Question

Solve each of the following pairs of equation by reducing then to a pair of linear equations.
$\frac{5}{x+y}-\frac{2}{x-y}=-1\frac{15}{x+y}+\frac{7}{x-y}=10$ $Wherex\ne 0,y\ne 0$

Answer 1

$\frac{5}{x+y}-\frac{2}{x-y}=-1\frac{15}{x+y}+\frac{7}{x-y}=10$

$\frac{1}{x+y}=a$ $and\frac{1}{x-y}=b,then5a-2b=1.....\left(1\right),15a+7b=10....\left(2\right)Equation\left(1\right)\times 3,gives$ $15a-6b=-\mathrm{3.......}\left(3\right)$ $Equation\left(3\right)-\left(2\right),givens$ $-13b=13\Rightarrow b=\frac{-13}{-13}=1$ $Substituteb=1inequation\left(1\right),given$ $5a-2\left(1\right)-1\Rightarrow 5a-2=15a=-1+2\Rightarrow a=\frac{1}{5}$ $Buta=\frac{1}{x+y}\Rightarrow \frac{1}{5}=\frac{1}{x+y}\Rightarrow x+y=\mathrm{5......}\left(4\right)$ $b=\frac{1}{x-y}\Rightarrow 1=\frac{1}{x-y}\Rightarrow$ $\Rightarrow x-y=\mathrm{1....}\left(5\right)$ $Equation\left(4\right)+\left(5\right),gives$ $2x=6\Rightarrow x=\frac{6}{2}=3$ $Substitutex=3inequation\left(4\right),gives$ $3+y=5\Rightarrow y=5-3=2$ $Solution(x,y)=(3,2)$