Show that the line of intersection of the planes \[x+2y-z=2\]and\[3x+2y+2z=7\] is parallel to the line \[\langle x,y,x\rangle =\langle 1+6t,3-5t,2-4t\rangle\] Find the equation of the plane determined by these two lines.

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I can do the first part, the trouble I am having is finding the plane. How can I do this?

first, solve the system \[ \large \begin{cases} x+2y-z=2\\ 3x+2y+2z=7 \end{cases} \]

I would do it a bit differently I would cross the normal vectors of the two planes. The resulting vector should point in the same direction as the line.

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