For the following system of equations determine the value of `k` for which the given system has infinitely many solutions: `5x+2y=k ,\ \ \ \ 10 x+4y=3`

Explanation

The given system of equations is
5x + 2y - k = 0 and 10x + 4y - 3 = 0
Here `a_(1) = 5, b_(1) = 2` and `c_(1) = - k`
`a_(2) = 10, b_(2) = 4` and `c_(2) = - 3`
The system has infinitely many solutions if
`(a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2))`
i.e., `(5)/(10) = (2)/(4) = (-k)/(-3)`
When `(1)/(2) = (k)/(3) implies k = (3)/(2)`
Hence, for `k = (3)/(2)`, the given system of equations will infinitely many solutions.