Question

For what value of c does the pair of equations $cx-y=2\mathrm{and}6\mathrm{x}-2\mathrm{y}=3$ have infinitely many solutions?

(a) c = 3
(b) c = −3
(c) c = −12
(d) not possible for any value of c

Answer 1

The correct option is (d).

The given system of equations can be written as follows:
cx − y − 2 = 0 and 6x − 2y − 3 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = c, b₁ = −1, c₁ = −2 and a₂ = 6, b₂ = −2 and c₂ = −3
∴ $\frac{a_1}{a_2}=\frac{c}{6},\frac{b_1}{b_2}=\frac{-1}{-2}=\frac{1}{2}\mathrm{and}\frac{c_1}{c_2}=\frac{-2}{-3}=\frac{2}{3}$
For an infinite number of solutions, we must have:
$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
⇒ $\frac{c}{6}=\frac{1}{2}=\frac{2}{3}$
The above condition is not possible for any value of c.