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Question
Find the value of k for which each of the following system of equations has infinitely many solutions
kx - 2y + 6 = 0
4x + 3y + 9 = 0
Solution
The given system of equation is
kx - 2y + 6 = 0
4x + 3y + 9 = 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 = k, b_1 = -2,c_1 = 6`
And `a_2 = 4, b_2 = -3,c_2 = 9`
For a unique solution, we must have
`a_1/a_2 = b-1/b_2 = c-1/c_2`
`=>k/4 = (-2)/(-3) = 6/9`
Now
`k/4 = 6/9`
`=> k/4 = 2/3`
`=> k = (2xx4)/3`
`=> k = 8/3`
Hence, the given system of equations will have infinitely many solutions, if `k = 8/3`
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