The people at the Expo love Grant's product, his glasses wipers. But they find that washing their glasses with water just isn't enough. There needs to be some sort of cleaning solution that will be the solution to their problem. Grant decides to get on it right away. He contacts the three biggest lens cleaning solution vendors and asks them how much they charge to buy their solution in bulk. He needs this stuff fast. So, he asks how much it would be to get these samples rush ordered. Grant takes careful notes when he's on the phone with these companies. Once he's done, he has the following information. Brand number brand charges $8 a gallon plus a $200 rush order fee. And the third brand charges $10 per gallon with a $150 rush order fee. Now, Grant has some decisions to make. He needs to figure out how much solution he needs to buy and who he should buy it from.
Grant expects that each order he receives will require about 0.1 gallons of solution, or 1/10 of a gallon of solution. Grant also wants to have 10 extra gallons of solution on hand, just in case of emergency. What we're looking for now is an equation that describes this situation. For this equation, let's allow n to equal the number of expected orders of solution, and let's also let g equal the total number of gallons of solution that Grant should order. Which of the five equations that I've written down here relates the number of expected orders n, to the number of gallons, g, that Grant should order?
The first option, g equals .1n plus 10, is the best. The first term .1n represents the gallons of solution Grant needs in order to meet the number of orders he's expecting, and the 10 tacked on the end represents the 10 extra gallons he wants to have on hand.
So, Grant expects to have 900 total orders by the end of the month. According to this equation then, how many gallons of solution should he buy?
The question tells us that Grant expects to get 900 orders. And this translates in the problem, n equals 900. If we want to find out how much solution he wants to buy, which means solving for g, when n equals 900, we get g equals 0.1 times
Now going back to the 3 pricing schemes for the 3 different cleaning solution companies, we want to figure out how much each of these solutions would cost Grant's cleaning glasses company if he were to buy them. You're going to want an equation for each solution option so that he can compare how much he would spend depending on how many gallons of solution he wants to buy. Let's start by creating a general word equation that we can use later to create specific equations for each of these brands. Here are some quantities. Gallons of solution ordered, price per gallon, shipping time, total cost, rush order fee, and distance shipped. I'd like you to place the letter corresponding to one of these quantities in each of these boxes. To create an equation that you think fits a situation Grant is up against.
The letters that belong here are, in order from left to right, D, B, A, and E. So let me translate that into a word equation by inserting these in the proper spots. We end up with total cost equals price per gallon, times number of gallons ordered, Plus the rush order fee. Remember, that Grant will have to pay the rush order fee, no matter how many gallons he orders. But the more gallons he orders, the more he will pay total, since he pays by the gallon.
Remember the information we were given about the three different brands of cleaning solution. Using this information, can you create an equation in the form of the word equation we just came up with, but make it so that it only involves numbers and variables? For those variables, I'd like you to let total cost equal y, and the number of gallons ordered equal x. So, create a different equation for each company.
For the first brand, we have y equals 10x plus 100. All we had to do to get this was substitute the new variables into their proper places, place the price per gallon with that listed under the brand, and do the same thing with the rush order fee. If we go through the same process, we get y equals 8x plus 200, we get y equals 10x plus 150 for the third brand.
Now, these three equations we came up with are great and all, but Grant is a visual person. And frankly, so am I. So, we can help Grant make a decision about which of these three brands of cleaning solution would be best to go with, let's make some xy tables. I've given you values of x already, for each of the tables. And I'd just like you to fill in the corresponding values of y, according to the equation above each table.
For the first brand, our y values are 100, 200, 300, 400, 500, and 600. For brand number 2, we have 200, 280, 360, 440, 520, and 600. And for a third and final company, our y values are 150, 250, 350, 450, 550, and 650.
Now that we have our x-y tables, graphing is super easy. In fact, we have way more information than we need. We know that we only need two points to draw a straight line. And, we have six for each of these equations. Anyway, please identify which of these three graphs goes with each number brand. Put the number of the brand corresponding to the line that ends first up here in this box. The line that goes with the second arrow in the second box and the company that goes with this third line in this third box. I know that the difference between these lines are subtle but just pay careful attention to the differences between the equations and how you think that affects the way that they're graphed.
Let's start by just looking at the first entry in each of the xy tables and seeing which graph it matches to. Brand number 1 should have a line that has the that up. So, brand 1 should go along with this line that comes in second here. Brand 2 has a 0.0200, which starts out highest and then ends up with this third arrow over here. That means brand 3 should be this graph. But let's just double check to make sure. This line should contain the 0.0150. And if we follow it down, sure enough, that's on the line.
Now, that we have our equations matched with the proper lines, I'd like to point out that I see three special spots on this plot, the three points that all lie right here on the y-axis. Looking at our three equations, which, of course, represent these three lines, can you pick the term in each equation that determines these three points?
If we look at the coordinates of these three points, you'll see that they all have an x value of 0, and their y values are 100, 150, and 200. This corresponds to these final constant terms added on to the ends of our equations. So, those are the three terms that you should have selected. This constant at the end of each equation tells us where the line will intersect to the y-axis This is called the y-intercept.
So a y-intercept is, first and foremost, a point. And in particular, it's a point where a graph intersects the y-axis. Let's say that we wanted to represent a y-intercept in point notation.
The x-coordinate of any y-intercept is, by definition, 0. Let's draw some random line. Here's my lovely line. The y-intercept is where this line intersects with the y-axis. Remember, we have our x-axis here and our y-axis here. At the y-intercept right here, that x coordinate is just 0, since we haven't moved horizontally away from the y-axis. In fact, the equation for the line that is the y-axis, is just x equals 0.
Returning to Grant's cleaning solution's solution, let's inspect our three potential brands of cleaning solution a little bit more. Each of these equation seems to have a different y intercept so they all, all different equations. But, two of these three equations are pretty similar to one another. Which two seem the most similar?
The first one and the third one, seem really similar to me, because they've the same coefficient here in front of the x.
Looking at lines 3 and 1 on our graph, this being line 3 and this being line 1, what quantity do these two equations share? Do they have points in common? Do they have the same slope? Do they have the same x-intercept? Or do they have the same y-intercept?
They have the same slope. Let's probe a little bit deeper.
To make it a little bit easier to visually differentiate between our lines, I've redrawn our graph and changed teh scale on either axis. So that we're only looking at a portion of the points that we were looking at before. I've also now color coded the lines to go with the different brands of solution. So brand one over here, which is written in dark grey, has the grey line on the graph. Brand two is in blue, and brand three is in orange. Now, using the techniques that we learned in the last lesson to calculate slope from points, can you calculate the slope of each of these three lines?
The slopes we end up with are 10 for brand 1, 8 for brand 2, and 10, again, for brand 3. This is really interesting. For both brand 1 and brand 3, when we move over 10 in the x-direction, we have to move up 100 in the y-direction. And, of course, 100 divided by 10 is just 10. I drew this right triangle on the gray line, which corresponds to brand 1, but this same exact triangle can be drawn for the line for brand 3. This has exactly the same proportions. It doesn't however work for brand 2. What's even more interesting, however, is how this ties in with our equations. We said before that brand 1 and brand 3 seem the most similar because they both had coefficients of 10 in their x terms. 10 is what we found for the slope of each of them as well. So, seems like this number, the coefficient of the x term in our equation corresponds to the slope of the line. This is also true for brand number 2, we see a coefficient of 8 in the term 8x, and a slope of 8.
But remember where all the numbers in these equations came from? Grant's glasses situation, of course. What did these numbers 8 and 10, that we now see in our equations, originally refer to? Did they come from each company's price per gallon of solution, the total price of cleaning solution, the number of gallons of solution Grant would buy, or the rush delivery fee?
The price per gallon. The price per gallon is measured in dollars over gallons. And this makes perfect sense if we think about what quantities we're using to calculate this slope on this graph. y here, stands for the total price of solution, which is, of course, measured in dollars. And x here, stands for the number of gallons of solution that Grant has bought, or is willing to buy. And that's measured in gallons. We know that slope is equal to the change in y over the change in x, which is, just as we found here, dollars over gallons.
Thinking about what price per gallon or the units of the slope, means in the context of Grant's story and about what we've learned about slope in general, what does slope mean overall for equations and graphs? Is it the factor by which y will, will scale as x changes? Is it the ratio of the change in x to the change in y? Is it the steepness of a line? Or is it the direction a line points in? You can pick as many of these as you feel are correct.
As you might remember from the last lesson, the slope is not, and I repeat not, the ratio of the change in x to the change in y. It's actually the other way around. If we switch this to a y and this to an x, then we have the slope.
We've encountered several situations in which the slope of a line was the coefficient of a term that an x to the first in our equation. But my question is, does this hold true in all cases? For which of these equations down here do you think that the slope of the line is the coefficient of the term with an x to the first in it? I know that you may not have an idea of what the graphs of each of these equations look like. But think about the form of the equations that we've looked at so far, and which equations down here might have the same graphical properties as those ones.
And the answer is that these two equations, y equals 5x and y equals 12 minus coefficient of the term with an x to the first power in it. The equations here which don't just make straight lines when we graph them, which, as we'll learn later on in the course, are those that have factors of x combined with other things besides constants. So, those would include this first equation, this very long, not very nice equation, this one and this one definitely don't have this property. In fact right now, we have no idea how to talk about the slope of line that isn't just a straight line. If we have some sort of funky curve, how can we talk about the direction that it's pointing? It's pointing in different directions at all these different points. So, we can rule all of those answers phases out. We'll talk more about the slopes of these four remaining equations in just a second.
I have compiled in this little chart over here, some of the equations for which we've already said that we could calculate the slope or rather just read the slope off of the equation by looking at the coefficient of the x term. And I've also included two equations in which we said that, that was something we could do. My question is, what is unique about these questions on the left that allows us to just read the slope off of their x term coefficient? Can we do this because of the form of the equation because there are two or fewer terms on the right side of each of these equations? Because there's nothing added to y on the left side? Or because they have a y and an x?
The answer is the form of the equation. All of the equations on the left-hand side start out with y equals, and then on the right-hand side, have a constant term and a term with x to the first in it. The two equations in this right-hand column do not fit the same form. We'll talk a lot more about the form that these equations are in as we move forward.
If I write a general equation, y equals mx plus b, which may look reminiscent of those equations in the left-hand side of our previous table, can you tell me what the slope of this line is?
Since this equation is in the same form as the equations for which we could read the slope as the coefficient of the x to the first term, the slope for this equation is m. In fact, this is the letter that we usually use to represent the slope of any line, m.
Still considering this general equation for a line in this form, which letters represent variables? This is actually kind of a tricky question. So think about on a line, which values you want to vary and which values we want to stay the same.
The variables here are y and x. m and b may be represented by letters in this general form, but they actually stand for constants, that for a given line, will just be numbers.
I've written down several of the equations that we've seen in this lesson so far. And for each of these equations, I would like you to identify what number is m, the slope, what number is b, the y coordinate of the y intercept, and how we could write this equation explicitly in the form y equals mx plus b.
For each of these equations, m is just the coefficient in front of the term with an x to the first. So, in the first equation, y equals 100 plus 50x, m is just the last equation, we have y equals 12 minus 3x, m is equal to negative 3. Since you will recall that in the form of the equation, the general form, we have y equals mx and no sign in front of the m, so we need to include the sign as part of the coefficient. The second potentially tricky thing is for the equation, y equals 5. We don't have a term here with an x. What that really means, is that we have the term plus 0x added onto the end of this. Clearly, 0x is just equal to 0, but writing this lets us see that m, the slope here, is actually equal to slope is 0. Moving onto our values of b, these are just the constant terms on the right side of each of these equations. Negative signs need to be included as part of the b term, and then the equation y equals 5x, since we don't explicitly see a constant term added out here, b must be equal to zero. Having this information written out explicitly like this, makes writing the equations in y equals mx plus b form pretty simple. Hopefully, now you feel very confident that all of these equations can be written in this precise form.
Which type of lines, or types of lines, can not be written in this form we have just been talking about, y equals mx plus b?
The answer is that vertical lines cannot be written in this form. Remember that the slopes of vertical lines are undefined. Because as we saw earlier, when we try to calculate the change in x in order to find the slope, we end up with a denominator of zero. And dividing by zero does not give us defined answers. Or in other words, answers that are numbers that we can work with. Since we can't come up with a number to represent m, we can't write vertical lines in this form. Furthermore, since vertical lines are parallel to the y-axis, they will never intersect it. That means that we can't find a value for b in this form of the equation either. Vertical lines, therefore, do not have y intercepts, and have undefined slopes. Which makes it impossible for us to write it in this form that involves slopes and y intercept's.
We solved these two equations earlier on. And now I'd like you to try to rewrite them in the form y equals mx plus b. So, use your understanding of how to manipulate things in Algebra to rearrange and modify the terms in the equation.
We find that for the equation, 5y equals 10x, we can end up with y equals 2x. And we can change negative 3x plus y plus 2 equals 0, so that instead it reads, y equals 3x minus 2. I just used the same tools that we used in previous lessons to isolate y in each equation. So, even if an equation isn't initially written in the form, y equals mx plus b, as long as its graph is a straight line and, of course, it's not a vertical line, it can be arranged so it does fit this form. And, of course, once we get it in that form, it's super-easy for us to find the slope and the y-coordinate at the y-intercept.
We've been talking a lot about equations that are in the form, y equals mx plus b. And we've talked about how m here represents the slope of this line, and b is the y-coordinate of the y-intercept, the place where the graph intersects the y-axis. When we write an equation in this form, we say that it is written in slope-intercept form. Since, as we've seen, this lets us just read off the values of the slope and the y-intercept. Understanding the rule that each of these constants play in this equation and its graph, can help us get to the stage of graphing a lot quicker. Let's see if we can figure out how to graph the equation y equals negative 2x plus 10 without creating an xy table first. Let's just move straight to the graph. Of course, we have to start by setting up our x and y axes. Notice here that the scale on either axis is different. Horizontally, we're counting by ones with each grid but vertically, we're counting by fours. I just want to make sure we have enough room to show as much of this line as I want to. One pretty easy way for us to start out would be by filling in a point that we can find out super easily. So, please begin by telling me where the y-intercept of this equation is. Please don't forget that the y-intercept is actually a point, a set of coordinates, not just one number.
We see in our equation that b as in y equals mx plus b, is the number 10. So this is the y coordinate of the y intercept. Since the y intercept has to lie along the y axis, its x coordinate must be 0. So the y intercept is at 0, 10. Let's graph that. It should be right around here. Equidistance between 8 and 12.
Now that we have one point, let's find some more. The slope of negative 2, since negative 2 is the coefficient of the x term, so it is m in this equation, tells us that for every one unit we move to the right, we need to move down negative 2 units. Thinking about the point that we already know in the slope, can you name another point that should lie on this line?
Your answer here can actually be any number of things. As long as the coordinates that you pick, x and y, can be substituted into this equation and make it true, they should work. We can play around in the graph to find a few of those points. Keeping in mind the scale of either of our axes, we can go over 1, and down 2 from our y intercept of course, to find the point 1,8. We can do that again, to the right 1, down 2, and that gives us 2,6. We can keep doing the same thing, over and over again. We probably want to move in the other direction as well. If we reverse what we are doing in x, so we move to the left instead of right, then we need to do the opposite in y as well. So we need to move up, instead of down. So to the left 1, and then up 2. That gives us negative 1,12, and so on and so forth. Once you draw enough of these points it's time to connect them with a line.
Now that we have this lovely line, we should be able to figure out a ton of different points that satisfy this equation. If we know that y equals negative
The x coordinate should be 9. We can follow the y axis down to negative 8, and then stay on that horizontal line, y equals negative 8, all the way over until it intersects our line. That happens at this point right here. If you go directly upward from that point to the x axis, you see that the x coordinate is coming from the origin outward 9. We could also check this by substituting negative 8 into our equation in the spot of y, and then solving for x.
Before we used the equation of a line to draw a graph, but how about this time we use the graph to find the equation? I've drawn three lines for you on the set of axes, and I'd like you to start by finding out the slope and the y-coordinate of the y-intercept for each of these lines. Please notice that I have labeled them f, g, and h. So, just fill in this table with the proper values for m and b.
Let's start by looking at line f, this turquoise line right here. I like to find the y-intercept first, which on line f, is at 0, negative 2. So b is just negative 2. Then I can use this point to help me find the slope. If I move over slope is going to be negative 1 over 1 or other words just negative 1. For line g, the y-intercept is actually the origin, 0, 0. This means that b has to equal right 2 units and then down 1 unit to reach the graph again. So, the slope is going to be negative 1, since we moved down 1 in the y direction, over positive we come to line h. This intersects the y-axis at 0, 3, b then is 3. Interestingly enough, just like with g, if we walk over 2, we have to walk down
Now that you've found the slope and the y-intercept of each of these three lines, please write an equation for each line for me.
Since we have m and b, we can just plug these in to slope intercept form to find the equation for each of these lines. Line f has the equation y equals negative x minus 2. Line g we can just write as y equals negative 1/2 x. You could, of course, also write plus 0 out here, but that's not necessary. And lastly, line h is y equals negative 1/2 x plus 3. As we saw before, lines g and h have the same slope, negative 1/2, and we can see on the graph that they look parallel. They don't have the same y intercept, but they do point in the same direction. Parallel, as we learned very early in the lesson, just means same slope.
As I talked about earlier, if 2 lines are parallel, that means that they have the same slope. Considering what parallel means, and your understanding of how to form equations of different lines, can you tell me, then, what the equation of a line parallels to the line, y equals 1/2x plus 3, but that has a y intercept of 0 negative 2 would be.
If the line we're looking for is parallel to the line y equals one-half x plus start off by writing y equals one-half x, since this is y equals mx. And then, we just need to figure out B, the y coordinate of the y intercept. However, we know the y intercept we're looking for. It's 0, negative 2. So we just add a minus 2 to the end of this equation. Let's graph this, just to see what it looks like. And sure enough, we can see that this orange line, which corresponds to our new equation, has the same slope as the original line. But it's translated down so it has a different y intersept. These lines are definitely parallel.
Now that you've seen a couple examples of the slopes of pairs of lines that are perpendicular to one another, it'll be great if we could create a general statement for how slopes of perpendicular lines are related. In one case, we had a pair of perpendicular slopes where one was 1/2 and the other was negative 2. And in the other case, we had the slopes 1 and negative 1 as perpendicular. So using these as guidelines, or rather just as examples that should fit our general rule, please create a general expression for m2 in terms of m1. This is definitely kind of tricky, so just try out a couple things, and then see if these pairs of slopes fit your formula.
And the answer is that m2 should equal negative 1 divided by m1, or the negative reciprocal of the slope that it's perpendicular to. You can see that both of our examples conform to this rule.
Which of these lines, whose equations are down here, are parallel to the line y equals 4 x minus 5?
And the answer is that these three, the ones that I've checked off, are parallel to y equals 4x minus 5. We can see this by doing a little bit of rearranging so we get each of these equations in the form y equals mx plus b. If we add y to both sides of this equation, we can rearrange this equation so that instead it reads y equals 4x. This one is already written in slope-intercept form, and we can see that it's slope is, in fact, 4. Just like that of our original line. If we divide both sides of this equation by 2, we end up with y equals 4x minus one-half. Here, m is the same as in this equation. For each of these lines, however, which are not parallel to this equation, in order to get y by itself, we would have to divide both sides by 2. This would change the coefficient in front of the x that it's no longer 4, which means that the slope of this line is actually 2. The same thing would happen for this equation. So, this is the perfect example of how sometimes, even if the coefficient in front of x, makes it look like a line could be parallel to another line. We have to make sure that the equation is actually in slope intercept form before we can make assumptions about that.
We've been talking about parallel lines a lot recently, but here I've drawn two lines that are clearly not parallel to one another. Rather, these lines are what we call perpendicular to one another. Making us some observations about these two lines that I've told you are perpendicular, what do you think that you can say about perpendicular lines in general? If two lines are perpandicular to one another do they have the same slope? Do they have different slopes? Do they have specially related slopes? Do they have slopes are related in a special way? Should they form a right angle with one another, or do they have slopes with oppisite signs? You can pick as many choices as you think are correct.
Two lines that are perpendicular to one another have different slopes. And not only that, but these slopes are different in a very special way. They also form a right angle with one another, as you can see that these two lines do right here. And lastly, they have opposite signs. For example, this light blue line in our graph has a negative slope. And this one has a positive slope. We'll talk more about these specific characteristics of perpendicular lines in the coming videos.
We know that the two lines on this graph are perpendicular to one another, and we just discussed that being perpendicular has a lot to do with slope. Now, I've named these two lines, p and q, and I'd like you to start off by telling me what the slope of each one of them is.
The slope of line p is 2, and the slope of line q is negative one-half. And I'd like you to reflect on this for just a second and think about what that special relationship between the slopes of perpendicular lines might be. You don't have to know for sure right now, but just start mulling it over.
Here we have another pair of lines that are perpendicular to another, line s in orange and line t in teal. Just like last time, please find the slope of each of these lines. See if after you find the slop of one of them, you can take a guess at the slope of the other one before you calculate it.
The slope of line s is just 1, and the slope of line t is just negative 1.
Using this new pattern that we've found, relating the slopes of perpendicular lines, what would be the slope of a line perpendicular to a line that has a slope of 10?
You know that in general, as we saw before, we just need to take the negative reciprocal of the first slope. So m is going to equal negative 1 divided by the first slope, which is 10. Or, negative 1/10.
Let's try another one. What if we have a line that has a slope of three quarters and we want to find the slope of the line perpendicular to that?
Just like before, we need to take negative 1 and divide it by this first slope. This is, of course, the same as multiplying by the reciprocal of the fraction and we get a final answer of negative 4 over 3.
For each pair of lines listed right here, please tell me whether they are parallel, perpendicular, or neither.
The answers are, from left to right, parallel, neither, perpendicular, and perpendicular. We can see right off the bat that for the first pair of lines, they have the same slope, 3. In the second pair of lines, it might at first look you have similar situation, but because of this coefficient of 7 in front of our y term in the 2nd equation, we actually end up seeing that we have a slope of 3 over 7 when we divide both sides by 7. So these lines are not parallel, nor are they perpendicular. For the third pair, negative 3 times 1/3 is negative 1. Or equivalently, 1/3 is equal to negative 1 over negative 3. Either way you look at it, these lines are perpendicular. And last but not least, to figure out the slope of this top of this equation, you need to divide both sides by two, which gives us a slope of 1/2. In the second equation you see a coefficient in from of the x term, of negative 2, and negative 1 over negative 2 is equal to 1/2. So these two lines are perpendicular.