Find the line of intersection between 2 planes

Question

Q: Determine the line of intersection S of the planes $E: x + 2y − 2z = 13$ and $F: x − y + z = 3$

Ok so I let $Z=t$ and then used the simultaneous equations to solve for x and y, however I think I made a mistake. This is how I wrote the process:

First equation $x+2y-2t=13$

Second equation $x-y+t=3$ \begin{bmatrix}x+2y-2t=13\\ x-y+t=3\end{bmatrix}

$x+2y-2t=13$

$x+2y-2t-\left(2y-2t\right)=13-\left(2y-2t\right)$

$x=13-2y+2t$

Solving $13-2y+2t-y+t=3$

...gets me $y=-\frac{-3t-10}{3}$

Then I subsitute Y into X

$y=13-2(\frac{-3t-10}{3})+2t$

Solving that gives me $13-2\left(-\frac{-3t-10}{3}\right)+2t=-\frac{20}{3}+13$

$x=-\frac{20}{3}+13$

My x does not have a t at all, how can I compute a line equation without a t?

I got

$y=-\frac{-3t-10}{3}$
$x=-\frac{20}{3}+13$

I don't think it's right because of how complicated it looks, I also don't think I can make a intersection line equation without a t for my x. Could someone tell me what I did wrong?

If what I did was right then my line of intersection would be

$r$=$$ \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} $$$=$$ \begin{pmatrix} \frac{20}{3}+13 \\ -\frac{-3t-10}{3} \\ t \\ \end{pmatrix} $$ $

Answer 1

You can replace equations E and F with any linear combination of themselves.
One is $E+3F$ which returns $x=7/3$.
Another is $E-F$ which gives $y-z=10/3$ .
Then $z=t,\; y=10/3+t, \; x=7/3$.
$ x$ does not have any $t$ , that is perfecty possible and licit, it means that the line has a constant $x$, i.e. that it lies on a plane orthogonal to x axis and intersecting it at $x=7/3$

Answer 2

I let $z = 0$, and solved this on pen and paper.

Answer 3

It’s perfectly reasonable for $x$ to be constant. That just means that the line is parallel to the $y$-$z$ plane, but it also means that you’ll have to describe the line with two equations instead of lumping everything into one expression as you might be used to doing, i.e., as $$x=x_0 \\ {y-y_0\over b}={z-z_0\over c}.$$ (Actually, the symmetric form ${x-x_0\over a}={y-y_0\over b}={z-z_0\over c}$ is is itself really a shorthand for two separate equations.)