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 equation is

4x - 5y - k = 0

2x - 3y - 12 = 0

The system of equation is of the form

`a_1x + b_1y + c_1 = 0`

`a_2x + b_2y + c_2 = 0`

Where `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`

`:. 4/2 != (-5)/(-3)`

⇒ k is any real number.

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