# Thread: Line of Intersection of Planes

1. ## Line of Intersection of Planes

Problem:

Find parametric and symmetric equations of the line of intersection of the two planes.

$2x+y+z-5=0$ and $3x+2y+2z-8=0$

$x=2$
$y=t$
$z=1-t$

No symmetric equations.

I'm stuck on this one.
I tried multiplying the first equation by $-2$ and did the elimination to solve for $y$ or $z$ but I get messed up.

2. Originally Posted by Morphayne
Problem:

Find parametric and symmetric equations of the line of intersection of the two planes.

$2x+y+z-5=0$ and $3x+2y+2z-8=0$

$x=2$
$y=t$
$z=1-t$

No symmetric equations.

I'm stuck on this one.
I tried multiplying the first equation by $-2$ and did the elimination to solve for $y$ or $z$ but I get messed up.
Two equations and three unknowns so infinite number of solutions (therefore there's a parameter). There are an infinite number of ways of expressing the answer. One such is to start by letting y = t, say, where t is any real number.

Then:

2x + z = 5 - t .... (1)

3x + 2z = 8 - 2t .... (2)

Solve (1) and (2) simultaneously for x and z in terms of t. I suggest 2 (1) - (2):

x = 2.

Now substitute x = 2 into (1) and solve for z.