Question

If $x=3,y=-4$ is the solution of the pair of linear equations $ax+by=4$ and $bx+ay=3$, then find the values of $a+b$ and $ab$.

Answer 1

$x=3,y=-4$ are the solutions of pair of linear equations $ax+by=4$ and $bx+ay=3$,

$\Rightarrow 3a-4b=4$ and $3b-4a=3$

$\Rightarrow 3a=4+4b$

$\Rightarrow a=\frac{4+4b}{3}$

$\Rightarrow 3b-4a=3$

$\Rightarrow 3b-4\left(\frac{4+4b}{3}\right)=3$

$\Rightarrow 9b-16-16b=3$

$\Rightarrow -7b=3+16$

$\Rightarrow b=\frac{-19}{7}$

$\Rightarrow 3b-4a=3$

$\Rightarrow 3\times \frac{-19}{7}-4a=3$

$\Rightarrow -4a=3+\frac{57‬}{7}=\frac{21+57}{7}=\frac{68}{7}$

$\Rightarrow a=\frac{-17}{7}$

$\therefore a=\frac{-17}{7}$ and $b=\frac{-19}{7}$

$\therefore a+b=\frac{-17}{7}+\left(\frac{-19}{7}\right)=\frac{-36}{7}$

$\therefore ab=\frac{-17}{7}\times \frac{-19}{7}=\frac{323}{49}$