Question

Write the set of values of a and b for which the following system of equations has infinitely many solutions.

2x + 3y = 7 

2ax + (a + b)y = 28

Answer 1

Given: 2x + 3y = 7, 2ax + (a + b)y = 28

Step-wise calculation:

1. For infinitely many solutions the two equations must be proportional, so `2/(2a) = 3/(a + b) = 7/28`.

2. Compute the common ratio:

`7/28 = 1/4`

3. From `2/(2a) = 1/4`

⇒ `1/a = 1/4`

⇒ a = 4

4. From `3/(a + b) = 1/4` and a = 4

⇒ `3/(4 + b) = 1/4`

⇒ 4 + b = 12

⇒ b = 8

5. Check: with a = 4, b = 8 the second equation is 8x + 12y = 28, which is 4 × (2x + 3y) = 28, so the lines coincide.

The system has infinitely many solutions exactly when a = 4 and b = 8.