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Question
Find the value of k for which each of the following systems of equations has infinitely many solutions :
4x + 5y = 3
kx + 15y = 9
Solution
The given system of equation is
4x + 5y -3 = 0
ks + 15y - 9 = 0
The system of equation is of the for
`a_1x + b_1y + c_1= 0`
`a_2x + b_2y + c_2 = 0`
Where `a_1 = 4, b_1 = 5, c_1 = -3`
And `a_2 = k, b_2 = 15, c_2 = -9`
For a unique solution, we must have
`a_1/a_2 = b_1/b_2 = c_2/c_2`
`=> 4/k = 5/15 = (-3)/(-9)`
Now
`4/k = 5/15`
`=> 4/k = 1/3`
`=> k = 12`
Hence, the given system of equations will have infinitely many solutions, if k = 12
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