The discussion revolves around solving a system of equations represented by the augmented matrix: $$\begin{bmatrix} h & 6 & | & 2 \\ 1 & (h+1) & | & 2k\end{bmatrix}$$. The determinant of the system, given by the expression \(h^2 + h - 6\), is critical for determining the nature of the solutions. The values of \(h\) that lead to no solutions are \(h = 2\) and \(h = -3\), with corresponding values of \(k\) being \(k = 0.5\) and \(k = -1/3\), respectively. For other values of \(h\) and \(k\), the system can yield either a unique solution or infinitely many solutions, depending on the conditions applied.
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Mark44 said: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.