Question

Solve for $xandy$

$\frac{2}{x}+\frac{3}{y}=2;\frac{1}{x}-\frac{1}{2y}=\frac{1}{3}$

Answer 1

Step 1:Solving the equation by assumption of values $\frac{\mathbf{1}}{\mathbf{x}}\mathbf{}\mathbf{=}\mathbf{}\mathit{A}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\frac{\mathbf{1}}{\mathbf{y}}\mathbf{}\mathbf{=}\mathit{B}$

Given equations,

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

$$\begin{gathered} \text{Let}\frac{1}{x}=A\text{and}\frac{1}{y}=B \\ \text{Equation}\left(1\right)\Rightarrow 2A+3B=2....................\left(3\right) \\ \text{Equation}\left(2\right)\Rightarrow A-\frac{B}{2}=\frac{1}{3} \\ \Rightarrow \frac{2A-B}{2}=\frac{1}{3} \\ \Rightarrow 2A-B=\frac{2}{3}..................\left(4\right) \end{gathered}$$

Step 2: To find A and B using elimination method

Equation $\left(3\right)-\left(4\right)$

$$\begin{gathered} \Rightarrow 3B-(-B)=2-\frac{2}{3} \\ \Rightarrow 4B=\frac{4}{3} \\ \Rightarrow B=\frac{1}{4}\times \frac{4}{3}=\frac{1}{3} \end{gathered}$$

Substituting $B=\frac{1}{3}\text{in equation}\left(3\right)$

$$\begin{gathered} \Rightarrow 2A+3\times \frac{1}{3}=2 \\ \Rightarrow 2A+1=2 \\ \Rightarrow 2A=2-1=1 \\ \Rightarrow A=\frac{1}{2} \end{gathered}$$

Step 3: To find $\mathit{x}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{y}$

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

Hence the solution is $x=2\text{and}y=3$