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Welcome to Algebra I: Representing a Direct Variation Algebraically and Graphically

33 minutes

Hi guys. Welcome to Algebra I. Today's lesson focuses on representing a direct variation algebraically and graphically. Your prior knowledge about the direct variation equation and graphing direct variation equations will help you get through this lesson. You ready to get started? Let's go. To jog your memory before we dive into this topic, recall what the equation of a direct variation looks like. There are two equations we can work with with direct variation. Either y equals kx or k equals y over x. Depending on the problem we're working with, sometimes one equation more helpful than the other. Remember that k was your constant variation, right? Keep that in mind. Let's keep going. Let's look at this one: example 1.

(female describer) Within a set of braces are three ordered pairs: minus 2 and minus 16, 3 and 24, and 4 and 32.

(teacher) Write an equation for the direct variation represented in the relation. In this situation, because we want to write that equation and we're not trying to determine if this is a direct variation-- we're not testing it out-- we're going to start by using-- We'll try to get to this. Let me scoot that so you can see. Our goal is to get to an equation in this form, y equals kx. What we need to determine is, what is k? Because we are told already that this is a direct variation, we don't need to test out those quotients like we do for each of these ordered pairs. We don't need to do negative 16 divided by negative 2, 24 divided by 3, and then 32 divided by 4 to ensure we get the same answer. We're told it's a direct variation situation, so we know we'll get the same answer. If you don't believe me, I'll show you just so you know. Negative 16 divided by negative 2 is 8; 24 divided by 3, it's 8; and 32 divided by 4-- let's get a little space here. It's 8, okay? When you're told that you have a direct variation situation, don't test your ordered pairs. You really just need to pick one--one ordered pair. What you want to find is, what is k? To write this equation, this y equals kx, I need to figure what is k? What is the constant variation? I proved already that this was a direct variation situation-- my testing out the ordered pairs-- so you can tell that k is 8. If k is 8 for this situation, then I can write the equation as y equals 8x. And I'm done, okay? To solve these problems where you write the equation, all you need to do is figure what is k and fill it in to that general format, that y equals kx, all right? Let's look at another one here. We have a table, but the directions are the same: "Write an equation for the direct variation represented in the relation." Kind of a mouthful.

(female describer) The ordered pairs in the table are minus 20 and minus 5, 28 and 7, and 44 and 11.

(teacher) What we want to figure is, what is k? Then we can write our direct variation equation. Because I'm already told this is a direct variation situation, I don't have to test every ordered pair. I'll just pick the first one. In order to figure out what k is, I need to do-- I'll write that here-- y equals k divided by x-- I'm sorry-- k equals y divided by x. For this one, k, if I pick the first ordered pair-- but you can choose any ordered pair-- y is negative 5; x is negative 20. I simplify this by dividing my numerator and my denominator by negative 5. Negative 5 divided by negative 5, positive 1. Negative 20 divided by negative 5 is positive 4. My k for this situation is 1 over 4. I want to write my equation in the format of y equals kx. Now that I know that k is 1/4, y equals 1/4x. Okay? And you're done. You have your equation for this direct variation situation, okay? If you chose to write your equation in this format-- the k equals y divided by x-- that's still acceptable. But when we're dealing with a table or set of ordered pairs, and eventually we get to the graphs, you want it in the y equals form most of the time. Both of these are acceptable if you were to have-- since your k is 1/4, you could say that 1/4 equals y divided by x. That is also true for this situation. Get in the habit of figuring what k is and then filling it in to that style of the equation. Okay, let's keep going here. I've got a couple for you. You have a set of ordered pairs and a table; you're asked to write that direct variation equation. Remember, figure what k is and then fill it in to that y equals kx equation. To check your answers, press play.

(describer) Relation A is a set of the three ordered pairs in braces-- 2 and 6, 5 and 15, and 9 and 27. Relation B is three ordered pairs in a table with two columns, headed x and y. The ordered pairs are 8 and 4, 10 and 5, and 18 and 9.

(teacher) Let's see how you did. For A-- Let's move this out of the way here. You should have got y equals 3x for your equation. For B, your equation was y equals 1/2x. Want to see how I got these? Stay with me, and I'll show you. Look at the first one. I first need to figure what k is. k, I can figure out-- let's get the pen-- by determining a quotient, y divided by x. Pick any ordered pair. I'll pick the middle one. I know that's my x; that's my y-- k would equal 15 divided by 5 for my y divided by x, which is 3. I want to write my final answer in the y equals kx form. Now that I know that k is 3, y equals 3x. You got it; you're done on that one. Figure out k and drop it in to that y equals kx equation. Let me show how I got that second one--my table here. Again, just figure out what k is first-- k equals y divided x. I picked the last ordered pair. You know this is a direct variation, so it doesn't matter which pair of points you pick. I see 9 is the y and 18 is the x. I can simplify this fraction by dividing both the numerator and denominator by 9. 9 divided by 9, that's 1; 18 divided by 9, that's 2. So, k is 1/2, right? Scroll down a little bit. I know I'm trying to fill in to the y equals kx equation. Now that I know k is 1/2-- a little more room-- y equals 1/2x. And you got it, okay? That's how I got those two answers for those two questions. Let's look at a word problem. You knew it was going to pop up eventually. "Example 3: The amount of money Jasmine earns varies directly "with the number of hours she works at the library. "Last week, Jasmine worked 12 hours and earned $111. How much will she earn next week if she works 15 hours?" Okay, now, there's some steps we can follow with a word problem like this, where we're given a set of information and then asked to actually generate another value for this situation, right? Here are the steps to take: The first thing is, we're going to determine what's x, what's y, what's k. We get those important values. Then we write the direct variation equation. We have all the information needed to write it. After we've got that equation, we use it to answer the question-- to figure out that value they want you to get for that answer. Remember those three steps: determine x, y, and k, write the equation, then use it to answer the question, okay? Back to that example, let's highlight the key information. Break this problem down a little bit. The amount of money Jasmine earns varies directly with the number of hours she works. The money varies directly with the number of hours. Last week, Jasmine worked 12 hours-- so 12 hours-- and earned--oop-- there we go-- and earned $111. How much will she earn if she works--next week-- if she works 15 hours? We want to know how much is she going to earn working 15 hours, all right? Step 1, let's figure out what's x, what's y, what's k. Let's get the pen back here. They're telling us-- I lost some highlighting-- there we go. They're telling us, the amount of money varies directly with the number of hours. That means, the amount of money depends on the number of hours. That means y, since it's our dependent variable, is the amount of money, and x, because it's our independent variable, that's the number of hours, okay? We've got the x; we've got the y-- y is the money, x is the hours. We've got to figure k. That's when we use that initial information we were given, that Jasmine worked 12 hours to earn $111, to be able to figure out what is k for this situation. Let's get that. Get a little more space. Going to have to scroll up and down here. For direct variation, k equals y divided by x. We know that y is the money. They told us she earned $111. x is the hours. She earned that when she worked 12 hours. Let's get some more space, okay? So k will equal that $111 divided by 12 hours, okay? For this, because this division isn't so pretty, we switch to the calculator to get our answer for this one. Let's get that calculator out. I like to clear the memory because, like we saw there-- clear all the information so you're starting with a clean slate. If we look at our problem to see what we were trying to determine, we want to figure out what is 111 divided by 12. We do that division instead of leaving fractions because we're dealing with money. If we have a decimal, that's like the coins for the money. We've used the fractions and just sticking with that, but we want the decimal for this situation. If you used fractions, you're not wrong, but it will just be easier to see exactly what our k is here. So 111 divided by 12; that's 9.25, all right. Let's get that down, and we'll interpret what that means exactly. Go full screen, okay. So we know that k is 9.25. That's our constant. If we go to step 2 now and write that equation-- I'm going to number this "step 2." Let me scroll up, and I'll label this "step 1" just to help us stay organized here. Now that we know k is 9.25, we fill that k in to our y equals kx equation, so y equals 9.25x. Basically, what that's telling us is that-- I believe it was Jasmine we were working with in-- that Jasmine earns $9.25 per hour. That's what that means. We used that given information that after working 12 hours, she earned $111 to figure that constant rate that Jasmine is being paid at. We've got step 2. We've got our equation, y equals 9.25x. Scroll up to see what we're being asked to figure. "How much will she earn next week if she works 15 hours." That 15 hours, that's our x. We've already identified that x is the number of hours. Let's scroll back down so we can do step 3, okay. We're going to determine what is y when x is 15? We're going to substitute that value in there for x. Once we figure out this product, this 9.25 times 15, we'll know how much Jasmine makes after working 15 hours. Again, this multiplication here isn't so pretty. We've got a decimal. I switch to the calculator to get us our answer quickly-- 9.25 times 15. "9.25 times 15." We've got 138.75. Let's get that down. Okay. That was 138.75 is what we got for y. So that means that after working 15 hours, Jasmine will earn $138.75. That's the answer to that question. To bring it together, I'm going to scroll again. Scroll up to the beginning of our work. We started out reading the problem. We highlighted those key parts of information in order to determine what was x, what was y, what was k. That was our step 1. After we figured out that k was 9.25, we wrote our equation that y equals 9.25x. We used that equation to answer the question, to figure out after working 15 hours, how much money Jasmine earned. We determined that it was $138.75, okay? Keep those steps in mind, and let's try another word problem here. Okay, let's do a little moving around here. "Example 4: Julio subscribes to an online streaming service. "The monthly cost varies directly "with the number of movies streamed. "Two months ago, Julio paid $18 to watch 4 movies. "If Julio's total cost last month was $27, how many movies did he watch?" We use our same strategies as we did before. We've read through it once. Now let's highlight that key information so that we can determine what's x, what's y, what's k. Okay, we know Julio subscribes to an online streaming service. "The monthly cost varies directly with the number of movies streamed." The monthly cost varies directly with the number of movies. So two months ago, Julio paid $18 to watch 4 movies. If Julio's total cost last month-- so cost last month--was $27, how many movies did he watch? We've got that key information. Let's use our steps to get to the answer. Step 1, we've got to figure out what's x, what's y, what's k. Let me switch to my pen. We were told in the problem that the monthly cost varies directly with the number of movies. The cost depends on the number of movies. So y... that is our dependent variable, and that's the cost. And x, that's our independent variable. That's what we can control. We can control the number of movies. We'll say, number of movies, okay. We've identified the y and x. We've got to get k. Let's look back up here. We'll use that first set of information that Julio paid $18 to watch 4 movies, right? Let's scroll down. We're doing a lot of scrolling because we've got a lot of work. To figure out k, we know that--oop-- that's step 1-- we know that k is y divided by x, right? y is our dependent variable, the cost. If you look up here, it said, Julio paid $18 to watch 4 movies. So the cost, 18... number of movies, 4, right. Get more space. I go to the calculator so I can get that rate, 18 divided by 4. Let's switch over here; let's clear that out. 18 divided by 4, enter-- 4.5, okay. We've got our k. Let's go back and write it down. All right, scroll down. Get some more space. So k equals 4.5. Because we're dealing with money here, that 4.5-- that's basically $4.50. I put the 0 to stay consistent because we're dealing with money in this case. If you had that k was 4.5, you're not wrong for that. You're right, okay. Now that we know that k is 4.5, we can fill it in to that direct variation equation-- Get some more room. I'm going to fill in that k equals 4.50-- I use the 0--x. To interpret, all that means is that Julio is paying $4.50 for every movie that he is streaming. That's all that this means, that equation, okay? We've got step 2; scroll up to see what we're being asked to find. The total cost last month was $27, so how many movies did he watch, okay? They've given you the cost. They have given you y, and they want you to determine what is x. How many movies did he watch? Okay. Let's look at exactly where we're going to replace those values. Last month, the cost was $27, and y represents the cost. So step 3, I'm going to have 27 equals 4.50-- 4.50 times x. So notice this time, I replaced that given value for y. They gave me the cost, and we already determined that y was our cost. This time, we know the cost was $27. We want to determine how many movies did Julio watch if that's what he paid. We have this one-step equation, 27 equals 4.50 times x. We will divide both sides by 4.50. That division isn't so pretty, so get the calculator out-- 27 divided by 4.50-- 27 divided by 4.50... 6, okay. So we got--for this situation, we go back to our work. Let's scroll down--I'm going to need some more space. That 27 divided by 4.50 equals 6. What this means is that if Julio paid $27, then he watched 6 movies, okay? Let's scroll back up just to bring that together. Make sure you got the steps. After you've read through the problem and highlighted that key information, start by identifying your x and your y and your k. Figure your independent and dependent variables so you can figure out what k is. After you have k, go ahead and write the equation. Use it to figure exactly what you are being asked to solve for to get that value. Those three steps will get you through these word problems. I want you to use those to solve this one, okay? Let me get a few things out of your way so you can see. Press pause and take a few minutes and work through this problem. To check your answer, press play.

(describer) The cost of Suzanne's data plan varies directly with the number of gigabytes she uses. Last month, Suzanne used 6 gigabytes of data and was charged $22.50. If Suzanne uses 8 gigabytes of data this month, how much will she be charged?

(teacher) All right, let's see how you did. Let's read this. "The cost of Suzanne's data plan varies directly "with the number gigabytes she uses. "Last month, Suzanne used 6 gigabytes of data and was charged $22.50." My highlighter out-- "If Suzanne uses 8 gigabytes of data this month, how much will she be charged?" Let's start highlighting. So the cost varies directly with the number of gigabytes she uses. Last month, she used 6 gigabytes "and was charged $22.50. "If Suzanne uses 8 gigabytes of data this month, how much will she be charged?" We've pulled out that key information here. Let's start with our first step to determine what's x, what's y, what's k, okay? Get some work space here; I have to do some maneuvering. First, it tells us that the cost varies directly with the number of gigabytes she uses. The cost depends on the number of gigabytes she uses. That means the cost is the dependent variable. Abbreviate it. The number of gigabytes is the independent variable. That's what she can control. Gigabytes, okay. That means that-- I need to scroll again. Last month, she used 6 gigabytes and was charged $22.50. We're going to use that information to get k, okay? Move that up. See if we can see so we don't have to scroll. Still on step 1. So k equals y divided by x. They told us, she used 6 gigabytes and was charged $22.50. So y is the cost. The cost, $22.50, and x is 6 gigabytes, the number of gigabytes that she uses. We need to determine, what is the value of 22.50 divided by 6. Let's go to our calculator-- 22.50 divided by 6. 22.50 divided by 6... 3.75. Let's write that down. Get some more space; here we go. So k equals 3.75. That is our k. We've got k. We've got y. We've got x. We go to step 2 and write the equation. Because we know it's y equals kx and we know that k is 3.75, y equals 3.75x. Basically, Suzanne is charged $3.75 per gigabyte she uses. That's what this equation is telling us, okay? Now we're ready for step 3. Let's see what they are asking us. "If Suzanne uses 8 gigabytes, how much will she be charged?" Replace 8 in the right spot-- substitute 8 in the correct location-- 8 gigabytes--gigabytes is my independent variable. That's my x. Let's scroll down. I'm going to need y equals 3.75 times 8. This time, they're giving me the x. They're giving me that independent variable's value and want me to figure out y-- figure out that cost. Let's go to the calculator: we need 3.75 times 8. 3.75 times 8... 30. Who knew it would be a whole number? Let's go back to our work here. We have figured out that y equals 30. That means that if Suzanne uses 8 gigabytes, she'll be charged $30 for that. Good job on that one. Let's keep moving here. It's time to attack the graphs. I move this up because our directions are underneath our problem, underneath our point. For example 5: "the given point is located on the graph of a direct variation." We have that point in our second quadrant. They're telling us, it's part of a direct variation. "Indicate another point contained on the graph of this direct variation." I want to jog your memory, and I want you to look at this. Remember that a direct variation, when you graph it, it's always a line that passes through the origin. It's not always this exact line. This is just an example of one direct variation, but every direct variation is a line that passes through the origin. It may have a positive or negative slope, but it must be a line that passes through the origin. Let's go back to that problem. They've given us one point that's on that line of this direct variation. They want us to find another point. Think about it-- the only point that you could always be certain is on a direct variation's graph is (0,0)-- right here at the origin. Because if you know nothing else, you know that for a direct variation, that line will pass through the origin graphed. You just need to plot your point there at the origin, and you're done. That's the only certain point on the graph of this direct variation. Okay? Let's keep going here. Look at this-- it's your turn. "Show an example of the graph of a direct variation equation. Indicate at least two points." There are several correct answers to this question here, okay? Figure out where are two points that would be on this direct variation. One certainly has to be there, and there another one that's up to you. So press pause. Get graph paper out. Get this one done. To check your answer, press play. Let's see how you did. Now, if this is a direct variation, I know, if nothing else, there is a point here at the origin. That's the one you knew would be there. The other point you plotted could be any of these four quadrants. Maybe you put a point here. That would have been fine. Maybe you didn't put a point there. Maybe--erase that. Maybe you put a point over here. That would have been correct. Or maybe you didn't put a point there. Maybe you put a point way over here. Where your other point was was completely up to you. If it was in these four quadrants, you were good. But you had to have one point at the origin in a direct variation situation. All right. Great job, guys, working on these problems, representing direct variations algebraically and graphically. Hope to see you soon for more Algebra 1. Bye.

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In this program, students learn to represent a direct variation algebraically and graphically. They also learn to identify the relationship based on what a variation looks like on a graph. Part of the "Welcome to Algebra I" series.

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Runtime: 33 minutes

Welcome to Algebra I
Episode 1
31 minutes
Grade Level: 7 - 12
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Episode 2
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Episode 3
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