Question

Solve the following systems of equations:
$\frac{1}{2(x+2y)}+\frac{5}{3(3x-2y)}=\frac{-3}{2}$
$\frac{5}{4(x+2y)}-\frac{3}{5(3x-2y)}=\frac{61}{60}$

Answer 1

Let $\frac{1}{x+2y}=m,\frac{1}{3x-2y}=n$

Equation (1) : $\frac{m}{2}+\frac{5n}{3}=\frac{-3}{2}$

Equation (2) : $\frac{5m}{4}-\frac{3n}{5}=\frac{61}{60}$

Equation (1) × $\frac{3}{5}$ : $\frac{3m}{10}+n=\frac{-9}{10}$

Equation (2) × $\frac{5}{3}$ : $\frac{25m}{12}-n=\frac{61}{36}$

Add two equations ;

$\Rightarrow m=\frac{1}{3}$

$\Rightarrow n=-1$

$\Rightarrow x+2y=3$

$\Rightarrow 3x-2y=-1$

Add two equations ;

$\Rightarrow 4x=2$

$\Rightarrow x=\frac{1}{2}$

$\Rightarrow y=\frac{5}{4}$