For this problem. We're being asked to maximize the profit for the mural manufacturing company, which is making plasma screen television sets. So let's see what our equation is for the profit that this company is going to make. Well, there are two types of television sets that it makes. One is a flex scan, which makes $350 of profit. So I'm just gonna mark here that my ex up one is going to be my flex scan model. And the second kind that it makes is the panoramic. And there's $500 of profit for that one. That's my panoramic in the ceramic. Okay, so this is my profit equation. $350 of profit for each flex scan. $500 of profit for each panoramic. Now, what are my constraints? Well, one constraint we have is the assembly line. Okay. Now, on the assembly line, we're told that the Flex scan requires five hours and the panoramic takes seven hours and at most I've got 3600 hours available on the assembly line. Our second constraint comes at the cabinet shop and there were told that the flex can only takes one hour in the cabinet shop, the panoramic takes too. Okay. And adding all of those up, I have 900 hours total available in the cabinet shop. My third constraint happens in testing and packing like that red. So in the testing and packing area, there were told that the flex scan takes four hours, as does the panoramic. So each one of those takes four hours altogether as 2600 hours. Now, I'm going to simplify this one slightly. I'm going to just divide everything by four just to keep my number's a little smaller. And this becomes 650. I do need to remember, though, that if something comes back with the slack variable for that equation, I need to remember that I made an adjustment. Will have toe, um, work on that at the end if something comes back for that one. Okay, Let's look at how I can add my slack variables so I could enter these into my grid. My first equation, my assembly. I'm just gonna write these in blue, so they kind of stand out a little bit. Five x up one plus seven x up to plus my first slack variable is going to equal 3600. My second equation. I'm going to add my second slack variable, and that will equal 900. And then for my third equation, I have my third slack variable. And that equals 650. Okay, Knowing all of those, I can now put together my grid. I have two X variables. I have three slack variables and I have ze. So let's enter our blue constraint equations in here. If I pull out all of my coefficients, I have 57 and one I don't have any of those variables and it equals 3600. My second equation is 12 It has the second slack variable in that equals 900. And my third one is one and one, and it has my third slack variable equaling 6 50 and my indicator row at the bottom I'm going to take and I'm gonna market with a blue arrow appear at the top, my equation for Z and I'm going to set everything equal to zero. So when I pull all of the terms over to the left hand side, I have negative 350 x of one minus 500 except to no slack variables and dizzy. Hey, so here's my grid. Okay, Now, I know that I'm going to be redoing my grid down here, So I'm going to going to set this up really quick. All of my same variables. Now let's take a look at where our pivot point's going to be. If I look at my indicator row, negative 500 is the biggest negative number I have. So we'll be looking at the ratio of my Constance to the X Up two column. And when I do that, 3 36 100 divided by seven is about 514. Yes, that's got it. With a decimal 900 divided by two is 450 and 6 50 divided by one is obviously 6 50. So the smallest one is the two. There's my pivot point. So the row with the pivot point my second row is not going to change. That stays the same to get rid of my other rose. Well, to get rid of the top row, I've kind of don't have a whole lot of room right there, so I'm just gonna put this off to the side. I'm going to take negative seven times the second row, plus twice the first row. And I'm going to put that back into the first row all the way across in my new grid. And when I do that, I end up with three 02 negative. 700 900. Remember, the goal was to make every number in the same row or same column with the pivot number all equal to zero. Okay, well, what about my third row? Well, in order to get that one that I'm circling here to get that to become a zero, what I'm gonna do is take negative the opposite of the second row, plus twice the third row, and that will go into the third row position. Doing that across the row gives me 100 negative. 1 to 0 and 400. Okay, finally, we need to dio are indicator row at the bottom. To get rid of that, I'll take 500 times the second row plus twice, the fourth wrote. And that's what I'll put in the fourth row all the way across. And when I dio, my output is negative. 200 00 500 02 and then 450,000. Hey, we are not done. We still have a negative in our indicator, wrote negative 200. So to find our pivot point, we're gonna look at the values in the x of one column or Exubera column and compare them to our constants. So no. 1 900 divided by three is 300 900 divided by one is 904 100 divided by one is 400. So this three right here gives me my smallest ratio. That's my new pivot point so and bring that down a bit. So I've got some room toe work and we make our next grid. So the road with the pivot point does not change. So I'm just going to copy Row one all the way across, as is. Hey, I want to get rid of every other number in that except one collar. I want them all to equal zero, so let's look at the second row to get rid of that. I will take the opposite of the first row, plus three times the second row, and that will go into the second row space across the board. When I do that, I end up with 06 Negative to 10. 00 1800 Right. Third row. Well, again, I have the same number in the second row and the third row. So my equation is gonna look the same. Negative are sub one plus three times the third row, and that will go into the third row position. Doing that gives me 00 negative. 2460 and 300. Now we just have to get rid of that negative 200 in the bottom row. To do that, I'm going to take 500 times the first row. I'm sorry. Timmy did write 500. I met, right, 200. Try that again. So 200 times the first row, plus three times the fourth row. And I'm going to put that into the fourth row position. Doing that all the way across that row gives me 00 401 100 06 and one million, 530,000. There are no more negatives in my indicator row, so we are complete so I can start reading the answer off of this grid. Uh huh. Except one just has one non zero. So I could say that three X up one equals 900 or X up one equals 300 except to also has only one non zero. So six X up to equals 1800 or accept to also equals 300. So without looking at any of the other variables, I know that the optimal number of units to make our 300 of each type of television set Now, I can also see that from my slack variables. S sub one and s up to are both going to be zero but s up. Three is not s up. Three. I could say six s up. Three equals 300 or as sub three equals 50. Now, I'm just gonna come back up to the top and remind you s up. Three. If you remember, I made those numbers a little bit smaller. I divided it by four. Everything in my first one divided by four. My second one. So I really need to multiply this by four to go back to the original equation. So s up three for my original equation is 200. So what does that mean? Well, that means that in that third constraint, which was my testing and packing facility, I have 200 unused hours. So I've got some, You know, everything else. I maxed out to capacity, but I still had room available here in that particular department. And what is? We come back so you can see this little bit better. What is my maximum profit? Well, six z equals 1,530,000 or Z equals 200 55,000. That is the maximum profit for selling 300 of each type of television set.