Question

Solve the following simultaneous equations.

`10/("x" + "y") + 2/("x" - "y") = 4; 15/("x" + "y") - 5/("x" - "y") = -2`

Answer 1

`10/(x+y) + 2/(x-y) = 4`

`15/(x+y) - 5/(x-y) = -2`

Substitution

Let `u = 1/(x+y) and v = 1/(x-y)`

10u + 2v = 4

10u + 2v = 4

Solve for u and v

Solving the system of linear equations:

From (1), 2v = 4 – 10u ⇒ v = 2 – 5u

Substitute into (2): 15u – 5(2 – 5u) = –2

15u – 10 + 25u = –2

`40u = 8 => u = 1/5`

Substitute u back:

`v = 2 - 5(1/5) = 2 - 1 => v = 1`

Solve for x and y

`1/(x+y) = 1/5 => x + y = 5`

`1/(x-y) = 1 => x - y = 1`

Adding the two resulting equations:

(x + y) + (x – y) = 5 + 1

2x = 6 ⇒ x = 3

Subtracting the two resulting equations:

(x + y) – (x – y) = 5 – 1

2y = 4 ⇒ y = 2

x = 3 and y = 2