Question

Show that the paths represented by the equations are parallel.

Answer 1

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