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

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Solution

For infinitely many solutions:

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

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

$\Rightarrow m = 5 n$

$\frac{3}{6 n} = \frac{12}{25 n - 1}$

$\Rightarrow 25 n - 1 = 24 n$

$\Rightarrow n = 1, m = 5$

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Find the values of m and n for which the following system of linear equations has infinitely many solutions :3x+4y=12(m+n)x+2(m−n)y=(5m−1)

For infinitely many solutions:⇒3m+n=42(m−n)=125m−13m−3n=2m+2n⇒m=5n36n=1225n−1⇒25n−1=24n⇒n=1,m=5

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

For infinitely many solutions:

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

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

$\Rightarrow m = 5 n$

$\frac{3}{6 n} = \frac{12}{25 n - 1}$

$\Rightarrow 25 n - 1 = 24 n$

$\Rightarrow n = 1, m = 5$