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Express a small number in scientific notation (example) Pre-Algebra Khan Academy.mp3 | 1 times 10 to the negative 1 is equal to 1 over 10, which is equal to 0.1. If I do 1 times 10 to the negative 2, 10 to the negative 2 is 1 over 10 squared, or 1 over 100. So this is going to be 1 over 100, which is 0.01. What's happening here? When I raise it to a negative power, I raise it to a negative 1 power, I've essentially moved the decimal from to the right of the 1 to the left of the 1. I moved it from there to there. When I raise it to the negative 2, I moved it 2 over to the left. |
Express a small number in scientific notation (example) Pre-Algebra Khan Academy.mp3 | What's happening here? When I raise it to a negative power, I raise it to a negative 1 power, I've essentially moved the decimal from to the right of the 1 to the left of the 1. I moved it from there to there. When I raise it to the negative 2, I moved it 2 over to the left. So how many times are we going to have to move it over to the left to get this number right over here? Let's think about how many 0s we have. We have to move it 1 time just to get in front of the 3, and then we have to move it that many more times to get all of the 0s in there. |
Express a small number in scientific notation (example) Pre-Algebra Khan Academy.mp3 | When I raise it to the negative 2, I moved it 2 over to the left. So how many times are we going to have to move it over to the left to get this number right over here? Let's think about how many 0s we have. We have to move it 1 time just to get in front of the 3, and then we have to move it that many more times to get all of the 0s in there. So we have to move it 1 time to get the 3. If we started here, we're going to move 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 times. This is going to be 3.457 times 10 to the negative 10 power. |
Express a small number in scientific notation (example) Pre-Algebra Khan Academy.mp3 | We have to move it 1 time just to get in front of the 3, and then we have to move it that many more times to get all of the 0s in there. So we have to move it 1 time to get the 3. If we started here, we're going to move 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 times. This is going to be 3.457 times 10 to the negative 10 power. Let me just rewrite it. 3.457 times 10 to the negative 10 power. In general, what you want to do is you want to find the first non-zero number here. |
Express a small number in scientific notation (example) Pre-Algebra Khan Academy.mp3 | This is going to be 3.457 times 10 to the negative 10 power. Let me just rewrite it. 3.457 times 10 to the negative 10 power. In general, what you want to do is you want to find the first non-zero number here. Remember, you want a number here that's between 1 and 10, and it can be equal to 1, but it has to be less than 10. 3.457 definitely fits that bill. It's between 1 and 10. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | In this video, I'm going to do a bunch of example slope problems. And just as a bit of review, slope is just a way of measuring the inclination of a line. And the definition, we're going to hopefully get a good working knowledge of it in this video, the definition of it is change in y divided by change in x. This may or may not make some sense to you right now, but as we do more and more examples, I think it'll make a good amount of sense. Let's do this first line right here, line A. Let's figure out its slope. And they've actually drawn two points here that we can use as the reference points. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | This may or may not make some sense to you right now, but as we do more and more examples, I think it'll make a good amount of sense. Let's do this first line right here, line A. Let's figure out its slope. And they've actually drawn two points here that we can use as the reference points. So first of all, let's look at the coordinates of those points. So you have this point right here. What's its coordinates? |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | And they've actually drawn two points here that we can use as the reference points. So first of all, let's look at the coordinates of those points. So you have this point right here. What's its coordinates? Its x-coordinate is 3, and its y-coordinate is 6. And then down here, this point's x-coordinate is negative 1, and its y-coordinate is negative 6. So there's a couple of ways we can think about slope. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | What's its coordinates? Its x-coordinate is 3, and its y-coordinate is 6. And then down here, this point's x-coordinate is negative 1, and its y-coordinate is negative 6. So there's a couple of ways we can think about slope. One is we could look at it straight up using the formula. We could say change in y. So slope is change in y over change in x. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So there's a couple of ways we can think about slope. One is we could look at it straight up using the formula. We could say change in y. So slope is change in y over change in x. And we can figure it out numerically, and I'll in a second draw it graphically. So what's our change in y? Our change in y is literally how much did our y values change going from this point to that point? |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So slope is change in y over change in x. And we can figure it out numerically, and I'll in a second draw it graphically. So what's our change in y? Our change in y is literally how much did our y values change going from this point to that point? So how much did our y values change? Our y went from here, y is at negative 6, and it went all the way up to positive 6. So what's this distance right here? |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Our change in y is literally how much did our y values change going from this point to that point? So how much did our y values change? Our y went from here, y is at negative 6, and it went all the way up to positive 6. So what's this distance right here? Well, it's going to be your endpoint y value. It's going to be 6 minus your starting point y value. Minus negative 6, or 6 plus 6, which is equal to 12. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So what's this distance right here? Well, it's going to be your endpoint y value. It's going to be 6 minus your starting point y value. Minus negative 6, or 6 plus 6, which is equal to 12. And you could just count this. You say 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So when we changed our y value by 12, we had to change our x value by what was our change in x over the same change in y? |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Minus negative 6, or 6 plus 6, which is equal to 12. And you could just count this. You say 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. So when we changed our y value by 12, we had to change our x value by what was our change in x over the same change in y? Well, we went from x is equal to negative 1 to x is equal to 3. Right? X went from negative 1 to 3. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So when we changed our y value by 12, we had to change our x value by what was our change in x over the same change in y? Well, we went from x is equal to negative 1 to x is equal to 3. Right? X went from negative 1 to 3. So we do the endpoint, which is 3, minus the starting point, which is negative 1, which is equal to 4. So our change in y over change in x is equal to 12 over 4. Or if we want to write this in simplest form, this is the same thing as 3. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | X went from negative 1 to 3. So we do the endpoint, which is 3, minus the starting point, which is negative 1, which is equal to 4. So our change in y over change in x is equal to 12 over 4. Or if we want to write this in simplest form, this is the same thing as 3. Now, the interpretation of this means that for every 1 we move over, we could view this, let me write it this way. Change in y over change in x is equal to, we could say it's 3, or we could say it's 3 over 1. Which tells us that for every 1 we move in the positive x direction, we're going to move up 3, because this is a positive 3, in the y direction. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Or if we want to write this in simplest form, this is the same thing as 3. Now, the interpretation of this means that for every 1 we move over, we could view this, let me write it this way. Change in y over change in x is equal to, we could say it's 3, or we could say it's 3 over 1. Which tells us that for every 1 we move in the positive x direction, we're going to move up 3, because this is a positive 3, in the y direction. You can see that. When we moved 1 in the x, we moved up 3 in the y. If you move 2 in the x direction, you're going to move 6 in the y. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Which tells us that for every 1 we move in the positive x direction, we're going to move up 3, because this is a positive 3, in the y direction. You can see that. When we moved 1 in the x, we moved up 3 in the y. If you move 2 in the x direction, you're going to move 6 in the y. 6 over 2 is the same thing as 3. So this 3 tells us how quickly do we go up as we increase x. Let's do the same thing for the second line on this graph. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | If you move 2 in the x direction, you're going to move 6 in the y. 6 over 2 is the same thing as 3. So this 3 tells us how quickly do we go up as we increase x. Let's do the same thing for the second line on this graph. Graph B. Same idea. And I'm going to use the points that they gave us. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Let's do the same thing for the second line on this graph. Graph B. Same idea. And I'm going to use the points that they gave us. But you really could use any points on that line. We have one point here, which is the point 0,1. And then the starting point, we could call this the finish point. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | And I'm going to use the points that they gave us. But you really could use any points on that line. We have one point here, which is the point 0,1. And then the starting point, we could call this the finish point. The starting point right here, we could view it as, let's see, x is negative 6, and y is negative 2. So same idea. What is the change in y given some change in x? |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | And then the starting point, we could call this the finish point. The starting point right here, we could view it as, let's see, x is negative 6, and y is negative 2. So same idea. What is the change in y given some change in x? Let's do the change in x first. What is our change in x? So in this situation, what is our change in x? |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | What is the change in y given some change in x? Let's do the change in x first. What is our change in x? So in this situation, what is our change in x? Delta x, we could even count it. It's 1, 2, 3, 4, 5, 6. It's going to be 6. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So in this situation, what is our change in x? Delta x, we could even count it. It's 1, 2, 3, 4, 5, 6. It's going to be 6. But if you didn't have a graph to count from, you could literally take your finishing x position. So it's 0, and subtract from that your starting x position. 0 minus negative 6. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | It's going to be 6. But if you didn't have a graph to count from, you could literally take your finishing x position. So it's 0, and subtract from that your starting x position. 0 minus negative 6. So when your change in x is equal to 6, what is our change in y? And remember, we're taking this as our finishing position. That's our finishing position. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | 0 minus negative 6. So when your change in x is equal to 6, what is our change in y? And remember, we're taking this as our finishing position. That's our finishing position. This is our starting position. So we took 0 minus negative 6, so then on the y, we have to do 1 minus negative 2. So what's 1 minus negative 2? |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | That's our finishing position. This is our starting position. So we took 0 minus negative 6, so then on the y, we have to do 1 minus negative 2. So what's 1 minus negative 2? That's the same thing as 1 plus 2. That is equal to 3. So it is 3 sixths or 1 half. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So what's 1 minus negative 2? That's the same thing as 1 plus 2. That is equal to 3. So it is 3 sixths or 1 half. So notice, when we moved in the x direction by 6, we moved in the y direction by positive 3. So our change in y was 3 when our change in x was 6. Now, one of the things that confuses a lot of people is how do I know what order to do the 0 first and the negative 6 second, and then the 1 first and then the negative 2 second? |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So it is 3 sixths or 1 half. So notice, when we moved in the x direction by 6, we moved in the y direction by positive 3. So our change in y was 3 when our change in x was 6. Now, one of the things that confuses a lot of people is how do I know what order to do the 0 first and the negative 6 second, and then the 1 first and then the negative 2 second? And the answer is you could have done it in either order as long as you keep them straight. So you could have also have done change in y over change in x. We could have said it's equal to negative 2 minus 1. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Now, one of the things that confuses a lot of people is how do I know what order to do the 0 first and the negative 6 second, and then the 1 first and then the negative 2 second? And the answer is you could have done it in either order as long as you keep them straight. So you could have also have done change in y over change in x. We could have said it's equal to negative 2 minus 1. So negative 2 minus 1. So we're using this coordinate first. Negative 2 minus 1 for the y over negative 6 minus 0. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | We could have said it's equal to negative 2 minus 1. So negative 2 minus 1. So we're using this coordinate first. Negative 2 minus 1 for the y over negative 6 minus 0. Notice this is the negative of that. That is the negative of that. But since we have a negative over a negative, they're going to cancel out. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Negative 2 minus 1 for the y over negative 6 minus 0. Notice this is the negative of that. That is the negative of that. But since we have a negative over a negative, they're going to cancel out. So this is going to be equal to negative 3 over negative 6. The negatives cancel out. This is also equal to 1 half. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | But since we have a negative over a negative, they're going to cancel out. So this is going to be equal to negative 3 over negative 6. The negatives cancel out. This is also equal to 1 half. So the important thing is if you use this y coordinate first, then you have to use this x coordinate first as well. If you use this y coordinate first, as we did here, then you have to use this x coordinate first as you did there. You just have to make sure that your change in x and change in y are using the same final and starting points. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | This is also equal to 1 half. So the important thing is if you use this y coordinate first, then you have to use this x coordinate first as well. If you use this y coordinate first, as we did here, then you have to use this x coordinate first as you did there. You just have to make sure that your change in x and change in y are using the same final and starting points. And just to interpret this, this is saying that for every minus 6 we go in x, so if we go minus 6 in x, so that's going backwards, we're going to go minus 3 in y. But they're essentially saying the same thing. The slope of this line is 1 half, which tells us for every 2 we travel in x, we go up 1 in y. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | You just have to make sure that your change in x and change in y are using the same final and starting points. And just to interpret this, this is saying that for every minus 6 we go in x, so if we go minus 6 in x, so that's going backwards, we're going to go minus 3 in y. But they're essentially saying the same thing. The slope of this line is 1 half, which tells us for every 2 we travel in x, we go up 1 in y. Or if we go back 2 in x, we go down 1 in y. That's what 1 half slope tells us. And notice, the line with the 1 half slope, it is less steep than the line with a slope of 3. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | The slope of this line is 1 half, which tells us for every 2 we travel in x, we go up 1 in y. Or if we go back 2 in x, we go down 1 in y. That's what 1 half slope tells us. And notice, the line with the 1 half slope, it is less steep than the line with a slope of 3. Let's do a couple more of these. Let's do this line C right here. I'll do it in pink. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | And notice, the line with the 1 half slope, it is less steep than the line with a slope of 3. Let's do a couple more of these. Let's do this line C right here. I'll do it in pink. Let's say that the starting point, I'm just picking this arbitrarily, I'm using these points that they've drawn here. The starting point is at the coordinate negative 1, 6, and that my finishing point is at the point 5, negative 6. Our slope is going to be equal to change in y over change in x. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | I'll do it in pink. Let's say that the starting point, I'm just picking this arbitrarily, I'm using these points that they've drawn here. The starting point is at the coordinate negative 1, 6, and that my finishing point is at the point 5, negative 6. Our slope is going to be equal to change in y over change in x. Sometimes it's said rise over run. Run is how much you're moving in the horizontal direction. Rise is how much you're moving in the vertical direction. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Our slope is going to be equal to change in y over change in x. Sometimes it's said rise over run. Run is how much you're moving in the horizontal direction. Rise is how much you're moving in the vertical direction. And then we could say our change in y is our finishing y point minus our starting y point. This is our finishing y point. That's our starting y point over our finishing x point minus our starting x point. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Rise is how much you're moving in the vertical direction. And then we could say our change in y is our finishing y point minus our starting y point. This is our finishing y point. That's our starting y point over our finishing x point minus our starting x point. And if that confuses you, all I'm saying is it's going to be equal to our finishing y point is negative 6 minus our starting y point, which is 6, over our finishing x point, which is 5, minus our starting x point, which is negative 1. So this is equal to negative 6 minus 6 is negative 12. 5 minus negative 1, that is 6. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | That's our starting y point over our finishing x point minus our starting x point. And if that confuses you, all I'm saying is it's going to be equal to our finishing y point is negative 6 minus our starting y point, which is 6, over our finishing x point, which is 5, minus our starting x point, which is negative 1. So this is equal to negative 6 minus 6 is negative 12. 5 minus negative 1, that is 6. So negative 12 over 6 is the same thing as negative 2. And notice, we have a negative slope here. That's because every time we increase x by 1, we go down in the y direction. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | 5 minus negative 1, that is 6. So negative 12 over 6 is the same thing as negative 2. And notice, we have a negative slope here. That's because every time we increase x by 1, we go down in the y direction. So this is a downward sloping line. It's going from the top left to the bottom right. As x increases, the y decreases, and that's why we've got a negative slope. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | That's because every time we increase x by 1, we go down in the y direction. So this is a downward sloping line. It's going from the top left to the bottom right. As x increases, the y decreases, and that's why we've got a negative slope. This line over here should have a positive slope. Let's verify it. So I'll use the same points that they use right over there. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | As x increases, the y decreases, and that's why we've got a negative slope. This line over here should have a positive slope. Let's verify it. So I'll use the same points that they use right over there. So this is line D. Slope is equal to rise over run. Now how much do we rise when we go from that point to that point? Well, let's see. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So I'll use the same points that they use right over there. So this is line D. Slope is equal to rise over run. Now how much do we rise when we go from that point to that point? Well, let's see. We could do it this way. We are rising. I could just count it out. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Well, let's see. We could do it this way. We are rising. I could just count it out. We are rising 1, 2, 3, 4, 5, 6. We are rising 6. And how much are we running? |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | I could just count it out. We are rising 1, 2, 3, 4, 5, 6. We are rising 6. And how much are we running? We are running, I'll do it in a different color. We're running 1, 2, 3, 4, 5, 6. We're running 6. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | And how much are we running? We are running, I'll do it in a different color. We're running 1, 2, 3, 4, 5, 6. We're running 6. So our slope is 6 over 6, which is 1. Which tells us that every time we move 1 in the x direction, positive 1 in the x direction, we go positive 1 in the y direction. So this is a, for every x, if we go negative 2 in the x direction, we're going to go negative 2 in the y direction. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | We're running 6. So our slope is 6 over 6, which is 1. Which tells us that every time we move 1 in the x direction, positive 1 in the x direction, we go positive 1 in the y direction. So this is a, for every x, if we go negative 2 in the x direction, we're going to go negative 2 in the y direction. So whatever we do in x, we're going to do the same thing in y in the slope. And notice, that was pretty easy. If we wanted to do it mathematically, we could figure out this coordinate right there. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So this is a, for every x, if we go negative 2 in the x direction, we're going to go negative 2 in the y direction. So whatever we do in x, we're going to do the same thing in y in the slope. And notice, that was pretty easy. If we wanted to do it mathematically, we could figure out this coordinate right there. That we could view as our starting position. Our starting position is negative 2, negative 4. And our finishing position is 4, 2. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | If we wanted to do it mathematically, we could figure out this coordinate right there. That we could view as our starting position. Our starting position is negative 2, negative 4. And our finishing position is 4, 2. And so our slope, change in y over change in x, I'll take this point, 2 minus negative 4, over 4 minus negative 2. 2 minus negative 4 is 6. Remember, that was just this distance right there. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | And our finishing position is 4, 2. And so our slope, change in y over change in x, I'll take this point, 2 minus negative 4, over 4 minus negative 2. 2 minus negative 4 is 6. Remember, that was just this distance right there. And then 4 minus negative 2, that's also 6. That's that distance right there. And we get a slope of 1. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Remember, that was just this distance right there. And then 4 minus negative 2, that's also 6. That's that distance right there. And we get a slope of 1. Let's do another one. Let's do another couple. These are interesting. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | And we get a slope of 1. Let's do another one. Let's do another couple. These are interesting. Let's do the line e right here. So change in y over change in x. So our change in y, when we go from this point to this point, I'll just count it out. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | These are interesting. Let's do the line e right here. So change in y over change in x. So our change in y, when we go from this point to this point, I'll just count it out. It's 1, 2, 3, 4, 5, 6, 7, 8. It's 8. Or you could even take this y coordinate, 2. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So our change in y, when we go from this point to this point, I'll just count it out. It's 1, 2, 3, 4, 5, 6, 7, 8. It's 8. Or you could even take this y coordinate, 2. This y coordinate, 2, minus negative 6, will give you that distance, 8. And then what's the change in y? Well, the y value here is, sorry, what's the change in x? |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Or you could even take this y coordinate, 2. This y coordinate, 2, minus negative 6, will give you that distance, 8. And then what's the change in y? Well, the y value here is, sorry, what's the change in x? The x value here is 4. The x value there is 4. x does not change. So it's 8 over 0. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Well, the y value here is, sorry, what's the change in x? The x value here is 4. The x value there is 4. x does not change. So it's 8 over 0. Well, we don't know. 8 over 0 is undefined. So in this situation, the slope is undefined. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So it's 8 over 0. Well, we don't know. 8 over 0 is undefined. So in this situation, the slope is undefined. When you have a vertical line, you say your slope is undefined because you're dividing by 0. But that tells you that you're dealing probably with a vertical line. Now, finally, let's just do this one. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So in this situation, the slope is undefined. When you have a vertical line, you say your slope is undefined because you're dividing by 0. But that tells you that you're dealing probably with a vertical line. Now, finally, let's just do this one. This seems like a pretty straight-up vanilla slope problem right there. You have that point right there, which is the point 3, 1. So this is line F. You have the point 3, 1. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Now, finally, let's just do this one. This seems like a pretty straight-up vanilla slope problem right there. You have that point right there, which is the point 3, 1. So this is line F. You have the point 3, 1. And then over here, you have the point negative 6, negative 2. So our slope would be equal to change in y. I'll take this as our ending point, just so you can go in different directions. So our change in y, so now we're going to go down in that direction. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So this is line F. You have the point 3, 1. And then over here, you have the point negative 6, negative 2. So our slope would be equal to change in y. I'll take this as our ending point, just so you can go in different directions. So our change in y, so now we're going to go down in that direction. So it's negative 2 minus 1. That's what this distance is right here. Negative 2 minus 1, which is equal to negative 3. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So our change in y, so now we're going to go down in that direction. So it's negative 2 minus 1. That's what this distance is right here. Negative 2 minus 1, which is equal to negative 3. Notice we went down 3. And then what is going to be our change in x? Well, we're going to go back that amount. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Negative 2 minus 1, which is equal to negative 3. Notice we went down 3. And then what is going to be our change in x? Well, we're going to go back that amount. What is that amount? Well, that is going to be negative 6. That's our end point, minus 3. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | Well, we're going to go back that amount. What is that amount? Well, that is going to be negative 6. That's our end point, minus 3. Negative 6 minus 3. That gives us that distance, which is negative 9. So for every time we go back 9, we're going to go down 3. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | That's our end point, minus 3. Negative 6 minus 3. That gives us that distance, which is negative 9. So for every time we go back 9, we're going to go down 3. If we go back 9, we're going to go down 3, which is the same thing as if we go forward 9. We're going to go up 3, all equivalent. And we see these cancel out, and you get a slope of 1 3rd, positive 1 3rd. |
Slope and rate of change Graphing lines and slope Algebra Basics Khan Academy.mp3 | So for every time we go back 9, we're going to go down 3. If we go back 9, we're going to go down 3, which is the same thing as if we go forward 9. We're going to go up 3, all equivalent. And we see these cancel out, and you get a slope of 1 3rd, positive 1 3rd. It's an upward sloping line. Every time we run 3, we rise 1. Anyway, hopefully that was a good review of slope for you. |
How to find equivalent expressions by combining like terms and using the distributive property.mp3 | All right, let's see if I can, if I can manipulate this thing a little bit. Let me just rewrite it. So I have x plus, x plus 2y plus x plus 2. So the first thing that jumps out at me before I even look at these choices here, I have an x over here, I have an x over there. Well, if I have one x and then I can add it to another x, that would be 2x. So I could rewrite this, let me do this in a different color. I could rewrite this x and this x, if I add them together, that's going to be 2x. |
How to find equivalent expressions by combining like terms and using the distributive property.mp3 | So the first thing that jumps out at me before I even look at these choices here, I have an x over here, I have an x over there. Well, if I have one x and then I can add it to another x, that would be 2x. So I could rewrite this, let me do this in a different color. I could rewrite this x and this x, if I add them together, that's going to be 2x. So this is, let me, actually let me, I don't want to skip any steps. So that's x plus x plus 2y, now I'm just switching the order, plus 2, and then these two x's right over here, I can just rewrite that as 2x. So I have 2x plus 2y, plus 2y, plus 2. |
How to find equivalent expressions by combining like terms and using the distributive property.mp3 | I could rewrite this x and this x, if I add them together, that's going to be 2x. So this is, let me, actually let me, I don't want to skip any steps. So that's x plus x plus 2y, now I'm just switching the order, plus 2, and then these two x's right over here, I can just rewrite that as 2x. So I have 2x plus 2y, plus 2y, plus 2. Now let's see, out of all of my choices, so this one, this is 2x plus 4y, plus 4. So that's not right, I have 2x plus 2y plus 2, so I can rule this one out. Now this one's interesting, it looks like they have factored out a 2. |
How to find equivalent expressions by combining like terms and using the distributive property.mp3 | So I have 2x plus 2y, plus 2y, plus 2. Now let's see, out of all of my choices, so this one, this is 2x plus 4y, plus 4. So that's not right, I have 2x plus 2y plus 2, so I can rule this one out. Now this one's interesting, it looks like they have factored out a 2. Let's see, if we factor out a 2 here, what happens? So we do see that 2 is a factor of that term, it's a factor of that term, and it's a factor of that term. So let's see if we can, if we can factor it out. |
How to find equivalent expressions by combining like terms and using the distributive property.mp3 | Now this one's interesting, it looks like they have factored out a 2. Let's see, if we factor out a 2 here, what happens? So we do see that 2 is a factor of that term, it's a factor of that term, and it's a factor of that term. So let's see if we can, if we can factor it out. So this is going to be 2 times x, I'll do the x in that same magenta color, 2 times x, plus, you have just a y left when you factor out the 2, and then if you factor out a 2 here, you're just going to have a 1 left. So 2 times x plus y plus 1, which is exactly what they have over here. And since I was able to find a choice, I will not pick none of the above. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | I'm actually going to show you two really equivalent ways of doing this, one that you might hear in a classroom, and it's kind of more of a mechanical, memorizing way of doing it, which might be faster, but you really don't know what you're doing. And then there's the one where you're essentially just applying something that you already know in kind of a logical way. So I'll first do the memorizing way that you might be exposed to. And they'll use something called FOIL. Let me write this down here. So you could immediately see that whenever someone gives you a mnemonic to memorize, that you're doing something pretty mechanical. So FOIL literally stands for First Outside. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | And they'll use something called FOIL. Let me write this down here. So you could immediately see that whenever someone gives you a mnemonic to memorize, that you're doing something pretty mechanical. So FOIL literally stands for First Outside. Let me write it this way. Let me write FOIL. Where the F in FOIL stands for First. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | So FOIL literally stands for First Outside. Let me write it this way. Let me write FOIL. Where the F in FOIL stands for First. The O in FOIL stands for Outside. The I stands for Inside. And then the L stands for Last. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | Where the F in FOIL stands for First. The O in FOIL stands for Outside. The I stands for Inside. And then the L stands for Last. And the reason why I don't like these things is when you're 35 years old, you're not going to remember what FOIL stood for, and then you're not going to remember how to multiply this binomial. But let's just apply FOIL. So FIRST says just multiply the first terms in each of these binomials. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | And then the L stands for Last. And the reason why I don't like these things is when you're 35 years old, you're not going to remember what FOIL stood for, and then you're not going to remember how to multiply this binomial. But let's just apply FOIL. So FIRST says just multiply the first terms in each of these binomials. So just multiply the 3x times the 5x. So 3x times the 5x. The Outside part tells us to multiply the outside terms. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | So FIRST says just multiply the first terms in each of these binomials. So just multiply the 3x times the 5x. So 3x times the 5x. The Outside part tells us to multiply the outside terms. So in this case, you have 3x on the outside, and you have negative 7 on the outside. So that is plus 3x times negative 7. The inside. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | The Outside part tells us to multiply the outside terms. So in this case, you have 3x on the outside, and you have negative 7 on the outside. So that is plus 3x times negative 7. The inside. Well, the inside terms here are 2 and 5x. So plus 2 times 5x. And then finally, you have the last terms. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | The inside. Well, the inside terms here are 2 and 5x. So plus 2 times 5x. And then finally, you have the last terms. You have the 2 and the negative 7. So the last term is 2 times negative 7. 2 times negative 7. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | And then finally, you have the last terms. You have the 2 and the negative 7. So the last term is 2 times negative 7. 2 times negative 7. So what you're essentially doing is just making sure that you're multiplying each term by every other term here. What we're essentially doing is multiplying, doing the distributive property twice. We're multiplying the 3x times 5x minus 7. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | 2 times negative 7. So what you're essentially doing is just making sure that you're multiplying each term by every other term here. What we're essentially doing is multiplying, doing the distributive property twice. We're multiplying the 3x times 5x minus 7. So 3x times 5x minus 7 is 3x times 5x plus 3x minus 7. And we're multiplying the 2 times 5x minus 7 to give us these terms. But anyway, let's just multiply this out just to get our answer. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | We're multiplying the 3x times 5x minus 7. So 3x times 5x minus 7 is 3x times 5x plus 3x minus 7. And we're multiplying the 2 times 5x minus 7 to give us these terms. But anyway, let's just multiply this out just to get our answer. 3x times 5x, the same thing as 3 times 5 times x times x, which is the same thing as 15x squared. You can use x to the first times x to the first. You multiply the x's, you get x squared. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | But anyway, let's just multiply this out just to get our answer. 3x times 5x, the same thing as 3 times 5 times x times x, which is the same thing as 15x squared. You can use x to the first times x to the first. You multiply the x's, you get x squared. 3 times 5 is 15. This term right here, 3 times negative 7 is negative 21. And then you have your x right over here. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | You multiply the x's, you get x squared. 3 times 5 is 15. This term right here, 3 times negative 7 is negative 21. And then you have your x right over here. And then you have this term, which is 2 times 5, which is 10 times x, so plus 10x. And then finally, you have this term here in blue. 2 times negative 7 is negative 14. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | And then you have your x right over here. And then you have this term, which is 2 times 5, which is 10 times x, so plus 10x. And then finally, you have this term here in blue. 2 times negative 7 is negative 14. And we aren't done yet. We can simplify this a little bit. We have two like terms here. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | 2 times negative 7 is negative 14. And we aren't done yet. We can simplify this a little bit. We have two like terms here. We have this. Let me find a new color. We have two terms with an x to the first power, just an x term right over here. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | We have two like terms here. We have this. Let me find a new color. We have two terms with an x to the first power, just an x term right over here. So if we have negative 21 of something and you add 10, or another way, if you have 10 of something and you subtract 21 of them, you're going to have negative 11 of that something. And we put the other terms here. You have 15x squared, and then you have your minus 14. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | We have two terms with an x to the first power, just an x term right over here. So if we have negative 21 of something and you add 10, or another way, if you have 10 of something and you subtract 21 of them, you're going to have negative 11 of that something. And we put the other terms here. You have 15x squared, and then you have your minus 14. And we are done. Now I said I would show you another way to do it. So I want to show you why the distributive property can get us here without having to memorize FOIL. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | You have 15x squared, and then you have your minus 14. And we are done. Now I said I would show you another way to do it. So I want to show you why the distributive property can get us here without having to memorize FOIL. So the distributive property tells us that if we're multiplying something times an expression, you just have to multiply it times every term in the expression. So we can distribute the 5x onto the 3x minus 7, this whole thing, onto the 3x plus 2. Let me just change the order, since we're used to distributing something from the left. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | So I want to show you why the distributive property can get us here without having to memorize FOIL. So the distributive property tells us that if we're multiplying something times an expression, you just have to multiply it times every term in the expression. So we can distribute the 5x onto the 3x minus 7, this whole thing, onto the 3x plus 2. Let me just change the order, since we're used to distributing something from the left. So this is the same thing as 5x minus 7 times 3x plus 2. I just swapped the two expressions. And we can distribute this whole thing times each of these terms. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | Let me just change the order, since we're used to distributing something from the left. So this is the same thing as 5x minus 7 times 3x plus 2. I just swapped the two expressions. And we can distribute this whole thing times each of these terms. Now what happens if I take 5x minus 7 times 3x? Well, that's just going to be 3x times 5x minus 7. So I've just distributed the 5x minus 7 times 3x. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | And we can distribute this whole thing times each of these terms. Now what happens if I take 5x minus 7 times 3x? Well, that's just going to be 3x times 5x minus 7. So I've just distributed the 5x minus 7 times 3x. And to that, I'm going to add 2 times 5x minus 7. I've just distributed the 5x minus 7 onto the 2. Now we can do distributive property again. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | So I've just distributed the 5x minus 7 times 3x. And to that, I'm going to add 2 times 5x minus 7. I've just distributed the 5x minus 7 onto the 2. Now we can do distributive property again. We can distribute the 3x onto the 5x. And we can distribute the 3x onto the negative 7. We can distribute the 2 onto the 5x over here. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | Now we can do distributive property again. We can distribute the 3x onto the 5x. And we can distribute the 3x onto the negative 7. We can distribute the 2 onto the 5x over here. And we can distribute the 2 on that negative 7. Now if we do it like this, what do we get? 3x times 5x. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | We can distribute the 2 onto the 5x over here. And we can distribute the 2 on that negative 7. Now if we do it like this, what do we get? 3x times 5x. That's this right over here. If we do 3x times negative 7, that's this term right over here. If you do 2 times 5x, that's this term right over here. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | 3x times 5x. That's this right over here. If we do 3x times negative 7, that's this term right over here. If you do 2 times 5x, that's this term right over here. If you do 2 times negative 7, that is this term right over here. So we got the exact same result that we got with FOIL. Now FOIL can be faster. |
Example 1 Multiplying a binomial by a binomial Algebra I Khan Academy.mp3 | If you do 2 times 5x, that's this term right over here. If you do 2 times negative 7, that is this term right over here. So we got the exact same result that we got with FOIL. Now FOIL can be faster. If you just want to do it, you kind of can skip to this step. I think it's important that you know that this is how it actually works. Just in case you do forget this when you're 35 or 45 years old and you're faced with multiplying binomial, you just have to remember the distributive property. |
Example two-step equation with numerator x Linear equations Algebra I Khan Academy.mp3 | So in order to do that, let's get rid of this 2. And the best way to get rid of that 2 is to subtract it. But if we want to subtract it from the right-hand side, we also have to subtract it from the left-hand side, because this is an equation. If this is equal to that, anything we do to that, we also have to do to this. So let's subtract 2 from both sides. So you subtract 2 from the right, subtract 2 from the left, and we get on the left-hand side, negative 16 minus 2 is negative 18, and then that is equal to x over 4, and then we have positive 2 minus 2, which is just going to be 0, so we don't even have to write that. I could write just a plus 0, but I think that's a little unnecessary. |
Example two-step equation with numerator x Linear equations Algebra I Khan Academy.mp3 | If this is equal to that, anything we do to that, we also have to do to this. So let's subtract 2 from both sides. So you subtract 2 from the right, subtract 2 from the left, and we get on the left-hand side, negative 16 minus 2 is negative 18, and then that is equal to x over 4, and then we have positive 2 minus 2, which is just going to be 0, so we don't even have to write that. I could write just a plus 0, but I think that's a little unnecessary. And so we have negative 18 is equal to x over 4, and our whole goal here is to isolate the x, to solve for the x. And the best way we could do that, if we have x over 4 here, if we multiply that by 4, we're just going to have an x. So we can multiply that by 4, but once again, this is an equation. |
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