Using Crammer's rule,

$\Delta =\left|\begin{array}{ccc}1& 2& 3\\ 2& 4& 1\\ 3& 2& 9\end{array}\right|=34-30-24=-20$

${\Delta }_x=\left|\begin{array}{ccc}6& 2& 3\\ 17& 4& 1\\ 2& 2& 9\end{array}\right|=204-3302+78=-20$

${\Delta }_y=\left|\begin{array}{ccc}1& 6& 3\\ 2& 17& 1\\ 3& 2& 9\end{array}\right|=151-90-141=-80$

${\Delta }_z=\left|\begin{array}{ccc}1& 2& 6\\ 2& 4& 17\\ 3& 2& 2\end{array}\right|=-26+94-48=20$

$\therefore x=\frac{{\Delta }_x}{\Delta }=1,y=\frac{{\Delta }_y}{\Delta }=4$ and $z=\frac{{\Delta }_z}{\Delta }=-1$

$\therefore x+y+z=4$