Question

Solve:
$\frac{5}{x-y}+\frac{3}{x+y}=8$
$\frac{3}{x-y}-\frac{5}{x+y}=4$

Answer 1

$\begin{array}{c}\frac{5}{x-y}+\frac{3}{x+y}=8\\ \Rightarrow \frac{3}{x-y}+\frac{5}{x+y}=4\\ \Rightarrow let\frac{1}{x-y}=4,and\frac{1}{x+y}=V\\ \Rightarrow 5u+3v-8=0\to \left(i\right)\\ \Rightarrow 3u-5v-4=0\to \left(ii\right)\\ Bysolvingbothequations.\\ \Rightarrow \frac{u}{\left(-12-40\right)}=\frac{v}{\left(-24+20\right)}=\frac{1}{-25-9}\\ \Rightarrow \frac{u}{-52}=\frac{v}{-4}=\frac{1}{-34}\\ \Rightarrow U=\frac{26}{17},V=\frac{2}{17}\\ \therefore \frac{1}{x-y}=\frac{26}{17},\frac{1}{x+y}=\frac{2}{17}\\ \Rightarrow x-y=\frac{17}{26},\to \left(iii\right)x+y=\frac{2}{17}\to \left(iv\right)\\ addingequation\left(iii\right)and\left(iv\right)\\ \Rightarrow 2x=\frac{17}{26}+\frac{17}{2}=17\left(\frac{1+13}{26}\right)=\frac{17\times 7}{13}=\frac{119}{13}\\ andy=x-\frac{17}{26}=\frac{119}{13}-\frac{17}{26}=\frac{24}{26}Ans.\end{array}$