Question

Solve for x and y:

`3/x + 8/y = -1, 1/x - 2/y = 2  (x ≠ 0, y ≠ 0)`

Answer 1

Given: `3/x + 8/y = -1, 1/x - 2/y = 2  (x ≠ 0, y ≠ 0)`

Step-wise calculation:

1. Set `u = 1/x` and `v = 1/y` to turn the system into a linear one.

Then the system becomes:

3u + 8v = –1

u – 2v = 2

2. From the second equation, u = 2 + 2v.

3. Substitute into the first: 3(2 + 2v) + 8v = –1

6 + 6v + 8v = –1

14v = –7

`v = -1/2`

4. Find u: u = 2 + 2v

= `2 + 2(-1/2)` 

= 2 – 1

= 1

5. Return to x and y:

`x = 1/u` 

= `1/1`

= 1 

`y = 1/v`

= `1/(-1/2)`

= –2