Question

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

$\frac{15}{x+y}+\frac{7}{x-y}=10$, where $x+y\ne 0$ and $x-y\ne 0$.

Answer 1

Let $\frac{1}{x+y}=u$ and $\frac{1}{x-y}=v$. Then, the given system of equations becomes

$5u-2v=-1$ .(i)
$15u+7v=10$ .(ii)

Multiplying equation (i) by $3$, this system of equations becomes
$15u-6v=-3$ .(iii)
$15u+7v=10$ .(iv)

Subtracting equation (iv) from equation (iii), we get
$-13v=-13\Rightarrow v=1$

Putting $v=1$ in equation (i), we get
$5u-2=-1\Rightarrow u=\frac{1}{5}$

Now, $u=\frac{1}{5}\Rightarrow \frac{1}{x+y}=\frac{1}{5}\Rightarrow x+y=5$ ..(v)

and, $v=1\Rightarrow \frac{1}{x-y}=1\Rightarrow x-y=1$ ..(vi)

Adding equations (vi) and (v), we get $2x=6\Rightarrow x=3$.

Putting $x=3$ in equation (v), we get $y=2$.

Hence, $x=3,y=2$ is the solution of the given system of equations.