Question

For what values of a and b does the following pair of linear equations have infinite number of solution ?
$2x+3y=7,a(x+y)-b(x-y)=3a+b-2$

Answer 1

$2x+3y=7$

$ax-bx+ay+by=3a+b-2$

(or) $2x+3y=7$ …………$\left(1\right)$

$(a-b)x+(a+b)y=3a+b-2$ ………….$\left(2\right)$

Given: the pair of linear equations have infinite number of solutions

$\frac{2}{a-b}=\frac{3}{a-b}=\frac{7}{3a+b-2}$

$\Rightarrow \frac{2}{a-b}=\frac{3}{a+b}$

$\frac{3}{a+b}=\frac{7}{3a+b-2}$

$\Rightarrow 2a+2b=3a-3b$

$\Rightarrow -a=-5b$

$\therefore a=5b$

$3(3a+b-2)=7(a+b)$

$9a+3b-6=7a+7b$

$2a-4b-6=0$

$a-2b=3$

Put $a=5b$ in $a-2b=3$ we get

$5b-2b=3$

$\Rightarrow 3b=3$

or $b=1$

$\therefore a=5\times 1=5$, $b=1$

Hence $a=5,b=1$.