Solving systems of three linear equations in three variables: substitution

0:00- [Instructor] What we're gonna do in this video is solve the system of three equations with three unknowns. And there's several ways we can approach this. You can use elimination, but here we're gonna use substitution. So pause this video, have a go at this before we do this together.

0:14All right, now let's substitute. Now, what we can do is we can pick any one of these equations, solve for one of the variables, and then substitute that expression for that variable into the other two and then we will have a system of two equations with two unknowns.

0:29Now I'm just gonna start with this top equation, solve for X, and then substitute for X in the other two, not just because this one's on top, I don't have to pick the one that's on top, but because it feels pretty easy to solve for X, I have no coefficient out here, so let me just do that.

0:43So if I were to solve for X, I could just subtract 2Y and Z from both sides. So I could get X is equal to -2Y minus Z plus 10. All I did is I subtracted that from both sides. And now I can substitute this in for X in these other two equations. So let's start with this equation in blue right over here.

1:09So, maybe I'll do that over here. So this is 2 times X, X is -2Y minus Z plus 10. And then let me close the parentheses. Plus Y plus Z is equal to 11. So all I did is instead of an X, I wrote this. So it's incorporating the constraint from that first equation.

1:32If we know that X has to be this then we can just say that let's just put that right there in that equation. Now let's solve, or at least simplify a little bit. So if we distribute this two onto all of these terms, we end up with -4Y minus 2Z plus 20 plus Y plus Z is equal to 11.

1:57And let's see, I can add -4Y and Y to get -3Y, and then I could add -2Z and Z to get a -Z. And then let me subtract 20 from both sides. And then I would get a negative, a -9. And if I want everything to be positive, I can multiply both sides by -1 and I would get 3Y plus Z is equal, 3Y plus Z is equal to, is equal to positive 9.

2:31Now lemme do the same thing with the red equation. For X, I will replace it with

2:36- 2Y minus Z plus 10. And then I have plus Y plus 3Z is equal to 14. So lemme try to simplify this.

2:49- 2Y plus Y is -Y, - Z plus 3Z is positive 2Z. And then if I were to subtract 10 from both sides, I would get this is equal to 4. Yes, I think I did that correctly. And now let me see, let me solve for Y here, and I can substitute back into that equation. So if I solve for Y I get

3:18- Y is equal to -2Z plus 4. Were multiplying both sides by -1. I get Y is equal to 2Z minus 4. Now, let me substitute this in for Y in that equation. So I get... So, just to be clear what I did. I just took this original solve for X, I substituted into that and that.

3:42I got two equations with two unknowns. I got this equation right here and I got this equation. I could say this equation, which is the same thing as this equation. And so two equations with two unknowns, I can still solve with substitution. I could solve with elimination now, but I'm gonna do it with substitution.

4:00So I'm taking this and I'm substituting back into that. And so if I were to do that, I am going to get 3 times 2Z minus 4, 2Z, whoop, different shade of red, 2Z minus 4, let me close the parentheses, plus Z is equal to 9. Distribute this 3. I get 6Z minus 12 plus Z is equal to 9.

4:30I can add that to that and I get 7Z. And if I add 12 to both sides, I get 21. Divide both sides by 7 and I get Z is equal to three. So now I'm in the home stretch. I've solved for one of the variables, now I just have to keep substituting back in to solve for the other ones.

4:48For example, if I wanna solve for Y, we know already that Y is equal to 2Z minus 4. So if we substitute Z equals 3 there we get Y is equal to 2 times 3 and then minus 4. Minus 4. So Y is equal to 6 minus 4, which is equal to 2. So we have our second variable there.

5:09And now we can substitute both of these in into this first equation. We can solve for X. X is equal to -2 times Y, which is positive two. I'm trying to keep everything color coded. And then we have minus Z. Z we solved is equal to 3. And then we have plus 10. Plus 10.

5:32And so this is going to be equal to -4, that's that part, minus 3 plus 10. That's -7 plus 10 is equal to 3. And so we got X equals 3, Y is equal 2, and Z is equal to 3. If we wanted to write that as coordinates, we could write that X is equal to 3, Y is equal to 2, and Z is equal to 3, if we assume that we're looking at X, Y, Z coordinates.

6:05That's where these three planes, if you were to visualize it in three dimensions, are intersecting in a point.