Question

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

Answer 1

The given pair of equations are $6x+3y=6xy,2x+4y=5xy$

Divide first equation by $3$ we get

$2x+y=2xy...\left(1\right)$ and $2x+4y=5xy...\left(2\right)$

Divide $\left(1\right),\left(2\right)$ by $xy$ we get

$\frac{2}{y}+\frac{1}{x}=2...\left(3\right)$ and $\frac{2}{y}+\frac{4}{x}=5...\left(4\right)$

Let $\frac{1}{x}=p,\frac{1}{y}=q$ we get

$p+2q=2...\left(5\right)$ and $4p+2q=5...\left(6\right)$

$\left(5\right)\Rightarrow p=2-2q$

Substitute $p=2-2q$ in $\left(6\right)$ we get

$4(2-2q)+2q=5$

$\Rightarrow 8-8q+2q=5$

$\Rightarrow 8-6q=5$

$\Rightarrow 6q=3$

$\Rightarrow q=\frac{1}{2}$

Hence $p=2-2q=2-2\times \frac{1}{2}=2-1=1$

Therefore $\frac{1}{x}=p=1\Rightarrow x=1$ and $\frac{1}{y}=q=\frac{1}{2}\Rightarrow y=2$

Solution: $x=1,y=2$