INSTRUCTOR: Find parametric equations for the line formed by the intersection of plains x plus y minus z equals 3 and 3x minus y plus 3z equals 5. First thing that I want to notice is because these are the intersection of two planes, there are infinitely many solutions. And because we want parametric equations anyways, I'm going to go ahead and make a choice that x is going to be my parameter. X equals t.

I could safely do this with any of the variables, but this is how I'm going to start. This changes my equations into t plus y minus z equals 3 and 3t minus y plus 3z equals 5.

Next, I'm going to try to eliminate a variable, which I can. I have a y and a minus y. And I might use substitution. And we'll get to that in a minute. And if we do this, we'll have 4t, the y's eliminate. If I add those equations, then I'm getting plus 2z equals 8.

So I can solve for z and say that z equals 8 minus 4t divided by 2, which will be 4 minus 2t. So z is equal to 8 minus 4t.

I need an equation for y. So I can take any of these that I have and replace my z in them. So I'm going to use the first equation. I have t plus y minus z. So I'm going to have minus instead of a z, 4 minus 2t equals 3.

Distributing and combining some like terms, I have y minus 4 plus 3t equals 3. And then adding subtracting some terms, I see that y equals negative 3t plus 7. So I'm going to write this as y equals 7 minus 3t. 7 minus 3t. So my parametric equations are x equals t, y equals 7 minus 3t, and z equals 8 minus 4t.