When a system of linear equations said to be consistent. For what value of k will the equations 3x + 4y + 2 = 0 and 9x + 12y + k = 0 are dependent ?

Explanation

The system of linear equations `a_(1)x + b_(1)y + c_(1) = 0` and `a_(2)x + b_(2)y + c_(2) = 0` is said to be consistent if it has a solution (either unique or infinite) i.e.,
if `(a_(1))/(a_(2)) ne (b_(1))/(b_(2))` (consistent and equations are independent)
or `(a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2))` (consistent and equations are dependent)
Here, `(a_(1))/(a_(2)) = (3)/(9) = (1)/(3), (b_(1))/(b_(2)) = (4)/(12) = (1)/(3)` and `(c_(1))/(c_(2)) = (2)/(k)`
For dependent equations
`(a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2))`
implies `(1)/(3) = (1)/(3) = (2)/(k) implies k = 6`