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Solve each problem by using a system of three linear equations in three variables.Milk, Coffee, and Doughmuts The employees from maintenance go for coffee together every day at 9 A.M. On Monday, Hector paid $\$ 5.45$ for three cartons of milk, four cups of coffee, and seven doughnuts. On Tuesday, Guillermo paid $\$ 5.30$ for four milks, two coffees, and eight doughnuts, On Wednesday, Anna paid $\$ 5.15$ for two milks, five coffees, and six doughnuts. On Thursday, Alphonse had to pay for five milks, two coffees, and nine doughnuts. How much change did he get back from his $\$ 10$ bill?

$$\$ 3.95$$

Algebra

Chapter 8

Systems of Equations and Inequalities

Section 2

Systems of Linear Equations in Three Variables

Graphs and Statistics

Equations and Inequalities

Rational Functions

Campbell University

Oregon State University

University of Michigan - Ann Arbor

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So for this problem, we're just flat out given three systems based off the price of three different orders. So going to have X represent milk I'm going to have. Why represent coffee Copely and C represents doughnuts. You just right that little better daamoth. And so we're given for Hector that three X plus four. Why a seven Z is equal to 5 45 and then for, um, the order on Tuesday were given that four X plus two y plus Beat Z is equal to 5 30 and then on Wednesday were given that two x plus find why I lost six. C is equal to 5 15 And then essentially, we're trying to, um, calculates the order are the pricey order on Thursday were asked if someone orders five X to why and nine Z Um, how much change will they get back on a $10 bills? So I'll put this I'm gonna put this weirdly just so it how it works mathematically, but you don't have to think or write this in terms of this problem. So we'll say that this is equal to W and that the answer our problem is what is 10 minus W. Okay, so now we're looking to eliminates, um X from two of our equations so that we can use substitution to solve for a variable. And the first thing I'm going to do for this is going to multiply Equation three by early March, our equations first. So we have equation one equation to in equation three. I'm going to multiply equation three by negative too. And then, uh, add that to equation to use a two plus negative two times three. And the reason I'm doing that as biz will cancel the Xer's. And then I'm gonna do the something similar for equation one to kill. So, uh, cancel the other ex. So when I do this, I'm going tohave for X plus two y plus eight z, uh, minus four X minus 10. Why? Minus 12 z That is going to be equal to 5.3 minus 10.3. And so I'll be left with negative eight y minus for Z is equal to negative five. And then I'm going to multiply equation one by, um, negative too. And equation three by three that both have an extreme of six and I can subtract that way. Um, doing the same promises as earlier. This will give that seven Y plus four z is equal to 4.55 So now we have two equations in terms of Y and Z. I'm going to use substitution to solve for why and then necessary sell frizzy and then substitute to solve for y. So let's mark this as equation for and equation five. So if I solve for Y from equation for Ah So I saw Rosie o get that Z is equal to 1.25 minus two. Why? I'm gonna substitute this into Equation five right there. Andi, this will give that why is equal 2.0 point 450.45 That's right. That little bit 0.44 and then from here, since Z is equal to 1.25 minus two. Why I can substitute 0.45 back in this would be 1.25 minus point line which would get that Z is equal 2.35 And now that we have the value of coffee and donuts weaken, use the's in any of the three original equations given to solve for X doing this will yield that X is equal to 40 cents. And now that we have the values of X, Y and Z, we are to plug Hm and to this equation up here because this is the equation we're going to use to solve for our answer. And when we do this will know that the price of the final order is 605 So W is equal to 605 We know that our answer is 10 minus w 10 minus w is 3 95 So on the order on Thursday, there change will be $3.95.

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