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Question
Find the value of k for which each of the following system of equations have no solution
x + 2y = 0
2x + ky = 5
Solution
The given system of equation may be written as
x + 2y = 0
2x + ky - 5 = 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 = 0`
And `a_2 = 2, b_2 = k, c_2 = -5`
For a unique solution, we must have
`a_1/a_2 - b_1/b_2 != c_1/c_2`
We have
`a_1/a_2 = 1/2`
`b_1/b_2 = 2/k`
And `c_1/c_2 = 0/(-5)`
Now `a_1/a_2 = b_1/b_2`
`=> 1/2 = 2/k`
`=> k = 4`
Hence, the given system of equations has no solutions, when k = 4
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