00:01
They want us to solve this system of linear equations.
00:04
Now, the first thing you might notice is that we end up with less equations than we do variables.
00:13
So we're going to need to figure out what our pivot columns are going to be.
00:17
So remember the pivot columns, when we have in this upper triangular form, is just whatever is the first in each of the rows.
00:25
So we have 7x, 2 .y, and 3w.
00:30
And what we get from this, since z does not have its own pivot column, that means z is going to be what's called a free variable.
00:41
So depending on what value for z will take, our solution will be different.
00:47
So it can essentially be anything for z.
00:50
And this kind of implies, or this does imply, that we have infinitely many solutions.
00:55
But we'll get to that when we get to it.
00:57
So let's just go ahead and solve for what we can first.
01:02
Alright, so just like we did before, we're going to start with our last equation.
01:07
So 3w is equal to 6, divided each side by 3.
01:10
So we get w is equal to 2.
01:13
Now we can go on to our next equation.
01:15
So we have 2y minus 3z minus 4w is equal to negative 2.
01:22
Now, whenever we have a free variable in the equation, what we're going to end up doing is solving for our pivot variable.
01:32
So we want to solve for y in this case.
01:34
But let's go ahead and first plug in w.
01:37
So doing that though, so this is times 2.
01:41
And let's simplify that down.
01:44
So that's going to be 2y minus 3 z minus 8 is equal 10 negative 2.
01:50
So what we're going to do is add 3z plus 8 to the other side.
01:54
So that gives us 2y is equal to 3z.
01:59
And then when we add 8, that should give us plus 6.
02:03
Now we're going to divide each side by 2.
02:05
So that's going to tell us that y is equal to 3 -half z plus 3...