Question

Solve: `(ax)/b - (by)/a = a + b, ax - by = 2ab`.

Answer 1

Given:

`(ax)/b - (by)/a = a + b`

ax – by = 2ab   ...(Assume a ≠ 0 and b ≠ 0 because of the divisions)

Step-wise calculation:

1. Multiply the first equation by ab to clear denominators:

a²x – b²y = a²b + ab²   ...(1)

2. Keep the second equation:

ax – by = 2ab   ...(2)

3. Multiply (2) by a:

a²x – aby = 2a²b   ...(3)

4. Subtract (3) from (1):

(a²x – b²y) – (a²x – aby) = (a²b + ab²) – 2a²b 

⇒ y × b × (a – b) = –ab(a – b)

5. If a – b ≠ 0 and b ≠ 0, cancel (a – b) and b: y = –a.

6. Substitute y = –a into (2):

ax – b(–a) = 2ab

⇒ ax + ab = 2ab

⇒ ax = ab

⇒ x = b

7. Special case a = b with a = b ≠ 0:

Both equations reduce to the same linear relation: x – y = 2a, so there are infinitely many solutions: x = y + 2a (parameterize by y).

8. Cases with a = 0 or b = 0 are not allowed division by zero in the original equation.

If a ≠ b and a ≠ 0 and b ≠ 0: x = b, y = –a.

If a = b ≠ 0: infinitely many solutions given by x – y = 2a (i.e., x = y + 2a).

a = 0 or b = 0: original system is not valid (division by zero).