Question

Solve:

$\frac{27}{x-2}+\frac{31}{y+3}=85$ ; $\frac{31}{x-2}+\frac{27}{y+3}=89$

Answer 1

Given,

$\Rightarrow \frac{27}{(x-2)}+\frac{31}{(y+3)}=85$; $\frac{31}{(x-2)}+\frac{27}{(y+3)}=89$

Let $\frac{1}{x-2}=a$ and $\frac{1}{y+3}=b$

$\Rightarrow 27a+31b=85$ …….$\left(1\right)$;

$31a+27b=89$ ……….$\left(2\right)$

$\Rightarrow$ Adding $\left(1\right)$ and $\left(2\right)$

$27a+31b+31a+27b=89+85$

$\Rightarrow 58a+58b=174$

$\Rightarrow a+b=\frac{174}{58}=3$ ……….$\left(3\right)$

$\Rightarrow$ Subtracting $\left(1\right)$ and $\left(2\right)$

$27a+31b-31a-27b=85-89$

$\Rightarrow -4(a-b)=-4$

$\Rightarrow a-b=1$ …………$\left(4\right)$

Solving $\left(3\right)$ and $\left(4\right)$

$\Rightarrow a+b=3$

$a-b=1$

______________

$2a=4$

______________

$a=2$,

$b=3-a=3-2=1$

Now,

$\frac{1}{x-2}=a$ $\frac{1}{y+3}=b$

$\Rightarrow \frac{1}{x-2}=2$ $\Rightarrow \frac{1}{y+3}=1$

$\Rightarrow 1=2x-4$ $\Rightarrow 1=y+3$

$\Rightarrow x=\frac{5}{2}$ $\Rightarrow y=-2$.