System of equations / matrices

[Mic :)]

OK. Can you summarize, in complete sentences, what you have found so far? Each sentence should show the specific values of h and k and should say whether there are a) no solutions, b) a unique solution, c) an infinite number of solutions.

Note that we still have some work to do, but I'm checking to see if you understand where we've gotten to at this point.

For h = 2, k = 1/2 and h = -3, k = -1/3, there are infinitely many solutions.

For h = 2, k ≠ 1/2 and h = -3 and k ≠ -1/3, there are no solutions.

I don't think we've agreed on something for b)

 

[Mark44]

What you have looks good.
Here's your augmented matrix, nicely formatted in LaTeX:
$$\begin{bmatrix} h & 6 & | & 2 \\ 0 & h^2 + h - 6 & | & 2hk - 2\end{bmatrix} $$

In your work above, you have said what the situation is if h = 2 or if h = -3. Now suppose that h ≠ 2, and h ≠ -3. What can you conclude about the augmented matrix, and from it, the system of equations?