Question

For what values of k is the system of equations $kx+3y=k-2,12x+ky=k$ inconsistent?

Answer 1

The given system of equations can be written as follows:
kx + 3y − (k − 2) = 0 and 12x + ky − k = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = k, b₁ = 3, c₁ = − (k − 2) and a₂ = 12, b₂ = k and c₂ = − k
∴ $\frac{a_1}{a_2}=\frac{k}{12},\frac{b_1}{b_2}=\frac{3}{k}\mathrm{and}\frac{c_1}{c_2}=\frac{-\left(k-2\right)}{-k}=\frac{\left(k-2\right)}{k}$
For inconsistency, we must have:
$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$
⇒ $\frac{k}{12}=\frac{3}{k}\ne \frac{\left(k-2\right)}{k}\Rightarrow k^2=\left(3\times 12\right)=36$
⇒ $k=\sqrt{36}=\pm 6$
Hence, the pair of equations is inconsistent if $k=\pm 6$.