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Furhiture Face Lift refinishes old wood furniture: Their uithin stanoneach worker works Independentl _ Process for refinishing chairs has 8 workers and _ Station h...

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Furhiture Face Lift refinishes old wood furniture: Their uithin stanoneach worker works Independentl _ Process for refinishing chairs has 8 workers and _ Station hishher_cwnchai . Assume inventory DUffes re alstatedd stations Each chair starts = Staffing Processing time (hours per betveen each station_ stripping stati= chair per worker) StrippingPrimingPaintingInspectionsuppose the start of the day there inventory of chaits the shop. That ( there are no chalrs within any of the stations between

Furhiture Face Lift refinishes old wood furniture: Their uithin stanoneach worker works Independentl _ Process for refinishing chairs has 8 workers and _ Station hishher_cwnchai . Assume inventory DUffes re alstatedd stations Each chair starts = Staffing Processing time (hours per betveen each station_ stripping stati= chair per worker) Stripping Priming Painting Inspection suppose the start of the day there inventory of chaits the shop. That ( there are no chalrs within any of the stations between tnam in any 0 chalrs? (The accuracy should be of two digits after the deciral Nlaces



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A furniture factory makes wooden tables, chairs, and armoires. Each piece of furniture requires three operations: cutting the wood, assembling, and finishing. Each operation requires the number of hours given in the table. The workers in the factory can provide $300 \mathrm{h}$ of cutting, $400 \mathrm{h}$ of assembling, and $590 \mathrm{h}$ of finishing each work week. How many tables, chairs, and armoires should be produced so that all available labor-hours are used? Or is this impossible? $$\begin{array}{|l|c|c|c|} \hline & \text { Table } & \text { Chair } & \text { Armoire } \\ \hline \text { Cutting (h) } & \frac{1}{2} & 1 & 1 \\ \text { Assembling } & \frac{1}{2} & 1 \frac{1}{2} & 1 \\ \text { (h) } & 2 & 1 & 2 \\ \text { Finishing (h) } & 1 & 1 \frac{1}{2} & \\ \hline \end{array}$$

Machine 73. Yeah. I'm gonna call a table of B because I don't want views in a t. I have one e for table plus C for chair. Plus a for armoire equals 300. I am 1/2 able. Plus I really have chair plus a equals 400. And then be blast Ready have seen. Plus two a equals 5 90 So get rid of my fraction in the 1st 1 I just don't find this fight too. I'm not him all by two. So two times 1/2 gives me one, two, two, 600. Multiply this by two, and I get one, three, 2 800 And I built my bottom one by two as well. And I get you breaking or 11 8 So now I'm gonna date road to Linus Roe one and bring it back in the road too. Wrote to minus row one back into row two. My top row stays thing You're lying. One, two, 2 600 by second row one minus one is zero three minus two is one to Linus to see round 800 minus 600 is too good. And then, or by I'm gonna take red. Um, negative, too. Times row one plus rose three back into row three. So, uh, I'm gonna put it in black. Negative. Too. Negative for negative. For negative. 1200. So zero negative. One zero negative. 20. And now I have to, uh, add road three and wrote to and put it back in row three. So Same broke free. Last road, too. Back in row three. So I get one damn. 2 600 010 200 00 01 80. So zero times chairs equals 1 80 I can't take anything. Time. Zero, get 1 80 So that's no solution, and we call that inconsistent.

Okay, So for this one, we're talking about a furniture factory that is going to produce be producing, um, chairs, cabinets and buffets. So we're going to say X's chairs. Why is buffets was very wise. Cabinets and Z is gonna be buffets on, although you can choose any three letters you want. Um, so we know that as far as like we have cutting, assembling and finishing. So it's gonna be 0.2 per chair for cutting 0.5 for buffets, points 3.5 for cabinets, 0.3 for buffets for total of 1900 50 hours. We know for assembly, it's gonna be 0.3 x plus point for why plus 0.1 z equals 14 4090. And then last but not least, this the finishing is 0.1 x plus 0.6. Why plus 0.4 z equals 2000, 160. So what we want to do is creator matrix off of this. But I'm gonna go ahead, multiply everything by 10 so I'm gonna have to five three in 19,500 three, four one and 14,900 and then 16 and four and tooth out. 21,600 for the So what? We have whole numbers instead of decimals. So the very first thing that I really want to dio has start creating this pivot pattern. So that way I have a number and then two zeros below it. So in order to do that, I'm gonna have to multiply the top by three. So I'm gonna have six, 15 nine and then 58,500 that I'm gonna I'm gonna multiply the second one by two. So it's going to be six eight to and in 29,800. Okay, so I'm gonna keep this as a to five. Three and in 19,500 for now. So then my first, that's going to be zero and then 15 minus eight s seven nine minus two is seven and then 58 500 minus minus 29 800. There's gonna be 28,700 now. Luckily, I can provide risk by seven to get 4100 and I can eliminate thes by seven to get one each. Okay, so then the next thing I need to dio is I need Teoh. Figure out what I need. Teoh at some knucklehead, multiply this bottom row by twos. That way I can eliminate it from the original top row. So to 12. Eight and 43,200 so to minus two is zero five. So 12 minus five will be seven eight minus three would be five. And then, of course, 43 1002 100 minus 19,500 is 23 700. So I have pivoted the first one. Now I am ready to pivot the second row. So what I want to do first is I want to multiply this by five. Then that's gonna be 20,500. So for now, I'm gonna leave the second row, as is, because it's about a simple fight. Is I'm going to get for this ground. So then I'm gonna have my to five minus 50 five minus three is two. And then I have 1000 left over. So now what I also notice I can do is I can divide by two now and just to make the math a little bit easier. Lips, That's what I'm going to dio. Okay, So now what I want to do is I'm gonna eliminate what? I just did this for us changing the the Matrix before it. And now I want to multiply this by seven. And this is gonna get me 28,700. So that way I can eliminate, so it's gonna be zero 07 minus five is, too. And then, of course, now I've got 5000, which I can eliminate this down by two to give me one and 2500, and then I have one final call him to pivot. So what I want to do is I want to go and just flat out subtract thes someone. I have 1001 And then, of course, 001 So I'm gonna do the first in the 3rd 1st. So this is obviously going to create zero, and then this is going to be 2000, cause it's 2500 minus 500 and then I'm going to go ahead and pivot the 1st and 2nd, so this is going to be zero, and the 4100 minus 2500 is going to be 1600. So now I have got everything down. Toe one and I have pivoted all three columns. So what this tells me is that there's gonna be 2000 chairs, 1600 buffets and 2500. I'm sorry. Um, 1600 cabinets and 2500 buffets.

So for this question, we have a table of difference desk models and the work that you have to put into manufacturer cuts in construction finishing. And we know that the company has assessment of ours for each type of step of stage. And we want to accounts how many of each desk we should be making. So if we let our variables be the different models say a secure fence. Model B is the office model. Nancy is very deluxe model, and we will make some equations out This so for cutting. We have 100 hours each week for cutting, so I work has to be 100. I have a Children's model. Takes two hours. The office three is it looks too. And the construction we have to Fay one for be on three, See? And we have 100 hours for this and for finishing. We have one, eh? One for bay hands to see. For this, we have 65 hours. Yes, So now we have our three equations, and when we saw this system, we will get the number of different models that were going to be producing. Now that's making augmented matrix to service. The difficult part is just passing the question and working out how to translate the information into these equations. From here on, it's quite standards. So I'm gonna let this be my room because it's got one here. This could be word suit on. This could be for three. Okay, so my first variable is going to be a We will have 1 to 2 second column. Could be bi one one three. See, it's going to be two three two ever written inside is 65 101 100. Okay. And that's straight in with some elimination. So things have minus two. Well are too. And minus 21 plus, uh, three to eliminate these two. So it's my real world is unchanged. My road suit is now zero minus one, minus one and minus. That's he answer. Three is zero one minus two minus 30. Okay, for my next step, I'm going to just eliminate this one hit. And Steven, I just stepped out my first ever. I'm just going to add word, too. Directory where? One still unchanged. Artoo is also entrenched. Hands were three zero zero minus three minus 60. Oh, yeah. So That's very easy gassing elimination. And I'm ready for back substitution. So for three is minus three C equals minus 60. So for Miss Aiken, C. C. Because 20 artsy is minus. B minus. C equals minus fancy. So be equals 10 and where one is a must be plus two c. He was 65 so a equals 15. So for miss, we have our losses were going to make 15 of the reference model 10 of the office model and 20 nobody looks small to use all the time available.

Well that a furniture factory makes wooden tables, chairs and he looks are more, I'm not sure what that is. It's a kind of closet basically didn't really pay attention. It's a wardrobe. I really We're told that each piece of furniture requires three operations cutting the wood assembling and finishing it. And each operation requires the number of hours h given in the table. The workers in the factory were told to provide 300 hours of cutting 490 hours of finishing each work week. Were asked how many table chairs and arm was or wardrobes should be produced So that all available labor hours are used? Or is this impossible? Is it not possible to use all the available labor hours? Right. Well, we're given a table and exercise 51. Now, we'll let X be the number of tables. Why the number of chairs nz the number of arm woz or wardrobes. That is the right schedules. Actually, no, that are produced. We know each table takes one half hour for cutting. Each chair takes one hour and each wardrobe takes an hour. Total number of hours required for cutting is therefore 1/2 X plus Y plus Z. Yes. And the workers can provide 300 hours of cutting. So our first equation is going to be When half x plus y plus C is 300. Was saying from the table, we see that each table takes one half hour for assembling each chair 1.5 hours and each arm wa an hour. And we're told that therefore the total number of hours required for assembling Is 1/2 X plus three halves. Why plus Z. Kids from Parkland? And we're told the workers can provide 400 hours of assembling. So we have one half X plus three halves, Y plus C equals 400. Finally, from the table we see that it takes an hour to assemble the table 1.5. Start to finish table 1.5 hours to finish the chair and to our hours to finish the wardrobes. So the total number of hours to finish is X plus three halves Y plus two Z. He and his dad got yes. The workers can provide 590 hours of finishing. It follows that X plus three hats, Y plus two, Z Equals 590. So we want to see if there is a solution to this system of equations, right, do this. I'll first write this in matrix form. So in matrix form on the left hand side we have our coefficient matrix one half, 1, one, one half. Yes, three halves one and 13 halves two. And the augmented matrix includes the right hand which is 300 400 And 590. But like oh now we'll use rural operations to write the matrix in row echelon form The first time. Going to interchange rows one in 3. So I get 1/3/2 90. One half, 3/2 1 400 and 1/2.11. Yeah. 300. Mhm. To say. Or next I'm going to replace row three By Row 3- Road to. Yeah. Yeah. It's not really thought of had access. Yeah it's true. And we end up with the Matrix 1/3/2 to 590. Yeah 1/2, 3/2. 400 finally one half minus one half is zero. 1 -3/2 is a negative 1/2 And 1 -1 is zero. Finally 300 -400 is negative. 100. Yeah. Yeah. Wait yeah. Is ready. Yeah. I'm sorry. Maybe instead of doing this I should have done replace row three. Bye negative. Well make games by road to minus row three. This is the same as negative row three plus route to if you like. And so this is actually zero positive one half zero and positive 100. Now I'll replace a row two By two. Row 2- Row one. Sure. Yeah. Work on the street now. So our matrix is 1/3/2 to 590. And then we have To go to let's see it's 1 -1 is zero. Yeah. 22 is three minus three halves is just three halves and then two times one is two minus two is zero, two times 400 800 -590 is 210. We can have some jewish And their third row is still 01/20 100. Well took. So next that's fine. Arthur we could use her to rate right? Never whatever plants are always come up with. That is what you've been talking about how your powers too old for this young. You think he's like oh promise talking my train friends home. Yeah. Eight years and then he's done just like so better start a podcast now. Friends come on, they're not yet. Yeah, we're all you can be good. There's not gonna be any time brutal. March put the Comanche which is we can break out Harvey Weinstein from prison. Yes, that's but you keep rearranging eventually you get the matrix 1/3/2 to 590 0 1/20. Thanks. 10 And 000 90. Yes. However, notice that in our last row You have zero equals 90 which is clearly a contradiction. I went there. Therefore, it follows that the system is inconsistent and therefore it's impossible to use all the labor hours. That's how they met does a big


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