Find three different systems of linear equation whose solutions are..

Question

Find three different systems of linear equation whose solutions are $x_1 = 3, x_2 = 0, x_3 = -1$

I'm confused, how exactly can I do this?

Answer 1

Note that your system is described by the matrix $$ \left[\begin{array}{rrr|r} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \end{array}\right] $$ Performing any row operation on this matrix yields a system with the same solutions. For example, you could add $\DeclareMathOperator{Row}{Row}\Row_1$ to $\Row_2$ to get the system $$ \left[\begin{array}{rrr|r} 1 & 0 & 0 & 3 \\ 1 & 1 & 0 & 3 \\ 0 & 0 & 1 & -1 \end{array}\right] $$ You could then add $\Row_2$ to $\Row_3$ to get $$ \left[\begin{array}{rrr|r} 1 & 0 & 0 & 3 \\ 1 & 1 & 0 & 3 \\ 1 & 1 & 1 & 2 \end{array}\right] $$ To get a third system, you could then add $2\cdot \Row_3$ to $\Row_1$ $$ \left[\begin{array}{rrr|r} 3 & 2 & 2 & 7 \\ 1 & 1 & 0 & 3 \\ 1 & 1 & 1 & 2 \end{array}\right] $$

Answer 2

come up with an a linear equation that includes the indicated point.

e.g. $x + y + 3z = 0$ Now come up with two more. If you want your solution to be unique you will need to make sure that the planes you have chosen are "linearly independent."