Problem Statement: Find the plane that passes through the point (-1, 2, 1) and contains the line of intersection of the planes:
$$x+y-z = 2$$
$$2x - y + 3z = 1$$
I understand there is a means of solving this with the cross product - but I am interested in whether or not I can solve this by using a matrix to represent the linear system.
$$ A = \left[\begin{array}{rrr|r} 1 & 1 & -1 & 2 \\ 2 & -1 & 3 & 1 \end{array}\right] $$
By row reducing the matrix we find:
$$ RREF(A) = \left[\begin{array}{rrr|r} 1 & 0 & -2/3 & 1 \\ 0 & 1 & -5/3 & 1 \end{array}\right] $$
Therefore, the system becomes:
$$x - 2/3z = 1 $$ $$y - 5/3z = 1$$
And by parametrizing $z = t$, and solving the system for $x$ and $y$:
$$x = 2/3t + 1$$ $$y = 5/3t + 1$$ $$z = t$$
Problem is, this resulting system does not actually match the line of intersection. Is there something I'm neglecting or doing wrong?
Show[ ContourPlot3D[ {x + y - z == 2, 2 x - y + 3 z == 1}, {x, -10, 10}, {y, -10, 10}, {z, -10, 10}, Boxed -> False, Axes -> True, AxesOrigin -> {0, 0, 0} ], ParametricPlot3D[ {(2/3) t + 1, (5/3) t + 1, t}, {t, -10, 10} ]]