Question

Find the values of ${}^{\prime }{a}^{\prime }$ and ${}^{\prime }{b}^{\prime }$ for which the given pair of linear equations $2x+3y=7;2ax+(a+b)=28$ have infinite number of solutions.

Answer 1

$2x+3y=7$

$2ax+\left(a+b\right)y=28$

For infinitely many solutions

$\frac{2}{2a}=\frac{3}{a+b}=\frac{7}{28}$

$\Rightarrow \frac{1}{a}=\frac{3}{a+b}=\frac{1}{4}$

$\Rightarrow \frac{1}{a}=\frac{1}{4}$ and $\frac{3}{a+b}=\frac{1}{4}$

$\Rightarrow a=4$ and $a+b=12$

Now $a+b=12$

$\Rightarrow 4+b=12$

$\Rightarrow b=8$

Therefore for infinite number of solutions $a=4$ and $b=8$