

College Algebra: Solving for Maxima-Minima



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About this Lesson

  • Type: Video Tutorial
  • Length: 11:06
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 119 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Linear Programming (2 lessons, $3.96)






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Review for Exam 1

~ Alexis7

Review for Exam 1

Now I want to take a look at how we could use these graphs of systems of inequalities to actually solve problems in the real world, believe it or not. These are known as linear programming questions. Here's the basic idea. Suppose you have like a business, for example, and you want to maximize profits or minimize costs. Well, that's going to require you to try to get the biggest value a certain function can take on. But there are constraints. You don't have sort of infinite resources. You don't have sort of infinite wealth. You don't have sort of infinite human power to do your stuff. So, you have these constraints, but given those constraints you want to, for example, maximize profit or minimize cost or maximize something else.
So, how do you actually maximize or minimize some linear function if you have constraints? Let me illustrate this with a particular numerical example and then we'll take a look at some actual real world problems. So, suppose that I want to maximize profit and suppose profit is given by x + 5y. So, if you give me a point x and y I could find the profit by just plugging in here and that tells me what the profit is. But there are some constraints. Here are the constraints. So, this is with constraints that 5x + 8y 120. So there's some constraint on something. Then 7x + 16y 192. There's another constraint. X cannot be negative. Maybe x is the number of workers you have and y cannot be negative. Maybe that's the number of buildings you rent or something.
Now with all these constraints the question is, "What are the values for x and y that make this as big as possible?" Well, I'm going to tell you how this works using linear programming. In fact, the first thing you have to do is graph this region and see what this region looks like. That's actually going to be a teeny bit involved because I've got to graph this inequality, that inequality, this inequality and that inequality, a lot of inequalities that need to be graphed. So, let's get going. Well, if I set up my axes just like this, then what I'm going to do is first I'll just graph this line and then figure out how the shading goes and so on and so forth. So what do I do here? Well, for this line what I might want to do is find the intercepts.
So, if I let y = 0, that's going to show me where the x intercept is. Now, watch this. I'll just put in 0 for y, so that goes away. I see 5x, let's just pretend it equals 120. So, therefore x would be 24. So, I see that x would be x 24. Which let's say is way over here, this is 24. So, this is the point of (24,0). That's the x intercept for this graph. How would I find the y intercept for that graph? I would let x = 0. So, look back at this picture. I let x = 0 and I see 8y would equal 120. So, I'd divide both sides by 8 and I would see that y would equal 15. So, in fact, I would see I go up to 15 here. So, this is a (0,15).
So, we could see what the line is. I can make it a solid line because I'm allowed to actually have equality. Now, I could draw the line all the way out to here, but I'm going to actually do something a little bit tricky. I'm going to notice that x has to be positive and y has to be positive. So, I'm not going to be looking in this region at all. I'm only looking at the first quadrant, positive x and positive y. So, I'm only going to connect these points right in between, since I know that's where my regions going to be living anyway. So, there's that picture. Of course, I keep going, but I know I'm going to get rid of those because of these conditions.
Now, do I go below or above? Well, I put in (0,0) and see. If I put in (0,0) I see that in fact 0 120. So, I'm going to color below. So, remember that. I'm not going to actually do that right now, but I'll remember I'm going to color below. Now, let's pick another color here and look at this line. Well, I'm going to do the same procedure. If I let x be 0, I'm going to find out where this intersects the y-axis. I just divide both sides by 16. I see y would equal in this case 12. So, we come over here and put a 12. So, this is the point (0,12). Now, if I want to find the x intercept, I put y = 0 and I see 7x would equal 192. If I divide both sides by 7 I would see that x would be around 27.4 something. So, let's put that over here. This would be (27.4,0).
Again, I can't go out of these regions so I might as well connect these two points right in the first quadrant because of these conditions, x has to be positive so I'm to the right of the y-axis, y has to be positive so I'm above the x-axis. Let's not even bother putting anything else. So, there it is. Now, do I color below or above? Putting (0,0) and I see 0 < 190. So, I want below. So, where is the intersection of all these things? I've got to be in the first quadrant and I have to be below the green and I have to be below the red. So, that region actually is this region right here, below both of them at the same time and in the first quadrant.
We can see that region actually is just a whole bunch of little line segments put together to make this shape. This line segment, this point of course is (0,0) that's the origin. Then line this line segment and then this line segment up to this point right there, the point of intersection of those two lines. Then this line segment up to this point. Now, here's a really cool fact. If you are trying to maximize or minimize a function that's linear, no squares, no cubes, no square roots, no logs, nothing, well then if you think about it, it would just be a plane. If I put a plane, I'm going to cover this up, but if I put a plane and tilt it on top of it, if you think about how that shape would be. It turns out that the minimum and maximum are actually going to occur at one of these vertices.
So, in fact, if you want to find the minimum or maximum of a linear function, linear programming tells you you just need to look at the vertices and plug those vertices in and the biggest one will give you the biggest value you can take. The smallest one will give you the smallest value you can take. So, one thing I need, by the way, is to find the intersection of those two lines. So, one thing I've go to do is actually solve these two equations. 5x + 8y = 120 and then 7x + 16y = 192. I have to actually solve them using any kind of method you want. In order to find out what this value is. One thing you can do, for example, is multiply this equation through by, let's say, two. If did that, I'm going to multiply this through by two. If I did that what would I see?
I would see 10x + 16y = 240. Now, if I subtract. Let's take this and subtract that, this and subtract that, this and subtract that. I see 3x and notice these drop out. See, I'm using the elimination method. This minus that equals 48. So that tells me that x would have to be 16. I can now take that value for x, plug it back in to one of these equations and solve for y. I would see, if you work this out that y = 5. I'll let you check that. So, it looks like x is 16, right here and y is 5. Does that make sense? Certainly that makes sense. This looks like 16 and this is 24. If this is 12 then that could certainly be 5. So, at least my picture is reasonable.
So, I actually had to solve those things simultaneously to find that particular point. Now, I've got how many points? Just look at the colored region. Don't look at these points out here. Just look at the colored region. I've got this point, that point, that point and that point. One of those vertices will represent the maximum value this linear function takes on. The other one will represent the minimum value. So, if you want the maximum and minimum value of a linear function with constraints that are straight lines like this, you just need to look at the vertices.
So, let's just plug in, (0, 12) into (x, y). If I do that, what would I see? Let me just draw you a little mock up of this, a little mock up of that region. This is supposed to be a mock up of this. I'm just going to put the values of P at each of these vertices. So, if I plug in 0 for x and 12 here for y, I would see 60 for P. So, P would equal 60 at this point. If I plug in (0,0), I just see 0. So, P would equal 0. Zero profit here. If I plug in (24,0), I see 24. If I plug in (16,5) and work that out we'd see 41. So, what's the maximum profit we could have given this constraint? Well, it turns out the answer seems to be right here where x = 0 and y = 12 the profit is 60. Where's the minimum profit? I can get that too. The profit is 0 if I have (0,0).
So, in fact, the way to sort of solve a linear programming problem, that's when you're given a linear function that you want to either maximize or minimize, make it really big or really small, with certain constraints is to first graph the constraints. You get some region. Then, take a look at all the vertices of that region. Find those coordinates. In which case, that sentence might require you to solve some equations simultaneously to find out where they meet. But once you have that then just plug in each of those vertices that we found into here and the biggest value is the maximum. The maximum can never happen inside here are anywhere else. It will happen at one of these points.
So, actually linear programming is pretty easy. You just find the vertices of the boundary region you have given by the constraints. Plug those values back into this thing you're trying to make big or small and the biggest value is the biggest value everywhere. The smallest value is the smallest value everywhere. We'll take a look at some really neat real world applications of this coming up next. So, I'll see you there.
Systems of Equations
Linear Programming
Solving for Maxima-Minima Page [2 of 2]

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