Question

For what value of c, the following system of linear equations has infinite number of solutions:

2x + 3y = 2, (c + 2) x + (2c + 1) y = 2 (c – 1)

Answer 1

Given, pair of equations

2x + 3y = 2

and (c + 2)x + (2c + 1)y = 2(c – 1)

On comparing the given equation with standard form i.e. a₁ x + b₁y + c₁ = 0 and a₂ x + b₂y + c₂ = 0, we get

a₁ = 2, b₁ = 3 and c₁ = – 2

and a_{2 =} c + 2, b₂ = 2c + 1 and c₂ = – 2(c – 1)

For infinitely many solutions,

⇒ 2(2c + 1) = 3(c + 2)

⇒ 4c + 2 = 3c + 6

⇒ 4c – 3c = 6 – 2

⇒ c = 4

Hence, the required value of c = 4.