Last time we got to know a little bit about the relationship between lines and their equations. Specifically we talked about this idea of intercepts, which are where our line intersects either the x axis or the y axis. Let's look at this relationship a bit more. Here's the graph of the equation y equals 2x plus this graph, but we're going to find out a little bit. So, from what you remember from the last lesson, what's the y-intercept for this graph? Enter the numbers for the coordinates here.
I can look at this graph and know that the y intercept or this point, is 2. So I move 0 in the horizontal direction since I see that's the origin and then I moved vertically up to. So our point is 0 comma 2. You might have also set the x value equal to 0. If you did that, you would still get y is equal to 2.
Let's look at some more y-intercepts. Here are two equations and here are two lines. This equation goes with the blue line and this equation goes with the yellow line. I want you to find the y-intercepts for each graph. You can list your coordinates here.
The first graph crosses the y axis at this point. We moved 0 horizontally and 3 up vertically, so our y-intercept is 0, 3. For the other graph, we moved 0 horizontally and then down 1 or negative 1 in the y direction, so 0, negative 1 is our y intercept. Great work if you got both of those.
Now that we know the y intercepts for these 2 equations. We can actually find out something interesting about these forms. So, here's an equation y equals mx plus b. You might have heard about it. Which part of this equation represents the y intercept? Is it the y, the m, the x, or the b?
We know the b represents the y-intercept. We can see the y-intercept was 3 or positive 3 and here the y-intercept was negative 1 just like in our equation. So the only thing that changed between these two equations was the y-intercepts and we can also see that on the graph.
But what about this m, now we know the y and x are variables. If I plug in a value for x, I get a value for y. Just like for intercept if x is zero then y would be equal to three. So what about this m, what is this thing we've labeled m correspond to? Let's see, instead of looking at these graphs I'm going to look at two different graphs. Graphs that have different m's.. Here is the line y equals 2x plus 3, and this is the line y equals one half x plus three. So what's different between these two graphs? Is it the steepness? The y-ntercept or the b in the equation y equals mx plus b.
Well, these graphs have different steepness. One rises really quickly, and the other one's a little bit more shallow, so this one was correct. If you got it, nice work.
If you have trouble picturing this think about a mountain climber. It would be harder for a mountain climber to walk from left to right across this graph than it would for someone to walk from left to right on this graph but we're mathematicians and it's not enough to know that this line is steeper than this line. I want to know how much steeper, I need a way of quantifying steepness In algebra we use something called slope. Slope compares the distant a line rises or falls to the distance aligned runs from left to right. We write slope as the ratio of rise divided by run. For example, let's look at this line. We'll look at the graph between x equals 0 and x equals 1, so we're between here. In this region we have a run of 1 so I'll put 1 in the denominator. How much would this rise be? Use the graph to help you out.
The graph has increased vertically 2 units. So, we'll call the rise 2. Slope is always rise divided by run, so the slope of this graph is just 2 divided by 1, or just 2. Excellent work if you figured that one out.
What do you think the slope is of this line? This is trickier, and you'll want to identify the rise and run for yourself.
The slope here is 1 half. I can start by choosing two points on the line. This one is 0,3, and this one is 2,4. We can see to go from this point to the next point, we should move up one. And then write 2. We moved one unit up in a positive direction. So we know the rise is positive 1. And then, we move two units to the right, in a positive direction. So, the run is positive 2. These direction arrows can help me associate the positive and negative signs for the rise and the run. There are, of course, other ways to find the slope. If you've found other ways to find the slope, share your thoughts on the forum. We'd love to see what you did.
Instead of choosing those two points, I'm going to choose two other points. We can see that to go from this point to the second point, we need to rise three units up and run six units to the right. Because we went up vertically three units, we know that this is a positive 3. And since we traveled six units to the right, we know that the run is really positive 6 in the positive direction. When we find the slope, or the m, we're really finding the ratio of the rise divided by the run. We already know that 3 6ths is really the same thing as 1 half. This is great, we got the same slope as from before. We can see that the slope of 3 can also read slope in other directions. I don't have to start from this point and go to this point, I could go in the reverse direction. If I started from this point, 6, 6. I would travel down three units, and then left six units. So here, the vertical distance is really a drop. It's a 3 unit distance down. So the rise would be negative 3. We move three units down in a negative direction. And 6 units left in a negative direction. Since both of these directions are negative, we know the rise is really negative 3. And the run is really negative or 1 half. That's pretty cool. Any way we try and find the slope, we always get the same ratio. We just need to be careful about our directions and our negative signs.
In general, when we want to compute the slope of a line, we look at two points. We can label these points P1 and P2. We use a subscript to indicate which point it is. Now, these points could have any coordinates. So it'd be really unspecific or general in mathematical speak. We're just going to say that this point has some x coordinate and some y coordinate. We'll label the point x sub 1 and y sub 1. The 1 is a subscript and it tells us which point we're on. Likewise, I could label this point x sub 2, y sub 2. We know slope is the ratio of the rise divided by the run. So this distance would just be the rise. If I wanted to find the rise between these two points, I would want to look at the y values. I would want to know how these y values changed. For example, if this y value was 5 and this y value was 3, I would just subtract them to figure out the rise. It'd be 2. We can see that the vertical distance or the rise is just the difference between the two y values, y2 minus y1. Knowing this information about the rise, which of the following do you think could represent the run? Is it the sum of the x coordinates, the difference of the x coordinates or the difference of the y coordinates?
The run is the difference between the x values. If we travel all the way over here to the right, we want to subtract off this distance so we're left with just the run. This is a really tough question. So great job if you got it right. If not, hang in there and we'll keep learning some more about slope.
So, we have an equation for slope between 2 points. Let's practice using it. What is the slope between these 2 points? This point is 4, 6. And this point is point is point 1 or which is point 2. Maybe it won't be a problem, though. Why don't you give it a shot? Write your final answer in this box.
Well, it turns out it doesn't matter which one you call p1, or p2. I'm going to find the slope with this one as p1, and this one as p2. I'm going to label this point with x sub 1, y sub 1, and this point with x sub 2, y sub 2. Y sub 2 is 3, and y sub 1 is 6, so I have 3 minus 6. X sub 2 is 7, and x sub 1 is 4, so I have 7 minus 4 for the run. I have negative 3 divided by 3 which makes negative 1, but let's see if this works the other way. I can label this point 2 and this point 1. So this time y2 is 6 and y1 is 3 and for the x values, x2 is 4 and x1 is 7. Here I have 3 divided by negative 3, and look at that, I still get negative 1.
Eventually, you'll be able to picture this in your head. We know the rise is the change in the y values. So we went down 3, or negative 3. And then run is the change in the x values. So we went 3 units to the right, from 4 to 7. If you can picture this stuff in your head, that's amazing. If you can't, that's okay too. Understanding what rise and run mean can really help you understand the slope formula.
Note that the slope from our last line was negative. As we look at this graph from left to right, the line goes down. If someone were walking or hiking along this line, they'd be going downhill, which corresponds to negative slope. Our slopes don't always have to be negative, though. If we turn the line slightly, we can see that we would have a flat line. We know a flat line has 0 slope, because it has lots of run, but no rise. And 0 divided by any non-zero number, would make 0. We could also turn our line a little bit more. This would produce a positive slope. So now we would be walking uphill. Positive slopes always point up, and to the right. The last type of line is a vertical line and these slopes are undefined. They have lots of rise but zero run. Remember, we cannot have zero in the denominator so the slope must be undefined. For our walking guy, we also remember he can't walk up a vertical wall. That's impossible, so the slope is undefined. So that might be a way you can remember the slope is undefined.
For these lines, I want you to indicate the type of slope it is. If it's positive, put a plus sign in the box, or if it's negative, use a negative sign. If the slope is zero type in the number zero in the box, or if it's undefined type the letter u.
The first sign point up and to the right so this must have positive flow. We also know if someone is walking on it, that they would be walking uphill, so we have positive slope. This line is vertical we can't walk up it, so the slope is undefined. This third line points down and to the right, so it must have negative slope. Here our line points down and to the right so its also negative slope. This one would be positive. And this last one would be 0. We have a horizontal line, so lots of run and no rise.
Now that we know how to calculate slope or just the steepness of a line, let's go back to our original lines, y equals 2x plus 3 and y equals 1 half x plus 3. Well they each have an intercept of 3. So 0 3 is on both of the lines. This blue line has another point 1, 5. And this yellow line has another point 2,4. Using this information, can you tell me what the slope of each line is?
Well, slope is run over run, or y2 minus y1 divided by x2 minus x1. So the slope of this first line is just 2. We rise 2 and then run 1 to the right. The slope of this line is one half. We rise up 1 unit and then 2 units to the right for the run. I'm noticing a pattern here. Do you see it?
So, we already found that the b corresponds to the y-intercept. What part corresponds to the slope? Is it the y, the m, the x, or the b?
And the answer is that, m represents the slope. We can see 2 is the slope for this equation and 1 half is the slope for this equation. So this particular form tells us a lot about lines. It tells us the y intercept and it tells us the slope.
Whenever we write an equation in this form, we say that it is in slope-intercept form. We say that, because we can read the slope and the y intercept or the b directly from the equation. Let's try putting this knowledge to use. Which of the following equations are in slope-intercept form? A word of warning. This is pretty tricky, but, I really want you to think about this. So, take your time and then go with the answer.
Well, the first equation is in slope intercept form. The slope is 4 and the y-intercept is 7. This second one is a little bit tricky,. It is in slope intercept form. I know the slope is 3 and the y-intercept is actually 0. If I add a 0 in the end, I don't change the equation, so this is also in the correct form. This third option is not in slope intercept form. It's actually in standard form. It's a form we're going to learn about soon. We know it's not in slope intercept form because we have a coefficient in front of the y. The y should be sought for, or it should be alone on one side of the equals sign. This equation is also in slope intercept form. The slope is 1 3rd. And the y-intercept is negative 4. Even though our form has a plus sign in it, I know I can rewrite subtraction as adding the opposite. So, this is definitely in slope intercept form. This fifth option is not in slope intercept form. I have a coefficient of 3 in front of the y, so the y isn't solved for. And finally, this last equation is in slope intercept form. Because remember, even though we don't write it, there's a 1 in front of a x. This is the coefficient, or the slope of our equation. If you got at least half of these right, excellent work. You've really got an understanding of slope intercept form.
Slope intercept form is a great way to write the equation of a line, but it isn't the only way. In the previous question, one of the equations was 2x plus form has the variables on one side of the equation with coefficients, and then other side is usually constant. In standard form, we actually usually list the x variable first, followed by the y term. I want to talk more about this equation, specifically the slope of this line. But first, let's think back to the previous lesson and see if you remember how to calculate the x and y intercepts. So fill in the missing numbers. Where does the x intercept occur and where does the y intercept occur?
To find the x intercept, we let y equal 0. If we solve the equation, we notice that x is equal to 3. So, the x intercept occurs at 3,0. To find the y intercept, we let x equal 0, solving for the variable y, we can see that y would equal 2. And, on a graph, we could see that the x intercept would be here and the y intercept would be here.There's out line.
So, we know two points on the line 0, 2 and 3, 0. We should be able to calculate the slope. You can put your answer in this box.
We know the slope can be calculated by the difference in the y values, divided by the difference of the x values. The rise divided by the run. I'm going to label this point x2, y2 and this point x1, y1. So, y2 is 0 and y1 is 2. Then x2 is 3 and x1 is 0. Notice again that I'm always using parenthesis when I substitute in numbers. This is so I don't make a mistake with negative signs. When I carry out the subtraction, I get negative two thirds. Negative two thirds was the slope. But wouldn't it be easier if we could just read it from the equation? It's not as easy to read off the slope from an equation written in standard form, as it was with an equation written in slope-intercept form. But you can still find the slope.
What I like to do with the equations written in standard form, is to first put them into slope intercept form. If I had an equation like this, I would want to change it to slope intercept form. I would want to get the y alone. So I'd have intercept form, that's the case. The x term comes first. I can make the switch here, and write my equation like this. Because of the commutative property of addition and subtraction. 1 plus 2 is the same as 2 plus 1. Likewise, positive 6 minus 2x is the same thing as negative 2x plus 6. Notice that the sign stays with the term. The 6 is positive. And the 2x is negative. Finally we divide every term by 3 to get y equals negative 2 3rds x plus 2. Now we can read that the slope is negative 2 3rds. Pretty great. I can also see the slope by looking at standard form. The slope is going to be equal to the value of a divided by the value of b. In this case a is positive to And b is positive 3. So I substitute 2 for a, and 3 for b. So, we get negative 2 thirds.
So what if you had the following equation, -5y plus 7x equals 3. What's the slope of this line?
Well, we can convert a slope-intercept form and do it that way. I have negative these because this doesn't have an x, these are not like terms. Next we divide every term by negative 5 to get y equals 7 5ths x plus negative 3 5ths.. I have a minus sign here, so I know the second term is really just a negative. Our equation is in slope-intercept form, so our slope must be 7 5ths. Well done if you got that one. If you thought about it in terms of standard form, you would have done negative A divided by B. You want to be really careful though. This is not in standard form. We need to write it in standard form first. The x term should come first. I can switch the place of the 7x and the negative 5y and I get this equation. Notice, the 7x is still positive and the 5y is still negative. So A is positive 7 and B is negative 5. A negative divided by a negative makes a positive, so I get 7 5ths, still the correct answer.
Wow, we've discussed a lot of stuff here. We know what the slope looks like, it's the steepness. We know what it means for slope to be positive, negative, zero, or even undefined. We can calculate the slope between two points using a fancy equation, and we've learned how to read an equation in slope intercept form, and a little bit in standard form. That's a ton of material. I know that we've covered a lot, so you make sure you take some time to practice these skills, and then come back when youre ready.
Well, I hope you understand slope and know how to find the slope given the equation of a line. I want you to practice finding the slopes of these three equations. You can put the slope for the first equation in this box, the slope for the second equation in this box, and the slope for the third equation in this box. Dont worry too much, just try your best.
Did you get those three slopes right? If you did, great job. For the first equation, we have y equals 3x plus 9. For the first equation, I have y equals 3x plus 9. This equation is in slope-intercept form. For slope-intercept form, we know the equation is y equals mx plus b, where b is the y-intercept and m is the slope. So, we know if the equation is in slope-intercept form. The coefficient to the x-variable is going to be the slope. So, in this case, the slope is equal to 3. The second equation is written in standard form, 3x plus 2y equals 7. It just takes two easy steps to put an equation from standard form into slope-intercept form. I find putting the equation in slope-intercept form, the easiest way to find the slope of an equation that is in standard form. It just takes two simple steps. The first step is to subtract 3x from both sides. When I subtract 3x from both sides, I get 2y equals negative 3x plus 7. I'm almost in slope-intercept form, but I have 2y instead of y equals. So, I have to divide everything through by 2. When I divide everything through by 2, I get y equals negative 3 over 2 is negative 3 halves x plus 7 halves. So, once again, this equation is in slope-intercept form, so we know the coefficient of the x-variable, which is negative 3 halves is the slope. So, the slope is negative 3 halves. The third practice problem was just a little bit trickier. To get this equation in slope-intercept form, I subtract 6x from both sides and I get negative 3y is equal to negative 6x plus 12. Now, here's where I have to be careful. I have to divide the whole equation through by negative 3. This can be a little confusing, but let's just take it one step at a time. On the left side of the equation, I have negative 3 over negative 3, which, of course, is 1, so that just leaves me with y. When I divide negative 6x by negative 3, negative 6 divided by negative 3 is 2x. And then I have 12 divided by negative 3, 12 divided by negative 3 is negative 4. So now, we have the equation in slope-intercept form. And the slope is going to be the coefficient of the x-variable, which is 2. If you got these answers correct, great job. If not, you might want to go back and review the videos on how to find slope from the equation of a line.