Question

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

(a – 1)x + 3y = 2, 6x + (1 – 2b)y = 6

Answer 1

The given system of equations can be written as

(a – 1)x + 3y = 2

⇒ (a – 1)x + 3y – 2 = 0   ...(i)

And 6x + (1 – 2b)y = 6

⇒ 6x + (1 – 2b)y – 6 = 0   ...(ii)

These equations are of the following form:

a₁x + b₁y + c₁ = 0

a₂x + b₂y + c₂ = 0

where, a₁ = (a – 1), b₁ = 3, c₁ = –2 and a₂ = 6, b₂ = (1 – 2b), c₂ = –6

For an infinite number of solutions, we must have:

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

⇒ `(a - 1)/6 = 3/((1 - 2b)) = (-2)/(-6)`

⇒ `(a - 1)/6 = 3/((1 - 2b)) = 1/3`

⇒ `(a - 1)/6 = 1/3` and `3/((1 - 2b)) = 1/3`

⇒ 3a – 3 = 6 and 9 = 1 – 2b

⇒ 3a = 9 and 2b = –8

⇒ a = 3 and b = –4

∴ a = 3 and b = –4