Question

Solve the following pair of linear equations:

`6/(2x + y) + 5/(2x - y) = 3, 9/(2x + y) - 10/(2x - y) = 1`

Answer 1

Given: `6/(2x + y) + 5/(2x - y) = 3, 9/(2x + y) - 10/(2x - y) = 1`

Step-wise calculation:

1. Let: `u = 1/(2x + y), v = 1/(2x - y)`

Then rewrite the system as: 6u + 5v = 3, 9u – 10v = 1

2. Solve the system:

From first, 6u + 5v = 3 

From second, 9u – 10v = 1

3. Multiply the first equation by 2:

12u + 10v = 6

4. Add this to the second equation:

(12u + 10v) + (9u – 10v) = 6 + 1

⇒ 21u = 7

⇒ `u = 7/21`

⇒ `u = 1/3`

5. Substitute `(u = 1/3)` into the first equation: 

`6 xx 1/3 + 5v = 3` 

⇒ 2 + 5v = 3

⇒ 5v = 1 

⇒ `v = 1/5`

6. Recall definitions:

`u = 1/(2x + y)`

`u = 1/3` 

⇒ 2x + y = 3 

`v = 1/(2x - y)`

`v = 1/5` 

⇒ 2x – y = 5

7. Add these two equations to eliminate (y): 

(2x + y) + (2x – y) = 3 + 5

⇒ 4x = 8

⇒ x = 2

8. Substitute (x = 2) back into (2x + y = 3): 

2 × 2 + y = 3 

⇒ 4 + y = 3

⇒ y = –1