Question

Find the values of a and b for which the following system of linear equations has an infinite number of solutions:

(2a – 1)x + 3y = 5, 3x + (b – 1)y = 15

Answer 1

Given: The system (2a – 1)x + 3y = 5, 3x + (b – 1)y = 15

Step-wise calculation:

1. A 2 × 2 system has infinitely many solutions when the coefficients and constants are proportional:

`(a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2)`

2. Compute the common ratio from the constants:

`5/15 = 1/3`

3. Set the ratios equal to `1/3`:

`(2a - 1)/3 = 1/3` 

⇒ 2a – 1 = 1

⇒ a = 1

`3/(b - 1) = 1/3` 

⇒ `3 = (b - 1)/3` 

⇒ 9 = b – 1

⇒ b = 10

4. Check: with a = 1 and b = 10 the first equation is x + 3y = 5; multiplying by 3 gives 3x + 9y = 15, which is the second equation coincident lines.

The system has infinitely many solutions when a = 1 and b = 10.