Question

The pair of equations $2x+y=5,3x+2y=8$ has
(a) a unique solution
(b) two solutions
(c) no solution
(d) infinitely many solutions

Answer 1

The correct option is (a).

The given system of equations can be written as follows:
2x + y − 5 = 0 and 3x + 2y − 8 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 2, b₁ = 1, c₁ = −5 and a₂ = 3, b₂ = 2 and c₂ = −8
∴ $\frac{a_1}{a_2}=\frac{2}{3},\frac{b_1}{b_2}=\frac{1}{2}\mathrm{and}\frac{c_1}{c_2}=\frac{-5}{-8}=\frac{5}{8}$
∴ $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
The given system has a unique solution.
Hence, the lines intersect at one point.