In Exercises 5–14, an objective function and a system of linear i...

Question

In Exercises 5–14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

Accepted Answer

Welcome back. I am so glad you're here. We are given this system of inequalities and we are to do several things with it. We need to graph the system of inequalities. We need to evaluate the objective function at each corner of the graft region. The objective function here is this Z equals four X plus six Y. And then finally, we will identify the values of X and Y for which the maximum value is found. So let's start with the graphing, going to draw a rough sketch of a coordinate plane. We have a horizontal X axis, the vertical Y axis, they come together at the origin in the middle. Now I'm going to focus on the first two inequalities. The first one is X is greater than or equal to zero? Where is it true that X is greater than or equal to zero? Or where is it true that X is positive? Well, that's anywhere to the right of and including the Y axis. Now, the second one we're looking at is why is greater than or equal to zero? Where is it true that why is positive? Well, that's anywhere above and including the X axis. So where is it true that both X and Y are greater than, or equal to zero? Well, that's going to be the first quadrant here. So I'm actually going to draw kind of a zoomed in coordinate plane where we still have our X and Y axis. But we're focusing just on this region, the first quadrant in colluding those parts of the X and Y axes that touched the first quadrant. Now we have two more inequalities to graph. So let's look at three X plus seven Y is greater than or equal to 21. Well, we recall from previous lessons that when we are graphing in inequality, we want to first set it equal to and equality or in equations. So three X plus seven Y equals 21. Then we ask ourselves is this line going to be included for our equality? And yes, this is greater than or equal to. And in fact, all of them are or equal twos. So there'll be solid lines, including those lines for everything that we're graphing here. Now we notice this is in fact an equation for a line. So we can plug zero in for X and zero in for Y to figure out the intercepts at the Y and X axis. So let's start off plugging zero in for X. We've got three times not X but zero plus seven Y equals three times 00. So we've got seven Y equals 21 divide both sides by seven and we get Y equals three. So we have a point at 03 from the origin. We are going neither left nor right for our X value of zero. And we're going up three units for our Y value of three. And we'll label that 03 right now. Let's figure out the X intercept. We're plugging zero for Y three, X plus seven not times Y, but now zero equals 21 7 times zero is zero. So we have three X equals 21 dividing both sides by three, we get X equals seven. So you point at 70 from the origin, we're going to the right seven units for our X value. And that's approximately here and neither up nor down for the Y value of zero. So 70, labeling this and now we can draw a line connecting those two points. And now we ask ourselves for this inequality. Three X plus seven Y is greater than, or equal to 21. Are we talking about this region down into the left of our line or up and to the right above our line? And in order to figure out which region we're talking about, we can choose a test point. I'm going to choose +00, which means we'll plug in zero for the X and zero for the Y values and see if this inequality is true. So we've got three times not expert zero plus seven, not times Y but zero is greater than, or equal to 21. The left hand side three times zero and seven times 00. That's all just zero. So is it true that zero is greater than, or equal to 21? It is not true. So we are not talking about the region That includes the test zero. We're not talking about the region below and to the left of the line, we're talking about the region above and to the right of the line. So now we can adjust what we've shaded in here. We've got one more inequality to graph. We've got 10, X plus seven Y is less than or equal to 70. Recalling that. Now we are going to set this equal to an equation. So we have 10 X plus seven Y equals and that line will be included. So we can make it solid. This is a line. So we can figure out the X and Y intercepts, let's figure out the Y intercept first plugging zero in for X. So 10 times not X but zero plus seven, Y equals 70 10 times 00. So we have seven Y equals 70 dividing both sides by seven, we get Y equals 10. So we have one point at 0 10 from the origin. We go neither left nor right for the X value zero. And we go up 10 units for Y value of 10. So we label this 0 10. Now we'll plug in zero for Y and get the X intercept 10, X plus seven times not Y but zero equals 77 times zero is zero as we have 10, X equals 70 dividing both sides by 10, we get X equals seven. So the 0.7 10, we already have that point or 70, not 7 10, we do have the 100.70 already marked. So we can draw a line connecting those two points 0 10 and 70. And now we've got that section, We need to figure out if we're talking about the section below into the left of this new line or above into the right of this new line. So again, we can choose the test point. The test point does not have to be in the region that fits all four of the inequalities. It just has to be either below into the left of our line or above into the right of our line. So I'm going to use 00 again, which is below into the left. So plugging zero in for X, we have 10 times zero plus zero and for Y will be seven times zero is less than or equal to 70. Again, the left hand side simplifies to zero. So is it true that zero is less than or equal to 70? Yes, that one is true. So this time, it does include the region that is down to the left of our new line. So we are talking about this region that forms a triangle. And now recall that we said earlier that we were including the AX and Y axis for these areas. So I'm going to connect the points 0 10 and 03 along the Y axis. There's our graph. Next, we need to evaluate the objective functions. Remember when we said we'd do that a long time ago. So the objective function formula here is the equals four X plus six Y. And what we do is we plug in the values for each of the corners of our graph. So the first one I'll choose is 03. And we'll be putting zero in for the X value because that's the X value here. So four times not X but zero plus and that will plug in three for the Y value. So six times not why, but three now simplifying four times zero is zero plus six times three is 18, zero plus 18 equals 18. Now the next corner I'll go up to 010. So we have plugging in for X four times again, zero plus six times. Plugging in for Y now 10 equals four times plus six times 10 is 60 0 plus 60 is 60. Now, the last corner here we've got 70. So plugging in four X will plug in a seven. So we have four times seven plus plugging in for why we have a zero. So we have six times 04 times seven is 28 plus six times +00, 28 plus zero is 28. And last, we just need to determine the maximum. So that's the highest value between 18 60 28 that one's 60. So where X equals zero and Y equals 10, our Z value is 60 and that is the maximum value. Alright, we've done it. We've got our graph, we have got our values for Z and we've got the maximum value. Now lets find the matching answer choice. So looking at our graphs, we want points at 030 10 and 70, we'll both A and B have those points. Now when Z the Z value for the 700.3 is going to be 18. Well, we've got for answer choice a we've got at 03, a value of 18. Alright. Now at the 0.0 10 we want a value of 60. We've got that. Now at the 600.70, we want a value of 28. We've got that to answer choice. A looking good. Now we want a maximum value of 60 where X equals zero and Y equals 10. That's exactly what we have. Answer choice. A works well done. We'll catch you on the next one.