Question

Solve the following pairs of equations by reducing them to a pair of linear equations:
6x + 3y = 6xy
2x + 4y = 5xy

Answer 1

$6x+3y=6xy$
$\Rightarrow \frac{6x}{xy}+\frac{3y}{xy}=6$
$\Rightarrow \frac{6}{y}+\frac{3}{x}=\mathrm{6...}\left(i\right)$
$2x+4y=5xy$
$\Rightarrow \frac{2x}{xy}+\frac{4y}{xy}=5$
$\Rightarrow \frac{2}{y}+\frac{4}{x}=\mathrm{5...}\left(ii\right)$
$Putting\frac{1}{x}=pand\frac{1}{y}=q\text{in (i) and (ii) we get,}$
3p+ 6q - 6 = 0
4p +2q - 5 = 0
$\text{By cross multiplication method, we get}$
$\frac{p}{-30-(-12)}=\frac{q}{-24-(-15)}=\frac{1}{6-24}$
$\frac{p}{-18}=\frac{q}{-9}=\frac{1}{-18}$
$\frac{p}{-18}=\frac{1}{-18}and\frac{q}{-9}=\frac{1}{-18}$
$p=1andq=\frac{1}{2}$
$p=\frac{1}{x}=1andq=\frac{1}{y}=\frac{1}{2}$
$x=1,y=2$
$Hence,x=1andy=2$