Question

Find the values of a and b for which the system of linear equations has an infinite number of solutions:
2x + 3y = 7, 2ax + (a + b)y = 28.

Answer 1

The given system of equations can be written as
2x + 3y - 7 = 0                         ….(i)
2ax + (a + b)y – 28 = 0             ….(ii)
This system is of the form:
`a_1x+b_1y+c_1 = 0`
`a_2x+b_2y+c_2 = 0`
where, `a_1 = 2, b_1= 3, c_1= -7 and a_2 = 2a, b_2 = a + b, c_2= – 28`
For the given system of linear equations to have an infinite number of solutions, we must have:

`(a_1)/(a_2) = (b_1)/(b_2) = (c_1)/(c_2)`
`⇒2/(2a) = 3/(a+b) = (−7)/(−28)`
`⇒ 2/(2a) =( −7)/(−28 )= 1/4 and 3/(a+b) = (−7)/(−28) = 1/4`
⇒ a = 4 and a + b = 12
Substituting a = 4 in a + b = 12, we get
4 + b = 12 ⇒ b = 12 – 4 = 8
Hence, a = 4 and b = 8.