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Question
Find the value of k for which each of the following system of equations have no solution :
3x - 4y + 7 = 0
kx + 3y - 5 = 0
Solution
The given system of equation may be written as
3x - 4y + 7 = 0
kx + 3y - 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 = 3, b_1 = -4, c_1 = 7`
And `a_2 = k, b_2 = 3, c_2 = -5`
For a unique solution, we must have
`a_1/a_2 - b_1/b_2 != c_1/c_2`
We have,
`b_1/b_2 = (-4)/3`
and `c_1/c_2 = (-7)/5`
Clearly, `b_1/b_2 != c_1/c_2`
So, the given system will have no solution
`if a_1/a_2 = b_1/b_2 => 3/k = (-4)/3 => k = (-9)/4`
Clearly, for this value of k, we have `a_1/a_2 = b_1/b_2 != c_1/c_2`
Hence, the given system of equations has no solutions, when `k = (-9)/4`
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