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Question
Find the value of k for which each of the following systems of equations has infinitely many solutions :
2x + 3y − 5 = 0
6x + ky − 15 = 0
Solution
The given system of equation is
2x + 3y − 5 = 0
6x + ky − 15 = 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 = 3, c_1 = -5`
And `a_2 = 6, b_2 = k,c_2 = -15`
For a unique solution, we must have
`a_1/a_2 = b_1/b_2 = c_1/c_2`
`=> 2/6 = 3/k`
`=> k = 18/2 = 9`
Hence, the given system of equations will have infinitely many solutions, if k = 9.
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