Question

Solve for $xandy$

$\frac{x+y-1}{x-y+1}=7;\frac{y-x+1}{x-y+1}=35$

Answer 1

Step 1 : Simplification of equation by cross multiplication method

$$\begin{gathered} \frac{x+y-1}{x-y+1}=7 \\ \Rightarrow x+y-1=7(x-y+1) \\ \Rightarrow x+y-1=7x-7y+7 \\ \Rightarrow 7x-7y+7-x-y-+1=0 \\ \Rightarrow 6x-8y+8=0 \\ \Rightarrow 6x-8y=-8.................................\left(1\right) \\ Also\frac{y-x+1}{x-y+1}=35 \\ \Rightarrow y-x+1=35(x-y+1) \\ \Rightarrow y-x+1=35x-35y+35 \\ \Rightarrow 35x-35y+35-y+x-1=0 \\ \Rightarrow 36x-36y+34=0 \\ \Rightarrow 36x-36y=-34......................................\left(2\right) \end{gathered}$$

Step 2: Solving for x and y using elimination method

$$\begin{gathered} equation\left(1\right)\times 6\Rightarrow 36x-48y=-48.........................\left(3\right) \\ equation\left(2\right)\Rightarrow 36x-36y=-34...................................\left(2\right) \\ \left(3\right)-\left(2\right)\Rightarrow -12y=-14 \\ \Rightarrow y=\frac{-14}{-12}=\frac{7}{6} \\ Togetx \\ substitutey=\frac{7}{6}inequation\left(1\right) \\ 6x-8x\frac{7}{6}=-8 \\ \Rightarrow 6x=\frac{28}{3}-8=\frac{28-24}{3}=\frac{4}{3} \\ \Rightarrow x=\frac{1}{6}x\frac{4}{3}=\frac{2}{9} \end{gathered}$$

Therefore the values of $x=\frac{2}{9}andy=\frac{7}{6}$