Question

Show that the system of equations $-x+2y+2=0\mathrm{and}\frac{1}{2}\mathrm{x}-\frac{1}{4}\mathrm{y}-1=0$ has a unique solution.

Answer 1

The given system of equations:
$-x+2y+2=0\mathrm{and}\frac{1}{2}\mathrm{x}-\frac{1}{4}\mathrm{y}-1=0$
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = −1, b₁ = 2, c₁ = 2 and a₂ = $\frac{1}{2}$, b₂ = $-\frac{1}{4}$ and c₂ = −1
∴ $\frac{a_1}{a_2}=\frac{-1}{\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}=-2,\frac{b_1}{b_2}=\frac{2}{\left(-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\right)}=-8\mathrm{and}\frac{c_1}{c_2}=\frac{2}{-1}=-2$
∴ $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
The given system has a unique solution.
Hence, the lines intersect at one point.