To determine the values of 'k' for which the pair of equations 3x + 4y = 12 and kx + 12y = 30 does not have a unique solution, we need to consider the determinant of the coefficient matrix.
The coefficient matrix for the system of equations is:
[
\begin{bmatrix}
3 & 4 \
k & 12 \
\end{bmatrix}
]
The determinant of this matrix is given by:
[
\text{det} = (3)(12) - (k)(4) = 36 - 4k
]
For a system of equations to have a unique solution, the determinant must be non-zero. In other words, the determinant cannot equal zero.
So, to find the values of 'k' for which the system does not have a unique solution, we need to solve the equation 36 - 4k = 0.
Simplifying the equation, we have:
36 - 4k = 0
4k = 36
k = 9
Therefore, the values of 'k' for which the pair of equations does not have a unique solution are k = 9.