Question

Solve the equations
$\frac{5}{x-1}+\frac{1}{y-2}=2$ and $\frac{6}{x-1}-\frac{3}{y-2}=1$

Answer 1

Let $\frac{1}{x-1}=X$ and $\frac{1}{y-2}=Y$

Hence, the given equations are reduced to:

$5X+Y=2$ ....$\left(1\right)$

$6X-3Y=1$ ....$\left(2\right)$

Multiplying $\left(1\right)$ by $3$, we get

$15X+3Y=6$ ....$\left(3\right)$

Adding $\left(3\right)$ and $\left(2\right)$, we get

$21X=7$

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

Substitute this value of $X$ in $\left(2\right)$, we get

$6\left(\frac{1}{3}\right)-3Y=1$

$\Rightarrow 2-3Y=1$

$\Rightarrow 3Y=1$

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

Resubstitute the values of $X$ and $Y$ to get $x$ and $y$,

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

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

$\Rightarrow x-1=3$

$\Rightarrow x=4$

$Y=\frac{1}{y-2}$

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

$\Rightarrow y-2=3$

$\Rightarrow y=5$

Hence, $x=4,y=5$ is the solution of the given simultaneous equations.