Question

The pair of equations $x+2y+5=0\mathrm{and}-3\mathrm{x}-6\mathrm{y}+1=0$ has

(a) a unique solutions
(b) exactly two solutions
(c) infinitely many solutions
(d) no solution

Answer 1

The correct option is (d).

The given system of equations can be written as:
x + 2y + 5 = 0 and −3x − 6y + 1 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 1, b₁ = 2, c₁ = 5 and a₂ = −3, b₂ = −6 and c₂ = 1
∴ $\frac{a_1}{a_2}=\frac{1}{-3},\frac{b_1}{b_2}=\frac{2}{-6}=\frac{1}{-3}\mathrm{and}\frac{c_1}{c_2}=\frac{5}{1}$
∴ $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$
Hence, the given system has no solution.