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In this video, we're going to cover linear programming.
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In this particular question, we're being asked, what are the optimal values of x and y, and what is the maximum value of our objective function? okay, so this pretty much states that we're trying to find out what is the maximum possible value of the objective function, which is our equation up here.
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I'm going to abbreviate with objective function, given our constraints.
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The constraints are the inequalities that you see below.
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So they're asking for the optimal value of x, the optimal value of y, and the maximum value of c, given this linear programming objective function.
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So we have our constraints.
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We have 4x plus 10y is less than or equal to 17, 15x plus 18y is less than or equal to 39, and then you have x is greater than or equal to 0.
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And then y is greater than or equal to zero.
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Now the cool trick with these last few constraints is that that pretty much means that we're operating solely in the first quadrant of the coordinate plane.
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All right.
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So this first quadrant where all values are positive on your coordinate plane because it states that if x has to be greater than or equal to zero, that means every value to the right of the y axis is in play to being a possible value.
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Same as if y has to be greater than equal to zero.
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That means no value below it, every value above the x axis is within play in consideration for our maximum values.
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What we have to figure out is how these other inequalities come into play, and we'll have to graph those.
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We'll grab them so we can see a visual representation of exactly what area we're working with within the first quadrant.
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So let's start with our first inequality.
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4x plus 10y is less than or equal to 17.
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Oops, not what i meant to do.
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We're going to pull this back row.
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So we're going to start with our first inequality, and we'll zoom out just to make sure we have enough space.
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So to solve, to be able to graph that inequality, the preferred method that i encourage is just set either x or y to zero and find the corresponding x or.
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Or y intercepts.
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So in that first video, excuse me, in that first inequality, you have 4x plus 10 y is less than or equal to 17.
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Just set one of these to zero.
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We'll start with the x.
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We'll set x to zero first.
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Four times zero plus 10y is less than or equal to 17.
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All right.
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This becomes zero, so you're left with 10y is less than or equal to 17.
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To 17, divide both sides of the inequality by 10, and you get y equals 1 .7.
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All right, 17 divided by 10 gives you 1 .7.
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All right.
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With that, you have your first point.
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All right, when x is 0, y is 1 .7.
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And these points will be helpful when we visualize what this means in the quadrant.
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For the next one, just set the other variable.
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In this case, y set that to 0.
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You have 4x plus 10 times 0 is less than or equal to 17.
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This becomes 0, so you're left with 4x is less than or equal to 17.
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Divide both sides by 4x is less than or equal to 4 .25.
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All right.
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So in this instance, you have a second point of your inequality, and it is 4 .25, when y is zero all right let's find the intercepts for our next inequality hopefully sorry won't drag like it's supposed to there we go our second inequality reads 15x plus crazy 18 y is less than or equal to less than or equal to 39 all same procedure.
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We're going to set each of these values to zero.
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We'll start with the x.
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15 times 0 plus 18y is less than or equal to 39.
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This becomes 0.
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18 y is less than or equal to 39.
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Divide both sides by 18.
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You get y is less than or equal to 2 .167.
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This is a repeating decimal that goes on and on on, but this question is asking us to round to the nearest third decimal place.
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So we stop it at 2 .167.
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So for this ordered pair of this inequality, you have when x is zero, y is 2 .167.
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All right.
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We're doing the same thing, just setting y equal to zero.
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So you're looking for 15x plus 18 times zero is less than or equal to 39.
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All right, at times 0 is 0.
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15x is less than or equal to 39.
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Divide both sides by 15, x is less than or equal to 2 .6.
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This one actually is a pretty clean integer.
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So with this one, your order pair, when x is 2 .6, your y value is then 0 .0.
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All right, so we have our points.
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We have our points now.
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Zoom out.
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And we're going to go ahead and visualize this up top.
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Okay? we're going to go ahead and draw our quadrant again.
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Just to show you where the answer should fall in this inequality or in this linear programmed model.
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All right.
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So we've got, we have a funky looking y -axis.
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Trying to get this straight.
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We have our y -axis.
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We have our x -axis.
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Again, this will not be to scale, so if it's a little off, do not fret.
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The values are still accurate, even though they don't look at it.
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All right.
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So we're going to graph our first inequality.
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All right.
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And we can graph it because we have two points.
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Our first point is 0 .1 .7.
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And eyeball in it, zoom in a little bit, so i can try to get as accurate as possible.
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1 .7 would be right here.
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And then we have our second point of this inequality is 4 .25 comma 0.
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So 4 .5 .0 sitting right there on the x -axis.
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This is my straight edge.
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Just try to get this as close to being a straight line as possible.
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All right.
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And there we have our first graft inequality.
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And remember, in all of these inequalities, they're both less than.
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All right, the 4x plus 10y is less than negative, or excuse me, less than 17.
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It means everything below that inequality is the possible constraints that we're looking for.
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All right, all those values have to be less than.
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Okay.
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For our second inequality, we have the points 0 .2 .167.
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So this one was, i want to say yellow.
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So for this one, we have 2 .167.
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This is our y intercept.
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And then we found our x intercept 2 .60.
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So 1 .2 .60.
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I can try to do my best to get this as accurate as possible.
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Looks pretty close.
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All right.
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All right.
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So right now, we've graphed.
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It looks super small, but we've graphed these points, these points corresponding to our first inequality and these points correspond to our second inequality for our constraints...