Question

Solve for x and y:

`3/(x + y) + 2/(x - y) = 2, 9/(x + y) - 4/(x - y) = 1`

Answer 1

Given: `3/(x + y) + 2/(x - y) = 2, 9/(x + y) - 4/(x - y) = 1`

Step-wise calculation:

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

2. The system becomes:

3u + 2v = 2

9u – 4v = 1

3. Multiply the first equation by 2:

6u + 4v = 4

Add this to the second equation:

(6u + 4v) + (9u – 4v) = 4 + 1 

⇒ 15u = 5

⇒ `u = 1/3`

4. Substitute `u = 1/3` into 3u + 2v = 2: 

`3(1/3) + 2v = 2`

⇒ 1 + 2v = 2

⇒ 2v = 1

⇒ `v = 1/2`

5. Return to x and y:

`1/(x + y) = u = 1/3`

⇒ x + y = 3

`1/(x - y) = v = 1/2`

⇒ x – y = 2

6. Solve the two linear equations:

Add: 2x = 5 

⇒ `x = 5/2`

Then y = x – 2

= `5/2 - 2` 

= `1/2`