00:01
Okay, so this question says, solve the linear programming problem using the simplest method.
00:05
I do not know if this is the simplest method, but i'm going to show you how to do it.
00:09
If there's an easier way, you can tell me.
00:12
So it wants us to maximize our objective function, which i've drawn here, subject to all of these constraints.
00:20
So what i like to do when solving these is i graph them.
00:24
I determine the vertices, and then i plug each of those vertices into the objective function and figure out which is the largest, the maximum, because that's what we were asked to do.
00:35
Maximize.
00:37
Okay, so to graph these, i went ahead and color -coded them.
00:41
I 100 % recommend color -coding your lines because it will get real confusing about which is which.
00:47
I've drawn my graph here.
00:48
This is my axis.
00:54
So, x2 equals zero is the first line i've drawn because this is the point where this up and down line is zero, and so it's going to be zero no matter how far left and right you go.
01:05
Same thing with this line.
01:08
So i've drawn both of those.
01:10
Next, this is x1 plus x2 is less than or equal to 80.
01:18
So to plot this, we would plug in zero for each other variable and solve for our intercepts.
01:29
So in this case, it's just 80 for both x1 and.
01:32
And x2 and then i'm going to connect that line as straight as i can.
01:38
Okay, so that one didn't take a lot of calculating.
01:41
I'm just working my way up here.
01:45
So to find the intercepts, put that down here, intercepts.
01:53
So we have 5x2 plus 2x, oops, 5x1 plus 2x2, and i can just say equals 91.
02:06
I'm dealing with where it crosses the axis.
02:11
Oh, before we do that, though, we need to talk about shading.
02:15
So these are both greater than, greater than for the red.
02:18
So i'm going to shade above and to the right of these lines here.
02:26
And then this equation is less than for the blue, so i'm going to shade below this line here.
02:33
Okay, we'll come back to shading after we find the next intercepts for green.
02:38
So in order to figure out where it crosses the x1 axis, i'm going to plug in 0 for x2 and solve, which essentially just gets rid of this term, and i'm left with 5x1 equals 90.
02:55
And that would be divide both by 5, and i get 18.
03:02
So on the x1 axis, i'm going to put a point at 18, which is approximately there.
03:07
I've just drawn these tick marks to be eyeball accurate.
03:11
They're not perfectly straight.
03:15
And then to solve for my x2 variable, i'm going to do the same thing, but instead i'm going to cancel the 5x term, so i'm just left with 2x squared equals 90.
03:27
Divide by 2, divide by 2, x2 equals 45.
03:33
So 1, 2, 3, 4, so halfway between the 4 and the 5.
03:37
And then i will connect those like so.
03:41
And this equation is also less than 90.
03:45
So i'm going to shade this region below.
03:49
Okay, now to find the intercepts of this other equation.
03:56
I've got 5x1 plus x2 equals 70.
04:02
So the x2 intercept is easy, it's just 70.
04:07
So on my x2 axis i'll put a dot at 70.
04:11
And then i'll do 5x1 equals 70, divide, divide, i get x1 equals 14.
04:20
So i'll put my dot somewhere past the 10.
04:24
I didn't mark these, but these go by 10s.
04:27
I did mark the 80 for each of them.
04:31
And then i will connect that line as best and as straight as i can.
04:40
And so it's below, once again, so i'm going to shade below, but not going to, of the region that's already been determined.
04:48
So i'm going to step below the green line and the yellow line.
04:53
Okay, so that was step one, graph.
04:58
Step two is determine the vertices.
05:01
So i'm going to get all this out of the way.
05:03
If you did not get a chance to write that down and you want to go back, pause the video, whatever you need to do.
05:08
But we need to give ourselves some room, but i'd like to still be able to see the graph as we work.
05:16
So getting all these little pieces out of our way.
05:18
Okay, determine the vertices next.
05:27
So the vertices are of the feasible region.
05:30
The feasible region is this little quadrilateral here.
05:35
So the corners would be this.
05:37
I'll call that a.
05:40
This guy up here is b.
05:43
We're gonna call this guy c and this guy d.
05:48
So remember it needs to be below all these, the green and the yellow and below the blue and above the two red lines.
05:58
So i colored it in with black here, this little quadrilateral piece.
06:06
So, determine the vertices.
06:09
A is at zero zero.
06:11
It's the intersection of these two lines...
