Question

Solve for $xandy$

$\frac{x-y}{xy}=9;\frac{x+y}{xy}=5$

Answer 1

Step 1: Simplification

Given,

$$\begin{gathered} \frac{x-y}{xy}=9..............................\left(1\right) \\ \frac{x+y}{xy}=5.....................................\left(2\right) \end{gathered}$$

$$\begin{gathered} \left(1\right)\Rightarrow \frac{x}{xy}-\frac{y}{xy}=9 \\ \Rightarrow \frac{1}{y}-\frac{1}{x}=9...............................\left(3\right) \\ \left(2\right)\Rightarrow \frac{x}{xy}+\frac{y}{xy}=5 \\ \Rightarrow \frac{1}{y}+\frac{1}{x}=5............................\left(4\right) \end{gathered}$$

Step 2 : Giving substitution

$$\begin{gathered} Let\frac{1}{x}=A\&\frac{1}{y}=B \\ 3\Rightarrow B-A=9.....................\left(5\right) \\ 4\Rightarrow B+A=5......................\left(6\right) \end{gathered}$$

Step 3 : To find A and B Using elimination method

$$\begin{gathered} \left(5\right)+\left(6\right)\Rightarrow 2B=14\Rightarrow B=7 \\ Therefore, \\ \left(6\right)\Rightarrow A=5-B=-2 \end{gathered}$$

Step 4 : To find x and y

We have,

$$\begin{gathered} A=\frac{1}{x}\Rightarrow -2=\frac{1}{x}\Rightarrow x=\frac{-1}{2} \\ B=\frac{1}{y}\Rightarrow 7=\frac{1}{y}\Rightarrow y=\frac{1}{7} \end{gathered}$$

Therefore the values of $x=\frac{-1}{2}\&y=\frac{1}{7}$