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Question
Find the value of k for which each of the following system of equations have no solution :
2x - ky + 3 = 0
3x + 2y - 1 = 0
Solution
The given system of the equation may be written as
2x - ky + 3 = 0
3x + 2y - 1 = 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 = 2, b_1 = -k, c_1 = 3`
And
`c_2 = 3, b_2 = 2, c_2 = -1`
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 = 2/3`
and `c_1/c_2 = 3/(-1)`
Clearly, `a_1/a_2 != c_1/c_2`
So, the given system will have no solution. If
`a_1/a_2 = b_1/b_2 i.e 2/k = (-k)/2 => k = (-4)/3`
Hence, the given system of equations has no solutions `k = (-4)/3`
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