Question

For what value of k is the pair of equations $8x+5y=9\mathrm{and}\mathrm{kx}+10\mathrm{y}=15$ inconsistent?

(a) $k=\frac{3}{5}$
(b) $k=\frac{5}{3}$
(c) k = 16
(d) k = −16

Answer 1

The correct option is (c).

The given system of equations can be written as follows:
8x + 5y − 9 = 0 and kx + 10y − 15 = 0
The given equations are of the following form:
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0
Here, a₁ = 8, b₁ = 5, c₁ = −9 and a₂ = k, b₂ = 10 and c₂ = −15
∴ $\frac{a_1}{a_2}=\frac{8}{k},\frac{b_1}{b_2}=\frac{5}{10}=\frac{1}{2}\mathrm{and}\frac{c_1}{c_2}=\frac{-9}{-15}=\frac{3}{5}$
For inconsistency, we must have:
$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$
⇒ $\frac{8}{k}=\frac{1}{2}\ne \frac{3}{5}\Rightarrow k=\left(2\times 8\right)=16$
Hence, the pair of equations is inconsistent if the value of k is 16.