Question

Solve for x and y:
$$\begin{gathered} 3\left(2x+y\right)=7xy \\ 3\left(x+3y\right)=11xy \end{gathered}$$

Answer 1

The given equations can be written as
$$\begin{gathered} \frac{3}{x}+\frac{6}{y}=7.....\left(i\right) \\ \frac{9}{x}+\frac{3}{y}=11.....\left(ii\right) \end{gathered}$$
Multiplying (i) by 3 and subtracting (ii) from it, we get
$$\begin{gathered} \frac{18}{y}-\frac{3}{y}=21-11 \\ \Rightarrow \frac{15}{y}=10 \\ \Rightarrow y=\frac{15}{10}=\frac{3}{2} \end{gathered}$$
Substituting $y=\frac{3}{2}$ in (i), we have
$$\begin{gathered} \frac{3}{x}+\frac{6\times 2}{3}=7 \\ \Rightarrow \frac{3}{x}=7-4=3 \\ \Rightarrow x=1 \end{gathered}$$
Hence, $x=1\mathrm{and}y=\frac{3}{2}$.