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Contents

Graphing Solution

We know that we can solve a system of equations graphically. Let's consider some rental car companies again. This time, we're going to look at a new one called Lucky Cars. Lucky Cars is going to charge $12 for a registration fee and $7 per hour. Car-nival, the one we seen before, charges $10 for a registration fee and $10 per hour. Let's look at these graphs. Here are the two equations for the cost of the rental car. And here are the two lines. So, when looking at this graph, we can see that the intersection lies somewhere in this region. Based on our graph, which of these points represents the intersection or when the companies cost the same?

Graphing Solution

It turns out for this graph, it's impossible to tell. We can see that the intersection occurs somewhere in this region and it looks like it's about right here. We're not on a grid line or a certain point, so it's really hard to tell or determine just from the graph. So we need some sort of other method to figure this out.

Two Thirds

Because the graph involved estimating, we can't find the exact coordinates of the point of intersection. So, here's another way that we can find it, substitution. For this method, we solve for one of the variables, and then we make a substitution for that variable into the other equation. Notice that for these two equations, both of them are solved for y. I have y alone on one side of the equation. Notice that both of these y's are the same variable. So, what we can do is replace this y with what this y is equal to. We're going to take substitution. Now, we have one equation with one variable and we can solve for this variable. We subtract 7x from both sides to get 3x plus 10 equals 12. Then we subtract 10 from both sides to get 3x equals 2. And finally, we divide both sides by three. So 1x is equal to 2 3rds. Remember, x is the variable that represents how long we're renting the car. So, we rented it for 2 3rds of an hour. Alright. Here's a throwback to fractions. What's 2 3rds of an hour in minutes? Write your answer here. And don't worry about the unit. I've already included it.

Two Thirds

We know 1 3rd of an hour would be 20 minutes, so 2 3rds of an hour would have to be 40 minutes. And if I go 3 3rds of an hour, well, that's 60 minutes, that's my full hour. If you put 40 minutes, nice work.

Substitution Expression

Here's another system of equations. The second equation already has one of the variables solved for or isolated, it's x it's a loan on one side of the equal sign. So what expression can be substituted in for this x, what could we put in this box?

Substitution Expression

Well, we know x equals negative 1 minus y, so we can take this expression and plug it in, or substitute it for x.

Making the Substitution

So, what does y equal in this case? Finish out this math. You can write your answer here.

Making the Substitution

First we distribute the positive 2 to negative 1 and negative y. So I get negative 2 minus 2y plus 5y equals 7. Next I combine the like terms to get positive 3y. We can add 2 to both sides to get 3y equals 9 and then divide both sides by 3 to get y equals 3. Way to go if you got 3.

Find the Other Variable

But remember, we're still not done. We only found one variable, and we need a point of intersection. So what does x equal? You can write your answer here.

Find the Other Variable

We know y equals 3, so which equation do you think it's easier to plug into, the first or the second? I would argue it's easier to plug into the second one. X is already solved for, so if I plug in 3 for y I can easily find the value of x. So I get x is negative 4. If you used this equation, that would have worked, too. You still should have gotten x equal to negative 4.

Check the Solution

We know negative 4, 3 is a solution to our system of equation. Well, we can also check it to be sure. Here are the 2 original equations, and here was our solution. We want to check our solution in each equation and make sure that we wind up with true statements at the end. Try doing that, and make sure these numbers match.

Check the Solution

We replace x with negative 4 and y with positive 3. 2 times negative 4 is negative 8 and 5 times 3 is 15. Adding these two numbers together, we get 7 is equal to 7. This is true. For the second equation, we replace x and y, and we get negative 4 equal to negative 4. This is also true. Great, another way that we can check to make sure we have the right answer.

Choosing the Variable

Let's consider another system of equations. When we perform substitution, we need to choose one variable and one of the equations that is easiest to solve for. For example, in this equation, we want to think, is it easier to get x alone, or y alone? Also, we could consider the other equation. Do we want to solve for this x, or this y? We need one of these equations to be in the form, x equals something, or y equals something in order to perform substitution. So, what do you think in this case? Which of these variables would be easiest to solve for? Choose the best one. If you actually solve for some of these variables, you might see that some are better than others. Think about why that is. Maybe some have negatives. Maybe some have fractions. You want the easiest one.

Choosing the Variable

We don't want to solve for x or y in the second equation, since we would need to move the terms around and divide by a coefficient in front of the variable. For example, if we try to solve for x, we would have to subtract 3y from both sides first, and then, divide every term by 5. That's kind of messy. A similar process happens if we try and solve for y, so these two are out. If we try to solve for x in this first equation we have to divide each term by 2, this would leave me with x is equal to 2 minus 1 half y. Remember, there's a 1 as a coefficient in front of the y, so this is really negative 1 half times y. I prefer to avoid working with fraction in my substitution process, so this x isn't necessarily the best choice. If I try and solve for y, I could add y to both sides, then, rewriting my problem up here, I could subtract 2x from both sides and I would get y is equal to negative 2x plus 4. Because I have integer coefficients like negative 2 and 4, it's easier to work with this in a substitution. If you want to solve for x and work with a fraction in your substitution, that's okay. Choose the method that you feel most comfortable doing.

Substitution Method

After we solve this equation for y, we'll have y equals negative 2x plus 4. Now we're ready for substitution. Using substitution, I want you to find the solution or the point of intersection for this system of equations, these two lines. Enter your numbers here, and here. And good luck.

Substitution Method

Notice that I have y and y in both of these equations. That means I can substitute negative 2x plus 4 in for y. This is my substitution step. Next we distribute a positive 3 to negative 2x and positive 4, so I'll have negative 6 plus 12. Combining like terms, I'll have negative 1x plus 12 equals 10. I have negative 6x's and positive 5x's. I have more negatives than I do positives, so when I simplify here, I know that I'm going to need a negative number, a negative 1x. So I have negative 1x equals negative 2. So if the negative of x equals negative 2, then the positive of x must equal positive 2. Some students think about multiplying both sides by negative 1 and others divide by negative to find y. To find y, I'm going to use one of my original equations. I've used the first one since it looks less complicated. Notice also that I didn't choose to use this equation. I might have made a mistake when trying to find this equation. That's why it's always best to use an original one. Plugging in 2 for x, I'll have 4 is equal to 4 minus y, then I can subtract 4 from both sides to isolate my y. So 0 equals negative y, which means 0 also equals positive y. Any number multiplied by 0 is still 0. So, if we multiply by negative 1, this should make sense. The point 2 comma 0 should be the point of intersection. If I use some mental math real quick, we can check it. 2 times 2 is 4, and 4 minus 0 is which is 10. This term would be 0, so 10 equals 10. That's true as well.

The Substitution Step

This is the last system of equation we'll work on together. I want you to perform substitution and the second equation. What expression should go in this box? Then, I want you to solve the equation. What expression would you get in this box after distributing the negative 2? And then, what does this all simplify to you? Put that in this box. Try solving the problem on your own and then see what should go in these boxes. Good luck.

The Substitution Step

We know we can perform a substitution with the y. I have y isolated on one side of the equation. So I can take 2x minus 2 and substitute it in for y. Next we distribute the negative 2. So I get negative 4x and negative 2 times negative positive 4. If you got those steps right, excellent work.

Substitution Non Sense

So what's going on here? Does 4 equal 6? Well no, this can't be right, we know 4 isn't equal to 6. So this substitution method isn't working, something else is going on with these two lines. If we remember that a solution to a system is a point of intersection, then we know that these two lines can't intercept, we don't get a true statement. In other words the lines have to be parallel. Let's make sure.

Parallel Lines No Solution

Here are the two original equations and let's try graphing them. The first equation is already in slope-intercept form, so I won't do anything with it. For the second equation, I'll change it into slope-intercept form. First, I subtract to be careful. I get positive 1y here, positive 2x here. And negative 3 for my constant. The y intercept is negative 2 for the purple graph and the slope is positive 2 over 1. For the green graph, the y intercept is negative 3 and our slope is 2. Notice that the slopes are the same, so we have parallel lines. Parallel lines will never intersect. So, we know that there can't be a solution here. Whenever we solve an equation and get a statement like 4 equals 6, we know there can't be a solution. The lines must parallel, just like in this case.

Practice 1

What's the solution to this system? You can enter your answer here.

Practice 1

The solution to the system is 3, 2. We want to start by solving for one of the variables and one of the equations. I think it's easiest to solve for y in this equation. You can also solve for x or y in this equation and still wind up with the same result. I can take the equation, subtract 3x from both sides, to get negative y equals negative 3x plus 7. Remember these are not like terms, so I just list them. Now I need the positive y value, so I divide every term by negative 1. We could also think about multiplying every term by negative 1, so y equals 3x minus 7. Now we can perform substitution, we can take 3x minus 7 and substitute it in for y. Since that's what it equals. Here I've replaced y with sides, so 8x is 24 and finally we divide both sides by 8, so x is equal to 3. So we found x, but remember we still need to find y. I can use this first equation to find y, since it's already solved for. I replace x with the value of 3, so y equals 3 times 3 minus 7, or 9 minus 7, which is 2. And remember, you can always make sure that you're right. Check the answer 3, 2 by plugging it into both equations. Do you come out with true statements? If so, you did great.