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Question
Find the value of k for which the following system of equations has a unique solution:
x + 2y = 3
5x + ky + 7 = 0
Solution
The given system of equation is
x + 2y - 3 = 0
5x + ky + 7 = 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 = 1,b_1 = 2,c_1 = -3`
And `a_2 = 5, b_2 = k,c_2 = 7`
For a unique solution, we must have
`a_1/a_2 != b_1/b_2`
`:. 1/5 != 2/k`
`=> k != 10`
So, the given system of equations will have a unique solution for all real values of k other than 10.
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