System of three linear equations

1

1. Show that the system of equations $$\begin{array} { r } x + 2 y + 3 z = 1 \\ 4 x + 5 y + 6 z = 1 \\ 7 x + 8 y + 9 z = 1 \end{array}$$ does not have a unique solution.
2. Show that the system of equations in part (a) is consistent. Interpret this situation geometrically.

3

1. 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.
2. Given that \(k = - 4\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
3. Given instead that \(k = - 2\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
4. 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]{7da7fa35-1b97-4708-a1a2-cba9e35c8bf0-06_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 \(\frac { 1 } { n }\).

3

1. 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.
2. Given that \(k = - 4\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
3. Given instead that \(k = - 2\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
4. 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 \(\frac { 1 } { n }\).

2

1. Show that the system of equations $$\begin{aligned} & x - y + 2 z = 4 \\ & x - y - 3 z = a \\ & x - y + 7 z = 13 \end{aligned}$$ where \(a\) is a constant, does not have a unique solution.
2. Given that \(a = - 5\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
3. Given instead that \(a \neq - 5\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.

1 Show that the system of equations $$\begin{aligned} 14 x - 4 y + 6 z & = 5 \\ x + y + k z & = 3 \\ - 21 x + 6 y - 9 z & = 14 \end{aligned}$$ where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically.