Question

$$\begin{gathered} x+y=5xy, \\ 3x+2y=13xy. \end{gathered}$$

Answer 1

The given equations are:
x + y = 5xy .......(i)
3x + 2y = 13xy .........(ii)

From equation (i), we have:

$\frac{x+y}{xy}=5$
$\Rightarrow \frac{1}{y}+\frac{1}{x}=5$ .............(iii)

From equation (ii), we have:

$\frac{3x+2y}{xy}=13$
$\Rightarrow \frac{3}{y}+\frac{2}{x}=13$ .............(iv)
On substituting $\frac{1}{y}=v\mathrm{and}\frac{1}{\mathrm{x}}=\mathrm{u}$ , we get:
v + u = 5 ...........(v)
3v + 2u = 13 ...........(vi)

On multiplying (v) by 2, we get:
2v + 2u = 10 ..............(vii)
On subtracting (vii) from (vi), we get:
v = 3
$\Rightarrow \frac{1}{y}=3\Rightarrow y=\frac{1}{3}$
On substituting $y=\frac{1}{3}$ in (iii), we get:
$\frac{1}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}+\frac{1}{x}=5$
$\Rightarrow 3+\frac{1}{x}=5\Rightarrow \frac{1}{x}=2\Rightarrow x=\frac{1}{2}$
Hence, the required solution is $x=\frac{1}{2}$ and $y=\frac{1}{3}$ or x = 0 and y = 0.