Question

Solve the following pairs of equations:

${\displaystyle \frac{xy}{x+y}=\frac{6}{5}}$

${\displaystyle \frac{xy}{y-x}}$ = 6
Where x + y ≠ 0 and y - x ≠ 0

Answer 1

${\displaystyle \frac{xy}{x+y}=\frac{6}{5}}$

⇒ ${\displaystyle \frac{x+y}{xy}=\frac{5}{6}}$

⇒ ${\displaystyle \frac{1}{y}+\frac{1}{x}=\frac{5}{6}}$     ....(i)

${\displaystyle \frac{xy}{y-x}}$ = 6

⇒ ${\displaystyle \frac{y-x}{xy}}$ = 6

⇒ ${\displaystyle \frac{1}{x}+\frac{1}{y}=\frac{1}{6}}$    ....(ii)
Adding eqns. (i) and (ii), we get
${\displaystyle \frac{2}{x}}$ = 1
⇒ x = 2
⇒ ${\displaystyle \frac{1}{y}+\frac{1}{2}=\frac{5}{6}}$

⇒ ${\displaystyle \frac{1}{y}=\frac{5}{6}-\frac{1}{2}}$

= ${\displaystyle \frac{5-3}{6}}$

= ${\displaystyle \frac{2}{6}}$

= ${\displaystyle \frac{1}{3}}$
⇒  y = 3
Thus, the solution set is (2, 3).