x+y / xy = 2 ; x-y / xy = 6

Question

Solve each of the following pairs of equations by reducing them to a pair of linear equations.

x+y / xy = 2

x-y / xy = 6

Answer 1

Given

\( \frac{x+y}{xy}\) = 2 

⇒ \( \frac{x}{xy}\)+\( \frac{y}{xy}\) = 2 

⇒\(\frac{1}{y}\) + \(\frac{1}{x}\) = 2 

\(\frac{x-y}{xy}\)= 6 

⇒ \(\frac{x}{xy}\)– \(\frac{y}{xy}\) = 6

⇒ \(\frac{1}{y}\) – \(\frac{1}{x}\)= 6 

Take \(\frac{1}{x}\)= a and \(\frac{1}{y}\) = b,

then the given equations reduces to

Substituting b = 4 in equation (1) we get 

a + 4 = 2 ⇒ a = 2 – 4 = -2 

but a =\(\frac{1}{x}\) = -2 ⇒ x =\(\frac{-1}{2}\)

b =\(\frac{1}{y}\)= 4 ⇒ y = \(\frac{1}{4}\)

∴ Solution (x, y) = \((\frac{-1}{2},\frac{1}{4})\)