Question

Find $x,y,z,w$ if $\left[\begin{array}{cc}x+y& x-y\\ y+z+w& 2w-z\end{array}\right]=\left[\begin{array}{cc}2& -1\\ 9& 5\end{array}\right]$

Answer 1

By using the property of equality of matrices, we have

$x+y=2$

$x-y=-1$

By the method of elimination we have $2x=2-1=1$ or $x=\frac{1}{2}$

and since $x+y=2\Rightarrow \frac{1}{2}+y=2\Rightarrow y=2-\frac{1}{2}=\frac{4-1}{2}=\frac{3}{2}$

Again $y+z+w=9\Rightarrow \frac{3}{2}+z+w=9$ or $z+w=9-\frac{3}{2}=\frac{18-3}{2}=\frac{15}{2}$

Solving $z+w=\frac{15}{2}$ and $2w-z=5$ we get

$3w=\frac{15}{2}+5=\frac{15+10}{2}=\frac{25}{2}$

$\Rightarrow w=\frac{25}{6}$

Substituting $w=\frac{25}{6}$ in $z+w=\frac{15}{2}$ we get

$z=\frac{15}{2}-\frac{25}{6}=\frac{45-25}{6}=\frac{20}{6}=\frac{10}{3}$

$\therefore x=\frac{1}{2},y=\frac{3}{2},w=\frac{25}{6},z=\frac{10}{3}$