Question

Solve for x and y:

`5/(x + 1) - 2/(y - 1) = 1/2, 10/(x + 1) + 2/(y - 1) = 5/2`, x ≠ –1 and y ≠ 1

Answer 1

Given: `5/(x + 1) - 2/(y - 1) = 1/2, 10/(x + 1) + 2/(y - 1) = 5/2`, with x ≠ –1 and y ≠ 1.

Step-wise calculation:

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

2. The system becomes:

`5a - 2b = 1/2`

`10a + 2b = 5/2`

3. Add the two equations:

`(5a - 2b) + (10a + 2b) = 1/2 + 5/2`

⇒ 15a = 3

⇒ `a = 1/5`

4. Substitute `a = 1/5` into `5a - 2b = 1/2`: 

`5(1/5) - 2b = 1/2`

⇒ `1 - 2b = 1/2` 

⇒ `-2b = -1/2`

⇒ `b = 1/4`

5. Return to x, y:

`1/(x + 1) = a = 1/5`

⇒ x + 1 = 5

⇒ x = 4

`1/(y - 1) = b = 1/4` 

⇒ y – 1 = 4

⇒ y = 5

x = 4, y = 5 valid since x ≠ –1 and y ≠ 1.