Question

Solve the following system of equations:

`2/x + 3/y = 13`

`5/x - 4/y = -2`

Answer 1

Given:

`2/x + 3/y = 13`

`5/x - 4/y = -2`

Step-wise calculation:

1. Let `u = 1/x` and `v = 1/y`, so the system becomes a linear system in u, v:

2u + 3v = 13 and 5u – 4v = –2

2. Eliminate v: multiply the first equation by 4 and the second by 3:

8u + 12v = 52

15u – 12v = –6

Add: 23u = 46

⇒ u = 2

3. Substitute u = 2 into 2u + 3v = 13:

4 + 3v = 13

⇒ 3v = 9

⇒ v = 3

4. Return to x, y:

`u = 1/x = 2` 

⇒ `x = 1/2`

`v = 1/y = 3`

⇒ `y = 1/3`