Question

Solve the following system of equations:

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

`15/(x + y) - 9/(x - y) = -2`

Answer 1

Given: `10/(x + y) + 2/(x - y) = 4, 15/(x + y) - 9/(x - y) = -2`

Step-wise calculation:

1. Let `a = 1/(x + y)` and `b = 1/(x - y)`. 

2. The system becomes the linear system:

10a + 2b = 4

15a – 9b = –2

3. Solve the linear system.

From the first equation divide by 2:

5a + b = 2

⇒ b = 2 – 5a

Substitute into the second:

15a – 9(2 – 5a) = –2 

15a – 18 + 45a = –2 

60a – 18 = –2 

60a = 16

`a = 16/60`

`a = 4/15`

Then b = 2 – 5a

= `2 - 5 xx (4/15)` 

= `2 - 20/15` 

= `2/3`

4. Return to x and y:

`1/(x + y) = a = 4/15`

⇒ `x + y = 15/4`

`1/(x - y) = b = 2/3`

⇒ `x - y = 3/2`

Solve: Add: `2x = 15/4 + 3/2`

= `15/4 + 6/4`

= `21/4`

⇒ `x = 21/8` 

Subtract or compute 2y: 

`2y = 15/4 - 3/2` 

= `15/4 - 6/4`

= `9/4` 

⇒ `y = 9/8`

The solution is `x = 21/8` and `y = 9/8`.