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, (a + b)x + (2a – b)y = 21

Answer 1

The given system of equations can be written as

2x + 3y – 7 = 0   ...(i)

(a + b)x + (2a – b)y – 21 = 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 = a + b, b₂ = 2a – b, c₂ = –21

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/(a + b) = 3/(2a - b) = (-7)/(-21)`

⇒ `2/(a + b) = (-7)/(-21) = 1/3` and `3/(2a - b) = (-7)/(-21) = 1/3`

⇒ a + b = 6 and 2a – b = 9

Adding a + b = 6 and 2a – b = 9, we get

3a = 15

⇒ a = `15/3`

⇒ a = 3

Now substituting a = 5 in a + b = 6, we have

5 + b = 6

⇒ b = 6 – 5

⇒ b = 1

Hence, a = 5 and b = 1.