So far, we've seen that when we have an equation. An equation like, for example, y equals 3 x minus 5, we can graph that equation. And when we graph that equation, any point that we can pick out that lies along this line will satisfy this equation. If we pick a point along here, like this one let's say, we can figure out its coordinates. Substitute them into x and y respectively, and then find the two sides of the equation are equal to 1 another. In principle then, we should be able to check whether or not a point is on the line of an equation without having to look at the graph at all. As great as visual representations are, algebra itself is super powerful. Let's test out this theory.
Does the point 3, 7 lie on the line y equals 3x minus 5? I'm going to walk you through some of the steps to do this, or at least give you some hints about how we should go about it. I'd like you to start by substituting in the proper value from the point into the left-hand side of the equation. L H S here is just shorthand for left-hand side. So type in the value here of the number that this side should simplify to. Then do the same thing for the right-hand side. That's what R H S stands for. Plug in the proper coordinate to the proper variable, and simplify this side of the equation. Once you've found the numbers that belong in these boxes. Tell me if they're equal to one another, and then, depending on how you answer that question, what does this say about whether or not this point lies on this line?
The left-hand side of the equation is just y, and in place of y, we're substituting in the y-coordinate at this point, which is 7. So, the left-hand side should just become 7. The right-hand side, as we can see over here, has an x in it. So, we need to substitute 3 into that spot. Simplifying that all the way, gives us 4. Unfortunately, 7 is not equal to 4. In answer to the question, are the sides equal, we have to say no. Since plugging in the coordinates this point makes it so that the two side of our equation are no longer equal, which we remember is the requirement of an equation, this point must not lay on this line. In other words, these values for x and y do not satisfy the equation, y equals 3x minus 5.
Here is for you to try on your own. Go through the same set of steps, but this time, let's ask whether the point 1, negative 2 lies on the same line.
The left-hand side just becomes negative 2, and the right-hand side also becomes negative 2. Negative 2 is, in fact, equal to negative 2. So, yes, these two sides are equal. And that means, yes, this point, 1 negative 2, does lie on the line y equals 3 x minus 5.
Let's try one that's a bit harder now. Does the point 3,4 lie on the line y squared plus 2x squared equals 34?
Just like before, we substitute in our x-coordinate in the spot of x and our y-coordinate in the spot of y, then we just simplify. For the left hand side, we end up with 34. For the right hand side, we start out with 34, and there aren't any variables involved in this expression. Since we don't have any place to substitute anything in We can just say 34 is equal to 34. And again, looking at these two sides and how they compare, 34 is still equal to 34. So, yes, these two sides are equal. That means that we can say, that even though we may not have any idea what this curve looks like, we know that this point lies on this line. You might see some equations like this later on in the course. Get excited.
Earlier on, we looked at solving equations which looked something like this. Here, we have 3x plus 4 equals 19. Let's consider something slightly different however. What if instead, we have the equation y equals 3x plus 4, let's say that we want know the value of x when y is equal to 19. Well, we know on one hand that we can just use substitution to replace y here with 19 and end up with the equation that we saw before. Then, we could isolate x and have an answer for it. We can also however use a graph to figure out the x-coordinate when y equals scale of the x-axis versus the, the y-axis. On the y-axis, every grid line up counts as 2, whereas, on the x-axis, it's only
If we go up the y-axis to find where y equals 19, we come just below the upper bound of this graph. Then, we need to slide over, staying at that horizontal position, until we meet our line. That gives us this point right here. To figure out that x coordinate at that point, we follow straight down from that point to the x-xis and see that we hit 5. This must mean that when y equals 19 somewhere along this line, x is also equal to 5.
Earlier on, we talked about how many gallons of cleaning solution Grant should buy depending on how many orders he received from customers. Initially, Grant had thought that he wanted to keep ten extra gallons of cleaning solution on hand at all times, but he's changed his mind a little bit. He's gotten a ton of orders, more than he was expecting. So, he's decided to adjust his surplus from gallons of cleaning solution Grant should buy, depending on many orders he's received? To be super clear, I've written down what each of our variables in the original equation stood for. Also, the surplus of cleaning solution is the only thing that Grant wants to change. He's not going to change the number of gallons that he buys per expected order.
The answer is, our new equation should be g equals 0.1n plus 20. Grant is not going to change the number of gallons that he buys, depending on how many orders he is expecting to get, the only thing that he is changing is the amount of extra solution he is buying, which is just this constant added out on the end.
Now, that we have equations for the total number of gallons that Grant will buy, let's compare the old equation we would have used to calculate how many gallons of solution Grant would buy to the new equation that we're going to use. How are these two lines related? Are they parallel? Are they perpendicular? Do they have different slopes? Do they have the same slope? Do they have different g-intercepts, or do they have the same g-intercept? Notice that we have g-intercept here instead of a y-intercept, because our variable over here is a g instead of a y. This is just different notation. Graphically, it really means the same thing as a y-intercept since we would create axes to graph these equations, with n as the horizontal axis, and g as the vertical axis.
These lines are both written in the same form, even though we have different variable names now than we normally do when we talk about slope intercept form. These equations are both in the same form as y equals m x plus b. We have a variable on one side, a constant times a variable, and then a constant term tacked on to the end. This is true for this equation as well. Becasue they are in slope intercept form, we can just read off their slopes from the equation directly. They do not have different slopes, they have the same slope. That's what being parallel means after all. What changed we moved from the old situation to the new situation with the surplus solution was the g intercept. We can find that g intercept need equation in the slide where b normally is in slope intercept form. If I wanted to graph these pretty informally I may little sketch up here, I would have our first line like this. In the second line, exactly parallel to it, but just shifted up on the vertical axis. The information we used to create the second equation then, came from us knowing the slope that this new line should have, and also one point that it should go through, 0, 20 which we reflected in the g-intercept part of the equation.
If Grant expects to receive 550 orders from customers, how many gallons of cleaning solution should he purchase? Please make sure that you have your answer reflect the new situation with the surplus of cleaning solution.
We need to start off with our new equation for g, the total number of gallons that Grant is going to buy. Since n stands for the number of orders that Grant is expecting to get, we need to substitute 550 in the spot of n. Then, we just need to simplify the right-hand side of the equation, to solve for g. 55 plus 20 is 75. So, the answer is 75 gallons.
Now that we've found out how many gallons of cleaning solution Grant's going to buy, we can look back to our equations that dealt with the amount that cleaning solution would cost for him, depending on which brand he decided to go with. And each do a little bit of variable switch up though. To get back in the paradigm that these equations were written in, I'll remind you of what each of the variables stood for, x here stood for the number of solution that Grant would buy, and y was equal to the total cost of Grant's order, taking into account both the solution that he bought in the rush order fee for each company. Setting up our coordinate plane with a scale of 10 on the x axis and 100 on the y axis, I'm going to graph these 3 lines again. To make this graph easier for you to read and understand, I've color coded the lines again just like we did before, and then written their corresponding equations in the proper colors. We've talked a ton in the past about how we can find values of variables that satisfy equations but we can also do this graphically. Think for a second, if I wanted to find out how much 75 gallons of solution will cost for each brand that Grant might buy from, what extra line should I add to this graph? Please write the equation for that new line in this box.
In the context of this problem and these equations, finding out that Grant wants to buy 75 gallons of solution means that we want to set x equal to 75. Now usually, we just think about something that looks like this as assigning a value to a variable and that is true. But don't forget, this is an equation of a line.
Since we're now talking about x equals 75 as a line, what kind of line is that? Do you think it's a horizontal line? A vertical line? A line with a positive slope? A line with a negative slope? Or do you think it's not a line at all, but it's a point instead?
x equals 75 is a vertical line. Let's graph it. The condition this imposes for every point along it is just that the x value of all of those points has to be rather all values of y from negative infinity all the way to infinity. So, here's our lovely new purple line, just to add some thing new to mix up our color scheme, that graphs x equals 75.
Now that we've graphed this new line, how should we actually use it? Why do they want us to do this in the first place? Remember that what we're trying to do in the end is find out how much Grant will pay to each of these companies if he decides to buy 75 gallons of cleaning solution from them. So should we compare the line x equals 75? Should we compare the slope of the line, x equals 75, to the slopes of these other three lines, the x intercept of this vertical line, to these line's x intercepts? Should we do the same thing with its y intercept instead? Or should we find out where this line intersects each of these three lines?
We should find out where x equals 75 intersects each of these three lines.
So, we want to find the places where this lovely purple vertical line intersects each of these 3 other lines. But why? What is the point where 2 lines intersect actually represent? Is it a point that lines on both of those lines? Does it tell us values for x and y that satisfy the equations of both of the lines? Is it a point where one of the lines turns into the other line? Or is it a solution to both equations?
These three choices are correct. At the spot where two lines cross, or intersect, they have to go through the exact same point, which is why they are able to touch in the first place. So this point of intersection lies both on this orange line and on this purple line. They both pass through this location. Remember, though, that a point on a line gives us x and y values that satisfy the equation for that line, or in other words, make it true. If a point lies on both lines, then it must do that for both of those equations. That means that for any two lines that intersect, there's some pair of x and y values that we could plug into either equation, and create a true statement for each. This is the same thing as saying that these values create a solution to both equations.
We know that Grant wants to buy 75 gallons of cleaning solution, which is why we graphed this line of x equals 75. But what happens if he decides to buy that from brand number 1? Can you use our graph to figure out how much he's going to pay? I've labelled each of the lines on our graph according to the number of the brand it corresponds to. So, the teal line is for brand 1, the orange line is for brand 2, and the light blue line is for brand 3.
What we're looking for is the point of intersection between our line x equals 75 and the line that corresponds to brand 1. So, we follow the teal line up until it intersects the purple line, which happens right here. We follow that over to the y-axis. That point looks like it's halfway between this grid line and this grid line, or halfway between 800 and 900, if we count up from 500. So, it looks like Grant will pay $850.
So, for $850, Grant can get 75 gallons of cleaning solution from brand number 1. But he's curious as to whether or not this is the best use of his money. If he's going to spend $850, should he really be going with brand number 1? In other words, could Grant get a better deal with his $850? Pick yes or no. Of course, we're only considering these three companies, and their different cleaning solution offers.
The answer is yes. Grant could definitely get a better deal.
Before we talk about the best way for Grant to spend $850, if that's how much he really wants to spend, let's talk about being fixed on buying 75 gallons of cleanser. So, what is the least amount that Grant could pay in order to get 75 gallons of solution? And in addition to this, which company would he have to buy the solution from in order to get this great price?
If we're talking about paying for 75 gallons of solution, we need to think about three points of intersection made by each of the three company lines, and our very familiar now vertical x equals 75 line. If we're curious about the least amount Grant could pay, we need to see which one of these three points of intersection has the lowest y value, or the smallest y coordinate. Of these three points of intersection, this one is the lowest, so it must be the one that we're interested in. The Y value of this point, if we follow this grid line over, looks to be 500, 600, 700, 800. So the best deal that Grant could get if he wants to buy 75 gallons of solution will be paying $800. This point that we're talking about is a point of intersection between the purple line and the orange line, and the orange line corresponds to company 2. So company 2 is the one that's going to give him the $800 deal on 75 gallons of solution.
So, $800 for 75 gallons sounds like a pretty good deal. >> But Grant really is willing to pay $850. >> He set aside that amount of money for cleaning solution, >> And he wants to get the most bang for his buck. >> What new line could we graph on this coordinate plane ... >> To help us figure out how many gallons of solution Grant could buy from each of these three companies ... >> For $850? >> Please put the equation for that line in this box right here.
Thinking back to what our variables x and y stand for, x is the number of gallons of solution that Grant will buy, and y is how much it's going to cost him to buy that much. If we fix the amount that Grant's going to spend, at $850, what that really means is fixing our y value at 850. So, the line we need is just y equals 850. Let's graph that. And here is our lovely horizontal line.
Let's quadruple check that you fully understand the connection between Grant's situation that we're talking about, the graph that we've drawn, and the equations that we're going to use. If we want to know how many gallons of cleaning fluid Grant can get for $850, what should we look for? The value of y when x equals 850? The value of x when y equals 850? The slope when x equals
We want the value of x when y equals 850, since remeber y in this situation stands for the total cost to Grant. And x stands for the number of gallons of cleaning fluid he'll get. If we follow the line y equals 850, over to where it intersects other lines, we'll be finding the points on those lines that have a y coordinate of 850. From there, we can find the x-coordinate that goes along with those points, and we'll have our solution.
Let's consider one of these points of intersection, the one between this new pink line and this light blue line. If we call the point where they intersect p, then what two equations can we use the coordinates of point p to relate to one another? Please fill in those two equations down here.
y equals 850 and y equals 10x plus 150. These are just the two lines that are intersecting at that point.
We know that at point p, these two equations should be satisfied by the same values of x and y. So, can you figure out a way to combine them into a single equation, where x is the only variable? We want to keep x as the variable because it's the value that we are looking for.
Since point p is a point that is shared, by both the line, y equals 850, and the line y, equals 10x plus 150, both of these lines have the same y value at that point. Since the y value of y equals 850 is always 850, that means that this the y value of 10x plus 150 at point p. So, we just need to substitute 850 into the spot of y, in the second equation. That gives us a single equation of 850 equals
Now that we have an equation with one variable in it, let's solve for that variable. What is x equal to?
With the information we've compiled over the last several quizzes, can you tell me the coordinates of point p? Please be sure to write in ordered pair notation.
We just solved for the x value of point p as 70, and we already knew that its y value is 850. So, those are its coordinates.
Finally, let's figure this out. How many gallons of solution will Grant get from each company if he pays them $850? Fill in each of the amounts in the proper boxes. And then lastly, based on your 3 answers up here, tell me which number brand is the best bargain?
To figure this out, we just need to find the points of intersection between y equals 850 and each of these three lines. So, they are here, here, oh, wow, three-way intersection, and here. Brand 1, as we've already figured out, would give Grant 75 gallons for $850. Brand 2 would give him 80 gallons and brand 3 would only give him 70 gallons. So clearly, brand number 2 is the best deal.
Grant has historically been pretty frugal, however, and it's pretty possible that he'll want to cut back on the money he devotes to spending on cleaning solution. That instead of $850, he decides that he only wants to spend $400. How many gallons could that get him from each of our 3 brands? Once you use our graph to figure that out Please tell me which brand is the best deal.
Each of the three solutions here that we're looking for is the x-coordinate of the proper line at the point where the y-coordinate is equal to 400. So, we're looking for where each of these lines intersects the line y equals 400. For line number one, this happens right here, at 30 gallons. Brand 2 and brand 3 actually both intersect our lovely green line at the same point. So, all three of these lines the orange, light blue, and the green, meet at whatever this point is. We know that the y-coordinate is 400 and the x-coordinate is, if you look down at the x-axis, 25. Taking into account how many gallons of solution each of these brands can give us for $400, it looks like now, brand 1 is actually the best deal. So, turns out that it's a bit trickier to choose which brand is the best to go with then Grant might have thought earlier. As you can see from our graph, the brand that's going to help him use his money most efficiently is going to be different depending on how much money he wants to spend. No matter how much money that Grant decided to spend on cleaning solution, this graph is definitely going to come in handy to help him make the best choice.
Through helping Grant mull over this cleaning solution issue, we learned about the use and the importance of points of intersections on graphs. When two lines cross one another, the coordinates of the point where that happens give us values of variables that satisfy the equations of all of the lines that pass through that point. So, we are able to use this knowledge to solve for different variables. We could just take the equations corresponding to either line, figure out how to set one side of one equation equal to one side of the other equation, and then solve for the unknown variable. That would give us a new coordinate of that point of intersection. But, what if instead we didn't have a graph? What if you just had an equation, and not just any equation but an equation with one variable, something like 3 x plus 4 equals 19? Now that we know we can find values for variables through finding points of intersection of lines, can you think of two lines we could graph whose point of intersection would help us solve this equation? Please put a two variable equation in each of these boxes that we could graph on the same coordinate plane to help us work with this equation. This definitely requires a little bit of outside the box thinking. So, just give it a try.
Our two equations are y equals 3 x plus 4, which of course incorporates the left side of this equation, and y equals 19, which deals with the right side. If we have these two seperate equations, but we know that at their point of intersection, they share a y value, then at that point, this y and this y will be the same. So we can set the other side of each equation equal to each other, which gives us 3x plus 4 equals 19.
Now that we figured out which two equations we need to graph, can you tell which two lines on this plot these equations correspond to? Please pick the two letters out of this list that correspond to the lines that you think we need.
y equals 3x plus 4 has a y-intercept of 0 4, which is this point right here and a slope of 3, which makes it really steep. So, it must be this line, which is line g, so I'll check that one off. y equals 19 is a horizontal line that is located 19 units above the x-axis. So, if you go 19 up, we hit line d. So, those are the two lines we need.
I've highlighted the two lines that we selected in the last quiz in this dark blue color. I'd like you to first to tell me what the point of intersection of those two lines is. And then using that, tell me what our solution for x in the earlier equation should be.
Our point of intersection is right here. Because this point lies on the line y equals 19, we already know that its y-coordinate must be 19. Its x-coordinate we can find by just tracing this down to the x-axis, straight below the point. And that is 5. So, our point of intersection is 5,19. The x-coordinate here is 5, so x must equal 5. Let's check this in our original equation. We started off with the equation 3 x plus 4 equals 19. Let's plug in our x value to see if this works. On the left-hand side, we'll have 3 times 5 plus 4, which simplifies to can just write 19. Sure enough, 19 equals 19. So, we're good. x equals 5 is the correct answer.
Let's start with a slightly more complicated equation. 3 x plus 4 equals 2 x minus 1. Just like I asked you before, if I want to find the solution to this equation through drawing two lines that intersect one another in order to find this value of x from their point of intersection.
Since at a point of intersection, the 2 lines that are crossing share both an x value and a y value, and we know that we want this x to be the x coordinate of some point of intersection, we can say that these two lines already share a y value. And we can let that y value equal this side and this side. If we set y equal to this side of the equation, we get a new equation y equals 3 x plus 4. So that could be this line for example. The line that it crosses at this point of intersection where we have this x coordinate is y equals 2 x minus 1.
I've graphed the lines of the two equations that you came up with in the previous quiz. And now, I would like you to find their point of intersection and from that, tell me what the solution for x was in the original equation.
Our two lines intersect at this point. We follow that up to the x-axis, looks to be halfway between these two grid lines, which are x equals negative 4 and x equals negative 6, which means, the x-coordinate is negative 5. The y-coordinate, if we go over the y-axis, is negative 11. So, this point is negative 5, negative 11. And the x-coordinate of that is negative 5. So, x equals negative 5. Remember that the original equation we started with was 3x plus 4 equals 2x minus 1. Let's make sure that negative 5 actually is a solution for this equation. If we plug in negative 5 to the left-hand side, we get 3 times negative 5 plus 4. And this simplifies to negative 11, which was, in fact, the y-coordinate that we found for our point. On the right-hand side, we have 2 times negative 5 minus 1, and that also equals negative 11. So, the solution for x works for our original equation. Awesome.
Now that Grant's finally made a decision about which cleaning solution brand to go with, he's decided to sell all the different products he's marketing in a package. He's going to sell nozzles and wipers and cleaning solution for one fabulous deal. Taking all the various costs of creating his products and shipping them and everything into account, Grant calculates that he'll make a profit of $1000 for every 60 glasses cleaning sets that he sells. And if he sells 100 glasses cleaning sets he'll make a profit of $1200, assuming that the relationship between number of cleaning sets sold and profits is linear, or in other words lies on a straight line, we should be able to predict, how much of a profit Grant will make, for any number of glasses cleaning sets that he might sell. It would be really great if we could plot this information on a graph. And then use that graph to find out about other points on the line. If we let x equal the number of cleaning sets that Grant sells and y equal his profit in dollars, can you write each of these pieces of information as a point on the line that we're going to be able to draw?
Our 2 points are 60, 1000, and 100, 1200. We can figure this out by just looking at which pieces of information will count as x values and which ones will count as y values.
Now that we have our two points, we need to find the equation of the line that goes through both of them. Let's start by finding its slope.
Remember that slope is the change in y over the change in x. So, we just subtract one of our y-coordinates from the other and one of our x-coordinates from the other, making sure to use the same point first in each case. I'm going to have this point give us our second coordinates and this point give us our first coordinates. This gives us a final answer of 5.
Now we have the slope of our line, and we still note two points that it passes through. Can you use this information to figure out the equation of our line? Please write it in this box. Just a hint, think about how you can use slope intercept form to do this. If you're having trouble remembering what slope intercept form is, you can just go back a few videos and review it.
An equation is in slope-intercept form if we can write it as something like y equals mx plus b, where m is the slope, which conveniently we already have for our equation, and b is the y-coordinate of the y-intercept. Since we have slots for x and y in our equations and we conveniently already have two points that we know can belong in those two spots, we can use either of these points and plug it into the slots for x and y. So we can start off with what we already know, using the slope. I am going to use this point, 100,1200 and plug 100 in for x and 1200 in for y. That's because we know that this point, or the coordinates of this point, when substituted into the proper places in this equation, satisfy this equation, or make it a true statement. That means we're allowed to use them. I still don't know what the y coordinate of our y intercept is, so I would really like to figure that out with the equation. So I want to get b by itself on one side of the equation, and I end up with 700 equals b. Now you can take this value and put it back in the equation we started with. That gives us a final equation of y equals 5 x plus 700. So this is our answer. You could have of course used the point 60, 1000 instead of 100,1200. And we can check to see this is right by plugging in each of the coordinates for the 2 points to it. And seeing if each side ends up equaling the other side.
Now that we have an equation for the profit Grant will make based on the number of glasses cleaning sets he sells, I'd like to know what this 700, this constant term at the end of the equation stands for in the context of Grant's money making endeavors. Is this how many cleaning sets he has to sell in order to break even? Just so you know, breaking even, means that profit equals zero. So Grant doesn't make any money but he also doesn't lose any money. Or is 700 his profit if he doesn't sell any glasses cleaning sets? Is it his lunch budget for the month? His profit for every extra glasses cleaning set he sells? Or is 700 the cost of one glasses cleaning set? Please pick what you think is the best answer.
700 is Grant's profit. If he doesn't sell any glasses cleaning sets. >> Since this equation is written in slope intercept form, this is the y-coordinate of the y-intercept, b, in our equation, y equals mx plus b form. >> That means there's a point on this line that has the coordinates 0,700. >> Since x in our equation stands for the number of glasses cleaning sets Grant sells, and y stands for his profit, >> This tells us that if Grant doesn't sell any glasses cleaning sets, he will make $700.
Based on what we found out in the last quiz, that Grant will make a profit of $700 if he sells no cleaning sets. What conclusion should we draw? Does this mean that Grant can pull money out of thin air? Does it mean that his initial predictions about profit based on the number of cleaning sets that he sold were not really very good, or does this mean that Grant will have lost $700 if he doesn't sell any cleaning sets? Please pick what you think is the best answer.
This question is really just asking us to use our common sense. Does is make sense that Grant will make money if he doesn't sell any products? I would argue no. If no one buys his products, he won't get any money back. And since he put money into making them, he'll be at a loss. And fortunately we have a positive y-intercept, not a negative one. So, this means this is positive profit. So, the information from our equation is not really in line with what we would think logically would happen. So, I would say that the two points that Grant gave us to start out, which you may remember were 60, 1000. And 100, 1200 were probably off. We based our entire equations just off the info we got from these two points to help us figure out our slope, and our y-intercept. So, this just goes to show you that if your initial data is wrong, your final conclusion will probably not make very much sense. I think this is also testament to the importance of being able to connect math with real world situations. And why we have to keep ourselves grounded in what our variables mean when we're solving problems.