Question

Solve for x and y:

`(2x + 5y)/(xy) = 6, (4x - 5y)/(xy) = -3`

Answer 1

Given: `(2x + 5y)/(xy) = 6, (4x - 5y)/(xy) = -3`

Step-wise calculation:

1. Let `u = 1/x` and `v = 1/y`.

2. Rewrite each fraction:

`(2x + 5y)/(xy) = 2/y + 5/x` 

⇒ 2v + 5u = 6

`(4x - 5y)/(xy) = 4/y − 5/x` 

⇒ 4v – 5u = –3 

So, the linear system is 5u + 2v = 6 and –5u + 4v = –3.

3. Add the two equations to eliminate u:

(5u – 5u) + (2v + 4v) = 6 + (–3)

⇒ 6v = 3

⇒ `v = 1/2`

4. Substitute `v = 1/2` into 5u + 2v = 6:

`5u + 2(1/2) = 6`

⇒ 5u + 1 = 6

⇒ 5u = 5

⇒ u = 1

5. Convert: 

`u = 1/x = 1` 

⇒ x = 1

`v = 1/y = 1/2` 

⇒ y = 2