Question

Solve for x and y:
$$\begin{gathered} \frac{5}{x+y}-\frac{2}{x-y}=-1 \\ \frac{15}{x+y}+\frac{7}{x-y}=10 \end{gathered}$$

Answer 1

The given equations are
$$\begin{gathered} \frac{5}{x+y}-\frac{2}{x-y}=-1.....\left(i\right) \\ \frac{15}{x+y}+\frac{7}{x-y}=10.....\left(ii\right) \end{gathered}$$
Substituting $\frac{1}{x+y}=u\mathrm{and}\frac{1}{x-y}=v$ in (i) and (ii), we get
$$\begin{gathered} 5u-2v=-1.....\left(iii\right) \\ 15u+7v=10.....\left(iv\right) \end{gathered}$$
Multiplying (iii) by 3 and subtracting it from (iv), we get
$$\begin{gathered} 7v+6v=10+3 \\ \Rightarrow 13v=13 \\ \Rightarrow v=1 \\ \Rightarrow x-y=1\left(\because \frac{1}{x-y}=v\right).....\left(v\right) \end{gathered}$$
Now, substituting v = 1 in (iii), we get
$$\begin{gathered} 5u-2=-1 \\ \Rightarrow 5u=1 \\ \Rightarrow u=\frac{1}{5} \\ x+y=5.....\left(vi\right) \end{gathered}$$
Adding (v) and (vi), we get
$2x=6\Rightarrow x=3$
Substituting x = 3 in (vi), we have
$3+y=5\Rightarrow y=5-3=2$
Hence, x = 3 and y = 2.