Question

Find the solution for the following equations:
$\frac{5}{2x}+\frac{2}{3y}=7;\frac{3}{x}+\frac{2}{y}=12(x\ne 0,y\ne 0)$

Answer 1

$\frac{5}{2x}+\frac{2}{3y}=7;\frac{3}{x}+\frac{2}{y}=12(x\ne 0,y\ne 0)$
Representing the above equations in the standard form and by substituting
$\frac{1}{x}=a;\frac{1}{y}=b$ we get
$\frac{5a}{2}+\frac{2b}{3}=7$
$\Rightarrow 15a+4b=\mathrm{42...}\left(1\right)$
$3a+2b=\mathrm{12....}\left(2\right)$
Multiplying equation $\left(2\right)$ by $2$ and subtracting from eq $\left(1\right)$
$15a+4b=42$
$6a+4b=24$

$-$ $-$ $-$
-------------------------
$9a=18$
$\therefore a=2$
Substituting value of $a$ in eq. $\left(2\right)$, we have
$3\left(2\right)+2b=12$
$b=3$
Then,
$\frac{1}{x}=a=2;\frac{1}{y}=b=3$
$\Rightarrow$ $x=\frac{1}{2};y=\frac{1}{3}$
$(x,y)=(\frac{1}{2},\frac{1}{3})$