Question

Find $x,y,a$ and $b$, if $\left[\begin{array}{ccc}3x+4y& 2& x-2y\\ a+b& 2a-b& -1\end{array}\right]=\left[\begin{array}{ccc}2& 2& 4\\ 5& -5& -1\end{array}\right]$.

Answer 1

Since, matrices are equal then, their corresponding elements are also equal.

$\Rightarrow$ $3x+4y=2$ ---- ( 1 )

$\Rightarrow$ $x-2y=4$ ----- ( 2 )

$\Rightarrow$ $a+b=5$ ----- ( 3 )

$\Rightarrow$ $2a-b=-5$ ---- ( 4 )

From equation ( 2 ),

$x-2y=4$

$x=4-2y$

Substituting value of $x$ in equation $\left(1\right)$,

$\Rightarrow$ $3(4-2y)+4y=2$

$\Rightarrow$ $12-6y+4y=2$

$\Rightarrow$ $-2y=-10$

$\therefore$ $y=5$

Substituting value of $y$ in equation $\left(2\right)$,

$\Rightarrow$ $x-2\left(5\right)=4$

$\Rightarrow$ $x=14$

From equation $\left(3\right),$

$a+b=5$

$\therefore$ $a=5-b$

Substituting value of $a$ in equation $\left(4\right),$

$\Rightarrow$ $2(5-b)-b=-5$

$\Rightarrow$ $10-2b-b=-5$

$\Rightarrow$ $10-3b=-5$

$\Rightarrow$ $-3b=-15$

$\therefore$ $b=5$

Substituting value of $b$ in equation $\left(3\right),$

$\Rightarrow$ $a+5=-5$

$\therefore$ $a=-10$

$\therefore$ $x=12,y=5,a=-10$ and $b=5.$