Question

The pair of equations $2x+3y=5\mathrm{and}4\mathrm{x}+6\mathrm{y}=15$ has

(a) a unique solution
(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 follows:
2x + 3y − 5 = 0 and 4x + 6y − 15 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁ = 3, c₁ = −5 and a₂ = 4, b₂ = 6 and c₂ = −15
∴ $\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2},\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2}\mathrm{and}\frac{c_1}{c_2}=\frac{-5}{-15}=\frac{1}{3}$
∴ ​$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$
Hence, the given system has no solution.