For what values of $m$ and $n$ following equation has infinitely many solution
$3 x + 4 y = 12$
$(m + n) x + 2 (m - n) y = 5 m - 1$

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Solution

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

If given equation has infinityely many solution
then,
$\frac{a}{a_{1}} = \frac{b}{b_{1}} = \frac{c}{c_{1}}$
$\Rightarrow \frac{3}{m + n} = \frac{4}{2 (m - n)} = \frac{12}{5 m - 1}$
$\frac{3}{m + n} = \frac{12}{5 m - 1}$ & $\frac{3}{m + n} = \frac{2}{m - n}$
$15 m - 3 = 12 m + 12 n 3 m - 3 n = 2 m + 2 n$
$3 m - 12 n = 3 [m = 5 n]$
$m - 4 n = 1$ ....(3)
then $5 n - 4 n = 1$
$[n = 1]$ & $[m = 5]$
$∴ n = 1$ & $m = 5$ given equation has infinitely many solution.

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For what values of m and n following equation has infinitely many solution3x+4y=12(m+n)x+2(m−n)y=5m−1

For what values of $m$ and $n$ following equation has infinitely many solution
$3 x + 4 y = 12$
$(m + n) x + 2 (m - n) y = 5 m - 1$