Question

Find the value of k for which each of the following systems of equations has a unique solution:
$4x-5y=k,2x-3y=12.$

Answer 1

The given system of equations:
4x − 5y = k
⇒ 4x − 5y − k = 0 ...(i)
And, 2x − 3y = 12
⇒ 2x − 3y − 12 = 0 ...(ii)
These equations are of the following form:
a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 4, b₁= −5, c₁ = −k and a₂ = 2, b₂ = −3, c₂ = −12
For a unique solution, we must have:
$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
i.e. $\frac{4}{2}\ne \frac{-5}{-3}$
$\Rightarrow 2\ne \frac{5}{3}\Rightarrow 6\ne 5$
Thus, for all real values of k, the given system of equations will have a unique solution.