Believe it or not, you're already a quarter of the way through the course. Excellent work. In the last unit, we learned about linear equations and inequalities. We looked at how they can be represented using tables, graphs, and equations. In this unit, we'll continue that approach. We'll look at not just one line, but two. Two lines or two linear equations are considered to be a systems of equations. Let's try to make meaning of these systems of equations by renting a car. You're going to rent a car from a car company called Car-nival. You have $45 to spend on the car rental and Car-nival is going to charge $5 for a registration fee and $10 per hour for renting the car. So we're going to figure out what's the longest time you can rent the car. You might already be able to solve this problem in your head, but we're going to try a different approach. We're going to use systems.
We know a system of equation involves two lines. So let's try and graph two lines about our problem. Here the x axis is going to represent the number of hours, and the y axis is going to represent our money. Knowing that you have $45 to spend on the car rental, which of these equations represents the amount of money you can spend? Choose the best answer.
If you chose y equals 45, excellent work. You have 45 hours to spend on the car rental, which means this amount can't change. On the graph, this means that the y value should be constant, it should always remain the same. If the y value is always 45 then that must mean I have a horizontal line. That would be this choice. The first choice would mean that I only have $5 to spend, which we know isn't true. The last choice would mean that, I drove the car for 45 hours, x is in hours, so x is 45. We know this isn't right. We know this equation isn't true. The amount of money we have doesn't depend on the number of hours, x. So this equation's out.
For renting our car we found an equation and a line that represents this statement. Now let's try and tackle the cost of the car rental. Car-nival will charge a $5 registration fee just for us to get the keys to the car, and that's not driving it anywhere. Then they'll charge an additional $10 per hour for renting the car. So let's come up with an equation for this statement. If x is the total number of hours for renting the car, and y is the total cost of renting the car, which of these equations would represent our total cost? Try creating a table if you feel stuck. Think about what you would pay for 1 hour of renting the car and 2 hours of renting the car. Maybe you can find the pattern.
This is the correct equation. If you got it, excellent work. If I don't drive the car anywhere or drive it for zero hours, I still have to pay $5. That's my registration fee. For driving it one hour, I pay ten more dollars. So I have to add ten to five, to get 15. Two hours would $25 and three hours would cost $35, and so on. So when x is zero, we get a y value of 5. That makes sense for this equation. When x is 1, we get a y value of 15. That also makes sense. This 5 is the fixed or constant part of renting the car. Anyone would have to pay this. The variable part of renting the car is the 10 times x, since you spend $10 for each hour that you drive the car. We know the cost of the car depends on the number of hours, x.
Let's take our 2 equations and plot them on the same coordinate grid. Now that we have 2 lines, we have a system of equations. We can represent the system using 2 equations or graphically using 2 lines. This first line indicates that you can spend $45 on the car rental, that's the most you're willing to pay. This second line represents the cost of the car, notice that the line continues to increase up into the right. This makes sense because the cost will continue to increase if we rented the car for more hours. You pay 10 more dollars for each hour you rent the car and that also is reflected on our graph. Knowing this information, what's the longest amount of time you can rent the car. Write your answer here.
The longest time that we can rent the car is 4 hours. The maximum amount of money you can spend is $45. And we can see that the cost of the car rental intersects or meets our maximum amount at 4 hours. That's when the cost of the car rental would be $45.
This system of equation has exactly one solution and it is a point of intersection. This point satisfies both of the equations. If I plug in an x value of 4 and a y value of 45, both equations should be true. Let's see if that works. Sure enough, in the end, we get true statements. If we can understand that the solution to a system is a point of intersection, then we can use this to solve all sorts of other problems. But before we get into solving other problems we're going to see if a point is a intersection to a system, and look at other types of systems. The x and y values make both equations true. This is how we can check whether or not the point is actually a solution to our system of equations.
Here are two other equations. I want you to determine if 20, 40 is the point of intersection for this system. What do you think?
No is the right answer here, if you got it, great work. We can check to see if this point is the solution to our system by plugging in the values of x and y, x is 20 and y is 40. Be careful with the second equation, we know that the x should be 20 so we want to make sure we plug that into the second term here. Just because the variables come in order in our point doesn't mean they'll appear in order in our equation. So be careful when you plug in. Our first equation checks, 20 equals 20. Our second equation doesn't check. We know 40 does not equal 15. So this point can't be a solution for our system. It's not the point of intersection.
Let's get back to renting cars. This time you want to rent a car, and you're comparing the cost between two companies, Car-nival and Rent-a-Car. Car-nival's the same as before. They're going to charge $10 for a registration fee and $10 per hour for renting the car. Rent-a-car is a little bit different. They're going to charge $30 for a registration fee and $6 per hour for renting the car. So, which company's the better deal? Is it Car-nival, Rent-a-Car, does it depend, do they cost the same, or would you rather teleport? This is one of those questions you really want to take your time on. Try to create a table of values for Car-nival and Rent-a-Car. You can let x be the number of hours and y be the cost of the rental car. You can use the table to investigate this or even create a graph. Try your best.
It turns out that it depends. Let's see why. We saw earlier that Carnival charges $10 for a registration fee, and $10 per hour for renting the car. Rent-a-car charges $30 for a registration fee, and $6 per hour, or x, that we rent the car. This would be the total cost. Let's see a table of values and these two equations graphs. Here are our two equations and the lines for our graph. And here's a table of values. X was the number of hours of renting the car, and this y represents the cost for a Carnival car rental. The second y represents the cost for a Rent-a-Car, car rental. We can see at one hour, that Rent-a-car cost more than Carnival. We also see that on the graph. Rent-a-car cost $36, and Carnival costs $20. In this region of the graph, we can see that Carnival is the better option. It's cheaper, or the costs, the y value, is below that of Rent-a-car. But, we can see at other times, like 7 hours, that Rent-a-Car is the better option. It's cheaper. We also see that on our graph. Rent-a-car cost $72, and Carnival cost $80. So, it depends. Carnival is usually better for the shorter car rentals, and Rent-a-Car is better for the longer car rentals.
So it turns out that the better deal depends on how long we want to rent the car for. Well, I want you to figure this question out. When would the car companies cost the same? How long would the rental take? And how much money would it cost? For the cost, don't include the dollar sign. I already have that listed for you.
We know the companies cost the same, at five hours, and the cost is $60. This also makes sense because this is our point of intersection. It's the point that makes both of these equations true. If x were equal to 5 we could plug that in here. 6 times 5 is 30 plus 30 would be $60 in total. That would be the cost for Rent-a-Car. For Car-nival we could also plug 5 in for x. 5 times 10 would be 50 plus 10 would be 60. So yes, this point is the solution or the intersection for the two equations.
Here are two other rental car companies, Mini-Hoopties and Rad Rentals. This time you want to compare the costs between the two companies. So let's start by comparing the registration fees. Based on the graph what are the registration fees for the two companies? You can enter your answer here, and don't include the dollar sign. I've already listed that.
Well, for Mini-Hoopties, if I drive the car for zero hours or if I just get the keys, I have to pay $20. And for Rad Rentals, I would have to pay $10. Looks like Mini-Hoopties is costing a bit more. But maybe that's not always the case.
Now that we know that information about the registration fees, what do you notice about the hourly car rental rate? Think about this for both of the companies and then answer this question. Which lengths of time could you rent a car from either company and pay the same amount? Check any of these answers that apply. Or maybe they never cost the same or maybe they always cost the same.
It turns out the two car companies would never cost the same. For a one hour car rental, Rad Rentals would cost $20 and Mini-Hoopties would cost $30. We know Rad Rentals is the cheaper option here. The same is true for a five hour car rental. Rad Rentals would cost $60. And Mini-Hoopties would cost $70. We can see on our graph that Rad Rentals, or its cost, is always below the cost of Mini-Hoopties. Getting a car from Rad Rentals will be always be less than one from Mini-Hoopties. So, we know they'll never cost the same.
And this system of equations is interesting, because we have two parallel lines. It turns out these lines will never intersect. We can see that the lines are parallel. And we also know that from their equation. The slope of each line is 10. That means we pay 10 more dollars for each hour we rent the car. The same is true for a car from Mini-Hoopties. We pay 10 more dollars for taking the car out for an hour, ten more dollars for another hour. If we look at the equations, we can see the slopes are the same and the y intercepts are different. Any two equations with these properties will be parallel lines and they'll never intersect. So there won't be a solution.
Let's look at one last example comparing car companies. Here's my coordinate grid. X is the number of hours or the time of the car rental, and y is the cost in dollars. This company, Numerator Cars, has already been graphed for you. We're going to compare Numerator Cars to Carz-to-Go. Carz-to-go charges $5 per hour, and a $10 registration fee. So what equation could you write to represent the cost of a car rental from Cars to Go. Write that equation in this box.
The equation should be y equals 5 x plus 10. If you got that right, excellent algebraic thinking. We know we pay $10, that's our fixed cost, the registration fee. The 5 times x comes from the $5 per hour. We pay $5 per every hour we drive the car. And, of course, if we add those two parts together we get y, our total cost.
I've graphed the cost of both Numerator Cars and Carz-to-Go on the same coordinate grid. Using this information I want you to figure out, when does a car from Numerator Cars and Carz-to-Go cost the same? Check all these answers that you think apply.
For one hour we can see that both the car companies would cost $15, so this one's correct. For ten hours, it's not on my graph, so I have to think a little bit harder. I could plug in the value of ten in for x. If x were equal to ten, Carz-to-Go would cost $60, and Numerator Cars would also cost $60. That's pretty simple because, well, they have the same equation. The same would be true for 15 hours and 20 hours and it turns out, this is true for every hour. We know the equations are identical so their cost should always be the same. This question was pretty tricky so don't worry if you didn't get it right on your first try.
So, this is our third and last type of system. When we have two equations that are identical, the lines overlap. They have an infinite number of intersections or solutions, so every x and y will satisfy both of the equations. For our car companies, this just meant that it didn't matter where I got my car from. I'd always have to pay the same amount.
Let's use our knowledge of types of systems to figure out what type of system this is? Do these two equations represent intersecting lines, parallel lines, or are they identical? Choose the best answer.
These two lines are parallel. We can solve each equation for y, so we can compare the two equations. Let's solve this one for y. I can subtract negative 2 x from each side of the equation, and I get negative 3 y is equal to negative 2 x plus 6. Then, I divide each side of my equation by negative 3 to get y is equal to two-thirds x minus 2. These two equations are in different forms, but represent the same equation. I can see that my original two equations have the same slope, and different y-intercepts. This must mean that they're parrallel.
Try figuring out what type of system this is. You can solve each equation for y and then think about their slopes and the y intercept. If the slopes are different they intersect, if they're the same they're parallel, or you might just have the same equation, which means you know you're in the last case. Good luck.
It turns out that these equations are identical. Let's see why. I can add x to both sides to get 2y is equal to x plus 14. Then I divide both sides of my equation by 2. So, I have 1y is equal to 1 half x plus 7. The second equation is more simple to solve for y. All we need to do is to divide each term by 6. So I have one y. 3 6ths of x is the same as 1 half x. And then seven. I know the left side equation is equal to the right side of the equation, so I can just rewrite my equation with both sides flipped. And sure enough, I have identical equations. They share all the same points.
What type of system do you think this one is?
These two lines are going to intersect. For the first equation, I just need to subtract 2x from both sides. And I get 1y is equal to negative 2x plus 3. For the second equation, I need to add 20 to both sides to isolate the 10y. So I have 5x plus 20 is equal to 10y. Finally, I divide each term by 10 and I get 1y is equal to the 2 plus 1 half x. I'm going to rewrite this equation flipped over, the y's in the left, and 1 half x plus 2 is on the right. When I look at the two equations, I notice that their slopes are different so they must intersect.