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Consigc the folloving Intermedlale tableau (not thc Initial Simnolertaocaui:RISDetrine the pivot clerncnt and perform all tne pivol opcrations for the entir pivot c...

Question

Consigc the folloving Intermedlale tableau (not thc Initial Simnolertaocaui:RISDetrine the pivot clerncnt and perform all tne pivol opcrations for the entir pivot colurn to oblain the next tablcau: In this next tableau that You oblaincd, In the objective rov:, Enter eacn box belov: thc number you have under Lach colurn_ Note _Ware ggplicoble_froctons must be entered 215 4JandsoonUnder columnthe objective row you nave:Under columnin the objective rowJavlUnder columnIn the objeclivl rov' YDu

Consigc the folloving Intermedlale tableau (not thc Initial Simnolertaocaui: RIS Detrine the pivot clerncnt and perform all tne pivol opcrations for the entir pivot colurn to oblain the next tablcau: In this next tableau that You oblaincd, In the objective rov:, Enter eacn box belov: thc number you have under Lach colurn_ Note _Ware ggplicoble_froctons must be entered 215 4Jandsoon Under column the objective row you nave: Under column in the objective row Javl Under column In the objeclivl rov' YDu havc: Under column In Lhe Objective TOw. you have: Under column [Nc objective row; You have: Under column RHS objcctive rov"_ you have: b) Given the tableau thal You oblaincd abovc LHree interpretalions are possible: In tne bux belov:, cOpy and paslc whichever answer shown boldfacc letters below thal You think is the Lonicct chcice_ Enter solved Ifyou think you have oblained tne final tablcau: Enter ready for another set of pivol operations If you tnink more pivot operations ar possible Enter solution If you think tne tableau has no solulion:



Answers

AGRICULTURE A fruit grower raises two crops, apples and peaches. Each of these crops is sent to three different outlets for sale. These outlets are The Farmer's Market, The Fruit Stand, and The Fruit Farm. The numbers of bushels of apples sent to the three outlets are 125, 100, and 75, respectively. The numbers of bushels of peaches sent to the three outlets are 100,
175,
and 125,
respectively. The profit per bushel for apples is $\$3.50$ and the profit per bushel for peaches is $\$6.00$.

(a)
Write a matrix $A$ that represents the number of bushels of each crop $i$ that are shipped to each outlet $j$. State what each entry $a_{ij}$ of the matrix represents.
(b)
Write a matrix $B$ that represents the profit per bushel of each fruit. State what each entry $b_{ij}$ of the matrix represents.
(c)
Find the product $BA$
and state what each entry of the matrix represents.

It's time. Thank you. Blow you one three. Negative too. To one negative. One. 131 Negative. And then we have a slack variable. No, no, There uncle alone? No, they want us to pivot and then read off the corresponding basics. Well, it you want us to pivot using? Listen, but actually the variable X three columns That's one column that do you like that? We're gonna when we multiplied by three bottom? Well, one thing have in mind multiplied by three. My one, You're out. Actually, one zero zero and 45 minutes. And now we're gonna want for no one. Thank you. No one Europe. You have one one? Yeah, it is attracting from it. I need you here. Sounds like a final country. Well, multiply by three and then add to the work. Fine. Zero three euro. Yeah. One. 30. Okay, we can read off our basic solution from that. You're one. Value is there are new value. Zero for extreme value. First black variable value zero Somewhere we have a typo commission Still be a zero. So we actually have that Are seconds like variable. Yeah, nine on our third slack variables. A value of is this is that three Columns one and these A row or column with only one non zero well equipped them to their appropriate right hand side unlamented row.

For two things to find. One to three. You and the slack. Variable No. 001 There. 0010 Unless they are objective. Variable. The arguments in column 20 board. Okay, right. They give us a choice of pivot. Really? Before that operation. Fine. Wait. Let me make it out. This is extinct one. Now we're gonna pivot. We would have wanted by the top five three. I don't our first the line, but fine your other My time. Time. Thank you. A lot of twenties address. That is 14. We'll have 10. We'll have 15 thing, which is zero love five. I'll have negative there, there. I'm like C 20. I'm 110. We'll have 110. Try only eight. Well, that any point, Let me make sure the and now only being second round the same since that's what we're pivoting from my no one. But now we're going to need, uh, multiplier third row by and then subtracted time. It will end up within our first thing. Fun which is one terms of drug port Handsome Thank zero. Oh, dear. I under negative to him. Let's stay in zero on a multiplying 45 by five 225 I believe you know 200. What's our homes like? Yeah, Minimalism tracked the double of coming. Now let's get our bottom room. Yeah, that we're gonna have to multiply all right in Rome by before and added five times that room. What thing to do? A negative negative. Plus eight is negative. Negative. 20 plus. Yeah. Here there are here. Five here for a time. Four times 28 should probably be 100. Well, we'll have 80 get behind. Not we can read off our basic solutions. This was extinct for being one non zero entry in our basic solution and then three transmission. Don't know who find a rallying from the column. What we can read off our solution That zero x went zero x two or x ray will be 28 over fun. First like variable will be £100 tracked 84 Not 20 30 over time. Second, slack variable zero. It's like very trying to. Well, however, 25 200 Another 25. 25. It was one of 69. We'll have a 1 69/5 and Leslie are value for that will be 100 tall over

This question is asking us to solve the given system of linear equations based on the reduced matrix out we have been given. So well, go ahead and first label our columns with the variables that the representing. So this is X one. This is X two, this is X three. So for the 1st 1 we can write out the equation with the coefficients for X one x two x three has given to us by this matrix. So the coefficient of X one is one. So we can just leave that out since it's a one. Or we could put that in if we want. And then the coefficient of X to an X three are both zero. So we can basically just ignore these and write X one equals negative three. Because these cancel out one times X one is just one X one. So we get that X one is equal to negative three. Uh, this same thing we're gonna bring you the same thing. For rose to one rose three. We have zero x one plus one. Next to zero x three is equal to zero. We simplify this, we get X two is equal to zero. Kathy's cancel out and one times X two is just x two. So lastly, will Sof uh, this final road here? So we have zero x one plus zero x two was one x three is equal to seven. We simplify that we get that extra is equal to seven so we can write our final answer in Dr notation. If we want, we can say this x factor of these three components x one x two x three is equal to negative three zero seven. Okay, so we're going to do the same thing for this. We'll label our variables, x one x two x three x four and also, uh, just tow. Remind ourselves this lost column is what the equations in this matrix are equal to. So for the first row B right x one zero x two plus x three plus zero x three was negative. Seven x four is equal to a and what we want to do before we solve. This is looking or matrix and see if any of these variables our free variables and what that means is that one of these variables x one x two of three or X four is not a leading injury. So here we see that this is a leading entry. This is a leading entry and this is weeding entry, which means that this is not and none of the entries in the X four column are reading injuries. So that means, uh, export is our free variable. And we want to solve all of our equations that we're gonna write down. We're gonna want to solve for X one, x two and x three in terms of x four. So we'll go ahead and do that. So he's cancel because these are both multiplied by zeros. We're left with X one minus seven. X four is equal to a And like we said before, scores are free variable. So we want to solve for X one in terms of export. So x one is equal to eight plus seven x for Okay, so now we'll go ahead and look at our second row. We have zero x one plus x +20 x three plus three of four is equal to two. He's cancel. We have x two plus three x four is equal to two and we solve for x two in terms of our free variable of exports, so actually was equal to two minus three x four. So then our last row, we're gonna want to solve for X three. So we have zero x one. The zero x two was x three plus explore is equal to negative five. He's cancel. We solve for X three in terms of explore, So x three is equal to negative five. Linus Export. Now what we're going to want to do is do what we did in the first problem and write that in vector notation. So we have the general expect, er that we're selling for which has components excellent X two and X three. And what we're going to do is we're going to May one. Dr. That's just these numbers that have no other variables involved. So we see our X one has a next to has it too. Exterior has the negative five. We're going to add that to explore times it's coefficient, so we can see that the coefficient for X one for X four, And, uh, this overall equation for X one is seven. So we're gonna write a seven here, uh, for X to the coefficient of X four is negative three. And for X three, it is negative one. And if this is looking a little bit confusing to you, we'll just go over here and know that we can rewrite this as we can. Reword right each of these lines the x one x two x three as x one It wa ce eight waas x four times seven, which is seven. Explore and we can see that that is exactly what we got here. We can do the same thing out for Ex tube just for practice. So x two is equal tune too. Shot number here. Plus that's four times negative three, which is negative three x four, which is the exact same thing that we have here. And lastly, X three is equal to negative five plus negative one x four, which is also the same thing that we see here, okay, onto the next one. So we'll do what we've been doing right down the different variables in these systems of equations. We have excellent Xbox three x four, index five and we then want to figure out what our free variables are, and we can see that This is a leading entry. This is leading entry. This is leading country. So that leaves next to an X five to be our free variables because they have no leading injuries. So we'll solve this just for myself. The other ones, uh, we'll start with the first column right here. You see that? This is X one minus six, x two plus three x five is equal to negative too. And remember, we don't have an extra year and explore because these are both coefficients of zero, which cancels he's out. So then we want to solve for X one in terms of our free variables, which our ex too and x five So we have X one is equal to negative two plus six x two was er sorry no plus minus three x five and then we'll do the same thing for the next row where we have, uh zero x one plus zero x two plus x three closed zero x four plus or expire is equal to seven. We want to solve in terms of our free variables which we have his x five and we get the, uh X three is equal to seven minus for X five. Um, and now we have our last row that we want to D'oh! And so this one is going to be solving for X four. So we are going to I'm gonna open up a other patriot. Quick. Um, so we're going to have X four five x five is equal to eight, and then we want to solve it in terms of our free variable, which is that's five, and we have explore is equal to eight minus five ex spies. So now we have the variable. So we want to solve for which is excellent x three, next four over leading injuries. We'll write down what we got for those. So we got X one is equal toe. Uh, negative two plus six x Q minus three x five. Well, go ahead and write that down for extremely. We know that that equals seven minus four x five. Then for last variable, which is explore, we have a minus five x five. If you want to put it in vector notation like we did with the other ones, we write x one. It's three that's for is equal to thinking of 27 a Waas X two, which is one of our free Barry Bulls time. Some doctor Waas That's five times some vector. So now let's go ahead and fill that in for X two. We see that X one has a coefficient of six. Since there's no X two terms in either extra explore, we know that these have a coefficient of zero because that'll cancel it out. And for X five, we can see the coefficients for X one. It's negative. Three for X two. It's negative for and for explore its negative five. Okay, so now we'll go ahead and go into the last one. So for this, before we even start solving for variables, we can kind of do a little shortcut and just take a look at this bottom row. So what this bottom row is saying is that zero x one. I'll label the columns again for the variables we have x one x two, the next three. It's saying zero x one plus zero x two zero Expiry is equal to one. So because he's a zeros zero times, any number is gonna be zero. So we have zero plus zero plus zero is equal to negative one. Er sorry is equal to one. And some of all zeros is going to Syria. We have zero equals one, which is obviously not true. So for this matrix because we have this which is invalid, we say that this matrix has no solution.

This time. Thank you. One, 21 10 on zero. Thank you, too. Negative one. Think it'll what? You know, So I mean it. Okay, The rest will just be like, were you 10 zero 01 There are 0010 on this one. Home. 50 engine. Yeah. What? They want us to pivot with choice. They want us to you on 103 it out. Let's just subtract from the top room. We'll end up with zero Mm. One. Yeah. Negative one here, then we can multiply by. Two is subtracted from the second row. 01 I 01 Negative, too. Zero number. We're leaving our pivot. Peru 10 10010 And now I'm multiplying by two to the bottom on it with their here. A negative one. Here, one here. Getting one here. 00 to those were multiple embedded. No one. Good. And now we'll have over here, I think. What's the basic solution here? Well, let's see. What is. Our BlackBerry will begin here, Lex. One is gonna be one of our basic variables. It's augmented value, uh, one being behind. All right. And that's all aware basic variables. Zero. Since we have only three country and there is the value of this, so are corresponding. Basic solution Near out zero zero four x two x three x one It's like wearable one. All right, variable too variable three. I really


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