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Let's break down the problem you're working on.

The system of equations is:

\[
-6a + (k - 3)b = 1
\]
\[
3ka - 5b = 4
\]

### Part (a): Find the values of \( k \) for which the system has a unique solution.

A system of linear equations has a unique solution when the determinant of the coefficient matrix is non-zero. The matrix for this system is:

\[
\begin{pmatrix}
-6 & k-3 \\
3k & -5
\end{pmatrix}
\]

The determinant of this matrix is given by:

\[
\text{det} = (-6)(-5) - (k-3)(3k)
\]
\[
\text{det} = 30 - (3k^2 - 9k)
\]
\[
\text{det} = 30 - 3k^2 + 9k
\]
\[
\text{det} = -3k^2 + 9k + 30
\]

To have a unique solution, the determinant must be non-zero, so:

\[
-3k^2 + 9k + 30 \neq 0
\]

This is a quadratic equation, and you would solve it for \( k \) to find the values where the system has a unique solution.

### Part (b): Find the unique solution in terms of \( k \).

Once you determine the values of \( k \) that yield a unique solution, you substitute them back into the system to find the solution for \( a \) and \( b \).

If you need further steps or clarifications, feel free to ask!

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Would you like more details on solving the determinant or substituting back into the equations? Here are some related questions you might find helpful:

1. How do we solve a quadratic equation?
2. What does it mean for a system of equations to be dependent?
3. How do we determine if a system has no solution or infinitely many solutions?
4. What is Cramer's rule for solving systems of equations?
5. Can a determinant of zero still provide a meaningful solution?

**Tip**: Remember, checking the determinant is crucial when determining the nature of solutions in linear systems!

Consider the system of equations: -6a + (k-3)b = 1 and 3ka - 5b = 4. (a) Find the values of the real parameter k such that the system has a unique solution. (b) Find the unique solution in terms of k.

Learn how to determine the values of k that allow a system of linear equations to have a unique solution. This problem involves calculating the determinant and analyzing quadratic expressions, suitable for Grades 10-12.

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