Question

Solve the pair of equations for $x\ne 0,y\ne 0:2x+3y=2xy$ and $6x+12y=7xy$.

Answer 1

$2x+3y=2xy$

$\frac{2x}{xy}+\frac{3y}{xy}=2$

$\frac{2}{y}+\frac{3}{x}=2$

$6x+12y=7xy$

$\frac{6x}{xy}+\frac{12y}{xy}=7$

$\frac{6}{y}+\frac{12}{x}=7$

Let $\frac{1}{x}=u,\frac{1}{y}=v$

$3u+2v=2$ ...(1)

$12u+6v=7$...(2)

Multiply eq. (1) by 4 and subtract it form eq. (2)

$12u+6v=7-12u+8v=8$

$\Rightarrow -2v=-1$

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

Put $v=\frac{1}{2}$ in eq. (1)

$\Rightarrow 3u+1=2$

$\Rightarrow 3u=1$

$\therefore u=\frac{1}{3}$

$u=\frac{1}{3}\Rightarrow x=3$

$v=\frac{1}{2}\Rightarrow y=2$

$\therefore x=3,y=2$