Applications of Linear Equations - Problem 1
To write an equation representing a word problem, start by defining what the variables, x and y, will stand for. "x" should represent the independent variable in the problem, and "y" should represent the dependent variable in the problem. In other words, the value of x will determine the value of y. If x changes, y will change accordingly. The value of y depends on the value of x. Set up an equation using x and y that represents the information in the word problem. To graph the equation using a table of values, choose a few x-values and plug them into the equation in order to find the corresponding y-values. Plot the coordinate points to draw the line of the equation.
A lot of you guys are already members of gyms. If not, one thing that stinks about joining a gym, is you usually have to pay an enrolment fee. And then on top of that, you have to pay like a monthly service to be a member of the gym. So at the "Pump-U-Up" Gym they charge $100 enrolment fee, plus a $35 per month to be a member. Write an equation that represents the cost as a function of the number of months enrolled. Make a table of values and draw the graph.
So the first thing I'm going to think about, is what I want my different letters, to stand for. I'm going to choose to let y equal the total cost and to let x equal number of months. The reason why I did that, is because I noticed that number of months is the independent variable. How much I pay depends on how many months I'm enrolled. Y is always the dependent variable, x is always the independent variable. Once I know that, let's think about how much you pay.
To start with, everybody has to pay $100. On top of that, in addition to that so I use plus sign, you have to pay 35 per month. That's why I'm using the letter x, x remember stands for number of months. Write an equation, done, that represents cost as a function of the number of months enrolled. That's good because I have cost, and I'm using the number of months enrolled as my independent variable.
Next thing I have to do is make a table of values. So keeping in mind, that x represents number of months, I'm just going to pick some months like 0, 1, 2, 3 and then I'm feeling kind of crazy. So I'm going to say like 8, I just made that up I don't know. I picked on a point that's kind of further out there, so that when I make my graph, it will be big. It won't just show like 1, 2 or 3 months. It will show 8 months which is probably more realistic for how long you might be a gym member.
So now what I need to do is, substitute in my x values one at a time and find the corresponding y or output values. So if x is 0, y would be 100 plus 35 times 0. So y would be 100. If x is 1, y would be 100 plus 35, if x is 2, y would be 100 plus 35 times 2, which is 70. So I'll have 170. For 3 I'll have 100 plus 105. And then for 8 months, I'm going to have to use my calculator, being really careful not to use the order operations.
First thing I'm going to do is 35 times 8. Once I get that done, I'm going add my 100. So 35 times 8 is 280 plus 100 will give me 380. So I've made a table of values that represents what I think is a real world like possibility in terms of how many months you might to be a member at a gym.
Next thing I need to do is draw the graph. So what I'm going to do is set up my axis, with x on the horizontal axis, representing months, y on the vertical to represent total cost. And now I'm going to put these dots on
there and connect them. So let’s come over to my graph so I can start drawing the picture. For my x axis for months, the highest number I picked was 8. So I'm going to start numbering my axis going all the way out to 8. 1, 2, 3, 4, 5, 6, 7, 8, I'm also going to be sure to label that as months, so that whoever looks at my graph knows that I’m talking about 8 months and not like 8 years or 8 days or whatever.
Next thing on my y or vertical axis, I want to represent money spent. I'm going to start at zero because I'm not spending any negative numbers. My highest value from my table is 380. So depending on how big your graph is, it affects how many lines you want to skip. Like if I want to go to 380, I'm going to skip, I don't know, maybe two lines is going to represent $50, 200, $250. I hope I don't run out of room. 300, 350, three a little bit higher, okay 400. Hopefully you can still see that.
Be sure to label your axis, this represents total cost in dollars. So I'm going to put a dollar sign there, so my viewer knows I'm not talking about some other money denomination. Next thing is to get these points on the graph. My first point is 0 for x, 100 for y, so that's going to go here. Then I have 1,135, I'm going to approximate because each line in my graph represents $25. So I want to go up $35 it's like kind of in between those two. Two goes with the 170, again approximating, 3 goes with 205, almost there, here we go. Mow I'm jumping out to 8 and 380. If I did this correctly, even though there's this big jump in there, I should still have a pretty darn ruler straight line, so let's see. Yeah, looks pretty good
There you go. So this graph represents how much money I would have to pay for different amounts of months that I'm enrolled at the Pump-U-Up Gym. And the neat thing about using the graph, is that, if the problem asked me how much would it cost for something like 10 months? Actually let's not use 10 because that's off my graph, let's say six months, I could just use my graph to answer that question. Here's six months on my x axis, I would go up to my line and see that 6 months costs about $315. That's an approximation because as you can see those aren't the most perfectly straight lines but it's a pretty useful tool.
Once you have a graph of a real life situation like this, you can use it to answer lots and lots of questions. I think the hardest part is setting up your axis. But if you can get some good practice doing that, you guys
will have a lot of success with these problems.