Question

Solve for x and y:

`1/(2x) + 1/(3y) = 2, 1/(3x) + 1/(2y) = 13/6 (x ≠ 0, y ≠ 0)`

Answer 1

Given: `1/(2x) + 1/(3y) = 2, 1/(3x) + 1/(2y) = 13/6 (x ≠ 0, y ≠ 0)`

Step-wise calculation:

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

2. The system becomes: `u/2 + v/3 = 2`

`u/3 + v/2 = 13/6`

3. Clear denominators by multiplying both equations by 6:

3u + 2v = 12

2u + 3v = 13

4. Multiply the first equation by 3 and the second by 2: 

9u + 6v = 36

4u + 6v = 26

Subtract: 5u = 10 ⇒ u = 2.

Substitute u = 2 into 3u + 2v = 12

⇒ 6 + 2v = 12

⇒ 2v = 6

⇒ v = 3

5. Return to x, y: 

`u = 1/x = 2` 

⇒ `x = 1/2` 

`v = 1/y = 3` 

⇒ `y = 1/3`