Intersection of planes and calculating the equations

Question

Two planes:

4a - 6b + 8c = 3

-9a + 12b -3c = -10

What steps are there to calculate the equation of the line when the planes intersect?

Answer 1

Multiplying the second equation by $4$ and adding to the first one we get $$x=1+\frac{13}{10}y$$ defining $$y=10t$$ we get $$x=1+13t$$ and plugging this in one of the equation above we get $$z=t$$ so our line is given by $$[x,y,z]=1,0,0]+t[13,10,1]$$ where $t$ is a real number.

Answer 2

I think a simple way it's the following.

Let $z=t$ and solve the following system: $$2x-3y=2-4t,$$ $$-3x+4y=-3+t$$ and $$z=t.$$ Can you end it now?

I got $$(1+13t,10t,t).$$