Now that you've seen how to convert to slope intercept form, I want you to try and graph this equation. I've marked up some points, and I want you to check all the boxes that make our graph. Take your time on this problem and think about it. Be really careful with your negative signs, and think about the directions you need to move with slope. As always, try your best. And if you get stuck, try watching the previous video again. Or, watch some of the answer and then pause it, until you get unstuck. This allows you to monitor your thinking, and continue solving the problem.
I'm going to convert this equation into slope-intercept form. So first I'm going to add 2x to both sides, because I want to isolate y. I get negative 3y equals 6 plus 2x and I can rewrite this side by switching the terms around. Then I divivde each term by negative 3 to get 1y is equal to negative 2 3rds x minus 2. Notice, I have one negative sign here, and one negative sign here, so both of these terms need to be negative. Alright, now, I'm ready to graph. I know my y intercept is at negative 2, so I go two units down on the y-axis. I know my slope is negative 2 3rd. I'm going to put the negative sign with the 2, which means I move down two units and right three units. If I move down two units and right three units, I end up here, at this point. This wasn't one of the boxes down here, so these are out. We could continue moving two units down and three units right to get other points down in this region. So, I have two points and I know my line should like this. But how do I get the other points up here? Well, I assign the negative to the 2. What if I put the negative instead with a 3? In this case, the 2 would be positive and I would move up two, and the 3 would be negative, and I would move left three. If I start from my y-intercept and move two units up and three units left, I end up at this point here. When working with slope, the positive and negative signs always indicate the directions we need to move. Two points determine line and the third one is like our insurance, so if I connect these, I should get a line. So here is the line that represents our linear equation, pretty cool.
If you took a break, welcome back. We've been talking a lot about lines and slopes and intercepts. And well, who cares? What good are these lines and equations anyway? It turns out we can use them. For example, let's say we're moving and we need to rent a truck to haul all of our stuff. I looked up prices to do this. And I found that doing this requires us to pay an initial fee of $20, plus $2 per mile. So, if I don't drive the truck at all, I still have to pay $20 just to get the keys. I'm going to list the dollar sign here, instead of the table for each value. So, if we rented this truck and drove it one mile, it would have cost $20 plus $2 for the extra mile. So that's $22 in total. Based on this price structure, how much would it cost to drive the truck 2 miles, 3 miles, and 4 miles? Put your answers in each of these boxes.
Well, I paid $2 per mile. So if I drive another mile, I pay 2 more dollars. So driving 2 miles would cost $24, and again, if I drive one more mile, I pay another $2. So it costs $26 to drive 3 miles and $28 to drive 4 miles.
Knowing this pattern how much do you think it would cost the truck and drive it so there's no need to write the dollar sign in your answer.
Well, it cost me $20 just to rent the truck, that's just to get the keys. Then I paid $2 per mile, so I'm traveling 20 miles and I have to pay $2 for each of those. So I have my initial fee of 20, $2 for my cost per mile, and then 20 miles, so I get $20 plus $40, which is $60. If you got $60, nice work.
When we answer these sorts of questions and make these types of calculations, we're actually using algebra. We're creating and using linear equations. But in algebra, we don't stop at the question, how much would 20 miles cost. We ask how much would x miles cost. I want to know the cost of my truck rental for any of the number of miles I drive it. So here's the big question. What expression can we write here that represents the cost of our truck rental in terms of the number of miles we drove x? Try thinking about the pattern in this table, and take some time. Don't worry if you don't know how to do this just yet. I want to get you thinking, and I want to try and give you a shot at it. Finally, when you enter your answer, don't use the dollar sign. Just write the expression.
Well the price of the truck will always include the $20 initial fee so let's write that first. Then whatever the number of miles we drive we have to multiply it by 2 since the cost is $2 per mile so we should add 2x to our expression and what's great we can check this. If we drive 20 miles we know the cost should be $60. Let's plug in $20 and for x to make sure that that's true. Sure enough, what do we find? The price is $60. This expression works.
This equation, let's us calculate the cost of the truck rental for any amount of miles that I drive it, x. This table provides a limited view of the cost of the truck rental. We can only see it for 0, 1, 2, 3, 4, and 20 miles. If I want a different representation of this data, I could use a graph. A graph can help us visualize the cost of the truck rental, say, for 6 miles, 10 miles or even 15 miles, we can see everything at once. Here is a graph with the positive x-axis and the positive y-axis. X is in miles and y is the cost in dollars. Here is some points on the graph. What I want you to do is to check the points that satisfy our equation. Good luck.
For the first point we know it's 0,20 so here's 0 and then I go up 20, so this one is on our line. The next point is 1,22. I notice that this box is at this is on our line. If I drive 3 miles, I pay $26. $26 would be halfway between equation. If I drive 4 miles, I pay $28. So this one is correct. If I were to drive 5 miles, I would have to pay two more dollars so, 30. But this point has a y value of 32, and I know if I drive 5 miles, I only pay $30. So, this isn't on our graph. And, finally if I 6 six miles, I know I would have to pay $32. So, this point is on our graph.
We can connect these points to form all the points that represent the miles driven and the cost of our truck rental. Here's what our graph would look like. And notice that the graph continues to increase up into the right, well that's because I can drive more miles and my cost would continue to go up. But wait, there's something else going here with this equation. If we switch these two terms around then the equation becomes y equals 2x plus 20. This is an equation in slope intercept form. So, what's the slope and the y-intercept for this equation? Put your answers here.
We know the slope is represented by the letter m, so that must be the 2. And the y-intercept is the number or constant on the end of the equation, so 20. That's our b.
We can actually use slope intercept form to graph this line as well. We can graph this equation by starting with the y-intercept or the b. So, I look on my y-axis, and I go to 20 units. 0,20 my y-intercept. You can start at this point because we begin graphing with the letter b. Now that we have the point of the y-intercept, we can graph using our slope. Positive 2. I know slope is rise divided by run. So, I'm going to write my number 2 as 2 divided by 1. Both of these numbers are positive, so I need to move in a positive direction. I'm going to go up two units and then right one unit. These are my directions for how I should move from this point on the graph. I should move up two units, and then right one unit. Notice that my y-axis is in increments of 4. So, I only need to move up half that amount. Half that amount would be two units. Then, on my x-axis, I move one direction to the right. So, I moved two units up, and then one unit to the right. That gets me to my next point on the graph. If I repeat this process, I can continue getting other points on the graph. I go up two units, right one unit. Up two units, right one unit. I know it looks a little funny on this graph, but it's because we've used a scale here. So the coordinate grid doesn't match these numbers. It looks a little different.
Let's try graphing this equation, y equals -3x minus 1. First, let's graph the y-intercept. Choose the point that corresponds to the y-intercept, one of these.
We know our equation is in slope intercept form, so the y intercept is b, or in this case, negative 1. So, our line has to cross the y-axis at negative 1. That would be this point here. We don't move horizontally on the x-axis, because the x coordinate is 0. The y value is negative 1, so we move down. Nice work if you got it.
Now let's use this slope to plot another point on the graph. Which of these points is on the graph? Check all of them that apply.
Our slope is negative 3, or negative 3 divided by 1. It terms of direction, we know we need to travel down 3 units and then right 1 unit. From our y intercept we travel down 3 units and to the right 1 unit. We wind up at 1, negative 4. So this point is certainly on our graph. Once we have two points, we can connect our line and draw it in. We can see we have some other points on our graph, like negative 1, 2 and negative 2, 5. But these aren't any of these other points, so they don't work. If you also plugged in these values for x and y into our equation, you would find that you'd get a statement that wasn't true. So these points aren't on the line because they don't satifsy our equation. If you're still having some trouble with slope, remember you can always refer back to the directions. You start at the point, and move down 3 and right 1. Drawing your arrows to help you out.
Notice that I put this negative sign with the 3 in the numerator. I could have also put the negative sign with the 1 in the denominator. Doing this allows us to make some other moves with slope. Instead of going down 3 units and then right 1, we're going to move up 3 units and left 1. We move left because the run is negative. So here is my y intercept, and I move up 3 and then left 1. Notice I still lined up with a point on my graph. And the same is true if I did it again. So either way that we use the slope we can still wind up with our line. Just be sure you use the negative in only one of the directions. In this case the negative was with the rise, which was down. And in this case, the negative sign was with the run, which is to the left.
Sometimes the equation isn't as easy to work with. Here this isn't in slope intercept form. I don't have y equals mx plus b. This one's actually in standard form. So we want to change it into slope intercept form so we can more easily graph it. If we don't want to create a table to graph this equation, then we can convert it to slope intercept form. If I subtract x from both sides I get negative 2 y equals 4 minus x. Im going to rearrange these terms so the x term comes first. So I have negative x first and then positive 4 right after it. Finally, to solve for y, we divide each term by negative 2, this isolates y. Remember this is a negative 1 as the coefficient in front of x. When we simplify this we have positive 1 half x minus 2, 4 divided by negative 2 makes negative there's our slope, our m 1/2. We can just graph this like before.
We've done a lot so far. We know how to graph lines using tables, and using slope intercept form. Let's look at one last method. We're going to use x and y-intercepts to make our graph. And you might be wondering, why all these different methods? Well, some ways are easier than others, and you might find that graphing using the intercepts will be more quick than solving for y. This is one of those cases. So for this equation, what's the x-intercept, and what's the y-intercept? Put the coordinates here.
For the x intercept we let y equal zero. So I get 4x is equal to negative 12 and then I divide both sides by 4 so x is negative 3. For the y intercept remember we are somewhere along this y axis so we haven't moved left and we haven't moved right. So the x cordinate must be zero. So when x is zero y is equal to four.
Now that we know the x-intercept and the y-intercept, we can graph. Choose the x and y-intercepts from the points on this grid. You should check two boxes, and that will determine your line.
For the x intercept, we're at negative 3, 0. We start at the origin and move three units left and zero units vertically, here. For the y intercept, we start at the origin. We move zero units horizontally and then four units up vertically, so here. We know two points to determine our line so we draw our line in here. Great work if you chose the correct intercepts.
Wow, that was a lot of algebra. We learned three different methods of graphing using a table, using self intercept form and using x and y intercepts. We looked at how math can help us rent a truck and we discovered what those numbers meant for the truck rental. Maybe you can find some other scenarios in your life where algebra can help you save or spend your money. If you have any ideas, share them on the forum. We hope to hear from you there. Now that we're done graphing equations, the next thing that we're going to do is write some equations. Stay with us.