Question

The pair of equations $2x-5y+4=0\mathrm{and}2\mathrm{x}+\mathrm{y}-8=0$ has

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

Answer 1

The correct option is (a).

The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁ = −5, c₁ = 4 and a₂ = 2, b₂ = 1 and c₂ = −8
∴ $\frac{a_1}{a_2}=\frac{2}{2}=1,\frac{b_1}{b_2}=\frac{-5}{1}=-5\mathrm{and}\frac{c_1}{c_2}=\frac{4}{-8}=\frac{-1}{2}$
⇒ $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
These graph lines will intersect at a unique point.
Hence, the given system has a unique solution.