Question

Find the value of k for which the following system of equations has a unique solution:

4x – 5y = k, 2x – 3y = 12

Answer 1

The given system of equations is

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:

`(a_1)/(a_2) ≠ (b_1)/(b_2)`

i.e., `4/2 ≠ (-5)/(-3)`

⇒ `2 ≠ 5/3`

⇒ 6 ≠ 5

Thus, for all real values of k, the given system of equations will have a unique solution.