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