Question

Find the values of a and b for which the system of linear equations has an infinite number of solutions:
(2a – 1) x + 3y = 5, 3x + (b – 1)y = 2.

Answer 1

The given system of equations can be written as
(2a – 1) x + 3y = 5
⇒(2a – 1) x + 3y – 5 = 0               ….(i)
and 3x + (b – 1)y = 2
⇒3x + (b – 1)y – 2 = 0                        ….(ii)
These equations are of the following form:

`a_1x+b_1y+c_1 = 0, a_2x+b_2y+c_2 = 0`
where, `a_1 = (2a – 1), b_1= 3, c_1= -5 and a_2 = 3, b_2 = (b – 1), c_2= -2`
For an infinite number of solutions, we must have:
`(a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2)`
`⇒ ((2a−1))/3 = 3/((b−1)) = (−5)/(−2)`
`⇒ ((2a−1))/6 = 3/((b−1)) = 5/2`
`⇒ ((2a−1))/6 = 5/2 and 3/((b−1)) = 5/2`
⇒ 2(2a – 1) = 15 and 6 = 5(b – 1)
⇒ 4a – 2 = 15 and 6 = 5b – 5
⇒ 4a = 17 and 5b = 11
`∴ a = 17/4 and b = 11/5`