Question

Find the values of a and b for which the following 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:

a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0

where, a₁ = 2, b₁ = 3, c₁ = –7 and a₂ = 2a, b₂ = a + b, c₂ = –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

⇒ b = 8

Hence, a = 4 and b = 8.