Question

Solve the following pair of linear equations:

`2/x + 2/y = 1, 6/x - 9/y = -2, x ≠ 0, y ≠ 0`

Answer 1

Given the system of equations:

`2/x + 2/y = 1, 6/x - 9/y = -2, x ≠ 0, y ≠ 0`

Step 1: Substitute `(1/x = u)` and `(1/y = v)` to transform the equations into linear form:

2u + 2v = 1 

6u – 9v = –2

Step 2: Simplify the first equation by dividing both sides by 2:

`u + v = 1/2`

Step 3: Solve the system:

From `(u + v = 1/2)`, express `(u = 1/2 - v)`.

Substitute into the second equation:

`6(1/2 - v) - 9v = -2` 

3 – 6v – 9v = –2 

3 – 15v = –2 

–15v = –5 

`v = 1/3`

Step 4: Find (u):

`u = 1/2 - 1/3`

`u = 3/6 - 2/6`

`u = 1/6`

Step 5: Recall the substitutions to get (x) and (y):

`u = 1/x`

`u = 1/6`

⇒ x = 6

`v = 1/y`

`v = 1/3`

⇒ y = 3

The solution to the system is x = 6, y = 3.