Question

Solve the following systems of equations:
$\frac{5}{x-1}+\frac{1}{y-2}=2$

$\frac{6}{x-1}-\frac{3}{y-2}=1$

Answer 1

The given equations are:

$\frac{5}{x-1}+\frac{1}{y-2}=2$

$\frac{6}{x-1}-\frac{3}{y-2}=1$

Let $\frac{1}{x-1}=u$ and $\frac{1}{y-2}=v$ then equations are

$5u+v=2\dots \dots \left(i\right)$

$6u-3v=1\dots \dots \left(ii\right)$

Multiply equation (i) by 3 and add both equations, we get

$15u+3v+6u-3v=6+1$

$\Rightarrow 21u=7$

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

Put the value of $u$ in equation (i), we get

$5\times \frac{1}{3}+v=2$

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

Then

$\frac{1}{x-1}=\frac{1}{3}$

$\Rightarrow x-1=3$

$\Rightarrow x=4$

And, $\frac{1}{y-2}=\frac{1}{3}$

$\Rightarrow y-2=3$

$\Rightarrow y=5$

Hence the value of $x=4$ and $y=5$.