Question

The values of $m$ for which the following system of linear equations has infinitely many solutions.
$3x+4y=12$
$(m+n)x+2(m-n)y=(5m-1)$ is ----- and ------.

Answer 1

For infinite solution

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

$\Rightarrow \frac{3}{m+n}=\frac{4}{2(m-n)}=\frac{12}{(5m-1)}$

from 1 & 2

$\Rightarrow 3m-3n=2m+2n$

$\Rightarrow m=5n$

from 2 & 3

$\frac{2}{m-n}=\frac{12}{(5m-1)}$

$\Rightarrow 5m-1=6m-6n$

$\Rightarrow 6n-1=m\Rightarrow 6n-1=5n$

$\Rightarrow n=1\&m=5$