Question

Solve for x and y:
$$\begin{gathered} \frac{1}{2x}+\frac{1}{3y}=2 \\ \frac{1}{3x}+\frac{1}{2y}=\frac{13}{6} \end{gathered}$$

Answer 1

Multiplying equation (i) and (ii) by 6, we get
$$\begin{gathered} \frac{3}{x}+\frac{2}{y}=12.....\left(i\right) \\ \frac{2}{x}+\frac{3}{y}=13.....\left(ii\right) \end{gathered}$$
Multiplying (i) by 3 and (ii) by 2 and subtracting (ii) from (i), we get
$$\begin{gathered} \frac{9}{x}-\frac{4}{x}=36-26 \\ \Rightarrow \frac{5}{x}=10 \\ \Rightarrow x=\frac{5}{10}=\frac{1}{2} \end{gathered}$$
Now, substituting $x=\frac{1}{2}$ in (i), we have
$$\begin{gathered} 6+\frac{2}{y}=12 \\ \Rightarrow \frac{2}{y}=6 \\ \Rightarrow y=\frac{1}{3} \end{gathered}$$
Hence, $x=\frac{1}{2}$ and $y=\frac{1}{3}$.