Question

Solve: 6x + 3y = 7xy and 3x + 9y = 11xy.

Answer 1

Given: 6x + 3y = 7xy and 3x + 9y = 11xy

Step-wise calculation:

1. Check zero-case:

If x = 0, first equation gives 3y = 0 → y = 0.

If y = 0, second equation gives 3x = 0 → x = 0.

So (0, 0) is a solution.

2. Assume x ≠ 0 and y ≠ 0.

Divide both equations by xy:

`6/y + 3/x = 7`

`3/y + 9/x = 11`

Let `a = 1/x` and `b = 1/y`.

Then: 3a + 6b = 7 (from 6b + 3a = 7) 9a + 3b = 11

3. Solve the linear system:

Multiply the first equation by 3:

9a + 18b = 21 

Subtract the second equation:

(9a + 18b) – (9a + 3b) = 21 – 11 

⇒ 15b = 10

⇒ `b = 2/3`

Substitute `b = 2/3` into 3a + 6b = 7: 

`3a + 6 xx 2/3 = 7` 

⇒ 3a + 4 = 7 

⇒ 3a = 3

⇒ a = 1

Thus, a = 1

⇒ `1/x = 1`

⇒ x = 1

And `b = 2/3` 

⇒ `1/y = 2/3` 

⇒ `y = 3/2`