Question

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

Answer 1

The given system of equations are
4x - 5y = k
⇒ 4x - 5y - k = 0 ….(i)
And, 2x - 3y = 12
⇒2x - 3y - 12 = 0 …(ii)
These equations are of the following form:
`a_1x+b_1y+c_1 = 0, a_2x+b_2y+c_2 = 0`
Here, `a_1 = 4, b_1= -5, c_1= -k and a_2 = 2, b_2= -3, c_2= -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