Question

If $\frac{2}{x}+\frac{3}{y}=\frac{9}{xy}\mathrm{and}\frac{4}{x}+\frac{9}{y}=\frac{21}{xy}$, find the values of x and y.

Answer 1

The given pair of equation is
$$\begin{gathered} \frac{2}{x}+\frac{3}{y}=\frac{9}{xy}.....\left(i\right) \\ \frac{4}{x}+\frac{9}{y}=\frac{21}{xy}.....\left(ii\right) \end{gathered}$$
Multiplying (i) and (ii) by xy, we have
$$\begin{gathered} 3x+2y=9.....\left(iii\right) \\ 9x+4y=21.....\left(iv\right) \end{gathered}$$
Now, multiplying (iii) by 2 and subtracting from (iv), we get
$9x-6x=21-18\Rightarrow x=\frac{3}{3}=1$
Putting x = 1 in (iii), we have
$3\times 1+2y=9\Rightarrow y=\frac{9-3}{2}=3$
Hence, x = 1 and y = 3.