A system of linear equations $a_{1}x + b_{1}y + c_{1} = 0$ and $a_{2}x + b_{2}y + c_{2} = 0$ will have a unique solution if the two lines represented by the equations $a_{1}x + b_{1}y + c_{1} = 0$ and $a_{2}x + b_{2}y + c_{2} = 0$ intersect at a point.
i.e., if the two lines are neither parallel nor coincident. Essentially, the slopes of the two lines should be different.

Condition for unique solution$:\dfrac{a_{1}}{a_{2}} \neq \dfrac{b_{1}}{b_{2}}$

Condition for no solution$:\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}} \neq \dfrac{c_{1}}{c_{2}}$

Condition for infinitely many solutions$:\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}} = \dfrac{c_{1}}{c_{2}}$

Now, question asked about not unique solution ,the the condition will be $\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}}$

Given the equations, $3x + 4y =12$ and $kx + 12y = 30$

Here, $a_{1} = 3,b_{1} = 4,a_{2} = k,b_{2} = 12$

Now, if above equation does not have unique solution , then $\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}}$