Question

Solve the following pairs of equations:

`(6)/(x + y) = (7)/(x - y) + 3`

`(1)/(2(x + y)) = (1)/(3( x - y)`
Where x + y ≠ 0 and x - y ≠ 0

Answer 1

The given equations are `(6)/(x + y) = (7)/(x - y) + 3` and `(1)/(2(x + y)) = (1)/(3( x - y)`

Let `(1)/(x + y) = "a" and (1)/(x - y) = "b"`
Then, we have
6a = 7b + 3
⇒ 6a - 7b + 3   ....(i)
And, `(1)/(2)"a" = (1)/(3)"b"`
⇒ 3a = 2b
⇒ 6a = 4b  ....(ii)
Substituting the value of 6a in eqn. (i), we get
4b - 7b = 3
⇒ -3b = 3
⇒ b = -1
6a = -4
⇒ a = `-(2)/(3)`

⇒ x + y = `-(3)/(2)` and x - y = -1
Adding both these eqns., we get
2x = `-(5)/(2)`

⇒ x = `-(5)/(4)`

⇒ `-(5)/(4) - y` = -1

⇒ y = `-(5)/(4) + 1`

= `-(1)/(4)`
Thus, the solution set is `(-5/4, -1/4)`.