Question

Find the values of a and b for which the following system of equations has infinitely many solutions:

x + 2y = 1 

(a – b)x + (a + b)y = a + b – 2

Answer 1

Given: x + 2y = 1, (a – b)x + (a + b)y = a + b – 2

Step-wise calculation:

1. For infinitely many solutions, coefficients must be proportional:

`1/(a - b) = 2/(a + b) = 1/(a + b − 2)`

2. From `1/(a - b) = 2/(a + b)`:

(a + b) = 2(a – b)

⇒ a = 3b

3. From `2/(a + b) = 1/(a + b - 2)`:

2(a + b – 2) = a + b

⇒ a + b = 4

4. Substitute a = 3b into a + b = 4:

3b + b = 4

⇒ 4b = 4

⇒ b = 1, so a = 3.