Question

Solve for $xandy$

$x+y=2xy;x-y=xy$

Answer 1

Step 1:Simplification

Given,

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

Dividing equation $\left(1\right)÷xyand\left(2\right)÷xy$

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

Step 2:Giving Substitution

$$\begin{gathered} Let\frac{1}{x}=A\&\frac{1}{y}=B \\ 3\Rightarrow B+A=2..............................(5) \\ 4\Rightarrow B\hspace{0.17em}-A=1................................\left(6\right) \end{gathered}$$

Step 3:Find A and B using elimination method

$$\begin{gathered} \left(3\right)+\left(4\right)\Rightarrow 2B=3 \\ \Rightarrow B=\frac{3}{2} \\ \left(1\right)\Rightarrow A=2-3=2-\frac{3}{2}=\frac{4-3}{2}=\frac{1}{2} \end{gathered}$$

Step 4 : To find x and y

$$\begin{gathered} A=\frac{1}{x}\Rightarrow \frac{1}{2}=\frac{1}{x}\Rightarrow x=2 \\ B=\frac{1}{y}\Rightarrow \frac{1}{y}=\frac{3}{2}\Rightarrow y=\frac{2}{3} \end{gathered}$$

Therefore the values of $x=2andy=\frac{2}{3}$

* Another method

Step 1

$$\begin{gathered} 1+2\Rightarrow 2x=3xy \\ \Rightarrow y=\frac{2x}{3x}=\frac{2}{3} \end{gathered}$$

Step 2 :To find x

$$\begin{gathered} 1\Rightarrow x+\frac{2}{3}=2x\times \frac{2}{3} \\ \Rightarrow x+\frac{2}{3}=\frac{4}{3}x \\ \Rightarrow 3x+2=4x \\ \Rightarrow 4x-3x=2 \\ \Rightarrow x=2 \\ Thereforex=2andy=\frac{2}{3} \end{gathered}$$