Question

If the pair of linear equations $2x-3y=10$ and $(m+n)x-(2m-n)y=6m$ has infinitely many solutions, then $m$ and $n$ satisfy the equation

• A. $5m+n=0$
• B. $5m-n=0$
• C. $m+5n=0$
• D. $m-5n=0$

Answer 1

The correct option is D $m-5n=0$

Given:-

$2x-3y=10$

$(m+n)x-(2m-n)y=6m$

For the given pair of linear equations have infinite many solution-

$\frac{2}{m+n}=\frac{-3}{-(2m-n)}=\frac{-10}{-6m}$

Case I:-

$\frac{2}{m+n}=\frac{-10}{-6m}$

$\Rightarrow$ $12m=10m+10n$

$\Rightarrow$ $2m-10n=0$

$\Rightarrow$ $m-5n=0$

Case II:-

$\frac{-3}{-(2m-n)}=\frac{-10}{-6m}$

$\Rightarrow$ $18m=20m-10n$

$\Rightarrow$ $2m-10n=0$

$\Rightarrow$ $m-5n=0$