Read a common line from simultaneous plane equations

Expand the plane equations:

$\begin{aligned}x+y-3z&=6,\\2x-y+z&=4.\end{aligned}$

The source uses the calculator’s PlySmlt2 simultaneous-equation application. Its solution screen uses $x_1,x_2,x_3$ for $x,y,z$, and displays

$\begin{aligned}x&=\frac{10}{3}+\frac23z,\\y&=\frac83+\frac73z,\\z&=z.\end{aligned}$

This can also be obtained by adding the equations to give $3x-2z=10$, then substituting for y. Put $z=\lambda$ to read the printed base point and direction. An equivalent direction is $(2,7,3)$, with the parameter rescaled consistently. Both the base point and direction satisfy the two respective constant/direction conditions, so every point of the line is in both planes. The free third variable is essential; setting it permanently to 0 would produce only one point.