Show that the system of equations
$$\begin{array} { r }
x - 2 y - 4 z = 1 \\
x - 2 y + k z = 1 \\
- x + 2 y + 2 z = 1
\end{array}$$
where \(k\) is a constant, does not have a unique solution.
Given that \(k = - 4\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
Given instead that \(k = - 2\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
For the case where \(k \neq - 2\) and \(k \neq - 4\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
\includegraphics[max width=\textwidth, alt={}, center]{23c7189f-850d-4745-a8ce-46a140ed0176-06_894_841_260_612}
The diagram shows the curve with equation \(\mathrm { y } = 1 - \mathrm { x } ^ { 3 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).