Question

Find the values of $m$ and $n$ for which the following system of linear equations has infinite many solutions.

$3x+4y=12$ and $(m+n)x+2(m-n)y=5m-1$.

Answer 1

It two linear equations has infinitely many solutions means

$a_1/a_2=b_1/b_2=c_1/c_2$

here two linear equations are

$3x+4y=12$ and

$(m+n)x+2(m-n)y=5\left(m\right)-1$

$i.e,\frac{3}{m+n}=\frac{4}{2(m-n)}=\frac{12}{5m-1}$

$6(m-n)=4(m+n).\frac{3}{m+n}=\frac{12}{5m-1}$

$6m-6n=4m+4n$ $3(5m-1)=12m+12n$

$2m-10n=0\cdots \left(1\right)$ $3m-12n=3\cdots \left(2\right)$

similarly $\left(1\right)\times 3$ & $\left(2\right)\times 2$

$6m-30n=0$

$6m-24n=6$

____________

$6n=6$ $\Rightarrow n=1$

now from $\left(1\right)2m-10=0$

$m=5$