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Scientific notation examples Pre-Algebra Khan Academy.mp3 | 0, 2, 7. And you want to write that in scientific notation. Well, you count all the digits up to the 2 behind the decimal. So 1, 2, 3, 4, 5, 6, 7, 8. So this is going to be 2.7 times 10 to the negative 8 power. Now, let's do another one where we start with the scientific notation value and we want to go to the numeric value, just to mix things up. So let's say you have 2.9 times 10 to the negative 5. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So 1, 2, 3, 4, 5, 6, 7, 8. So this is going to be 2.7 times 10 to the negative 8 power. Now, let's do another one where we start with the scientific notation value and we want to go to the numeric value, just to mix things up. So let's say you have 2.9 times 10 to the negative 5. So one way to think about it is this leading numeral plus all the 0's to the left of the decimal spot is going to be 5 digits. So you have a 2 and a 9, and then you're going to have 5, and then you're going to have 4 more 0's. 1, 2, 3, 4, and then you're going to have your decimal. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So let's say you have 2.9 times 10 to the negative 5. So one way to think about it is this leading numeral plus all the 0's to the left of the decimal spot is going to be 5 digits. So you have a 2 and a 9, and then you're going to have 5, and then you're going to have 4 more 0's. 1, 2, 3, 4, and then you're going to have your decimal. And how did I say no 4 0's? Because I'm counting this as 1, 2, 3, 4, 5 spaces behind the decimal, including the leading numeral. And so it's 0.000029. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | 1, 2, 3, 4, and then you're going to have your decimal. And how did I say no 4 0's? Because I'm counting this as 1, 2, 3, 4, 5 spaces behind the decimal, including the leading numeral. And so it's 0.000029. And just to verify, do the other technique. How do I write this in scientific notation? I count all of the digits, all of the leading 0's behind the decimal, including the leading non-zero numeral. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | And so it's 0.000029. And just to verify, do the other technique. How do I write this in scientific notation? I count all of the digits, all of the leading 0's behind the decimal, including the leading non-zero numeral. So I have 1, 2, 3, 4, 5 digits. So it's 10 to the negative 5. And so it'll be 2.9 times 10 to the negative 5. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | I count all of the digits, all of the leading 0's behind the decimal, including the leading non-zero numeral. So I have 1, 2, 3, 4, 5 digits. So it's 10 to the negative 5. And so it'll be 2.9 times 10 to the negative 5. And once again, this isn't just some type of black magic here. This actually makes a lot of sense. If I wanted to get this number to 2.9, what I would have to do is move the decimal over 1, 2, 3, 4, 5 spots, like that. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | And so it'll be 2.9 times 10 to the negative 5. And once again, this isn't just some type of black magic here. This actually makes a lot of sense. If I wanted to get this number to 2.9, what I would have to do is move the decimal over 1, 2, 3, 4, 5 spots, like that. And to get the decimal to move over to the right by 5 spots, I'll have to, let's say with 0, 0, 0, 0, 2, 9. If I multiply it by 10 to the 5th, I'm also going to have to multiply it by 10 to the negative 5. Because I don't want to change the number. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | If I wanted to get this number to 2.9, what I would have to do is move the decimal over 1, 2, 3, 4, 5 spots, like that. And to get the decimal to move over to the right by 5 spots, I'll have to, let's say with 0, 0, 0, 0, 2, 9. If I multiply it by 10 to the 5th, I'm also going to have to multiply it by 10 to the negative 5. Because I don't want to change the number. This right here is just multiplying something by 1. 10 to the 5th times 10 to the negative 5 is 1. So this right here, this part right here, is essentially going to move the decimal 5 to the right. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Because I don't want to change the number. This right here is just multiplying something by 1. 10 to the 5th times 10 to the negative 5 is 1. So this right here, this part right here, is essentially going to move the decimal 5 to the right. 1, 2, 3, 4, 5. So this will be 2.5. And then we're going to be left with times 10 to the negative 5. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | I think it's pretty easy to verify that b is a y intercept. The way you verify that is you substitute x is equal to 0. If you get x is equal to 0, remember x is equal to 0, that means that's where we're going to intercept the y axis. If x is equal to 0, this equation becomes y is equal to m times 0 plus b. m times 0 is just going to be 0. I don't care what m is, so then y is going to be equal to b. So the point 0, b is going to be on that line. So 0, b. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | If x is equal to 0, this equation becomes y is equal to m times 0 plus b. m times 0 is just going to be 0. I don't care what m is, so then y is going to be equal to b. So the point 0, b is going to be on that line. So 0, b. The line will intercept the y axis at the point y is equal to b. We'll see that with actual numbers in the next few videos. Just to verify for you that m is really the slope, let's just try some numbers out. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So 0, b. The line will intercept the y axis at the point y is equal to b. We'll see that with actual numbers in the next few videos. Just to verify for you that m is really the slope, let's just try some numbers out. We know the point 0, b is on the line. What happens when x is equal to 1? When x is equal to 1, you get y is equal to m times 1, or it's equal to m plus b. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Just to verify for you that m is really the slope, let's just try some numbers out. We know the point 0, b is on the line. What happens when x is equal to 1? When x is equal to 1, you get y is equal to m times 1, or it's equal to m plus b. We also know that the point 1, m plus b is also on the line. This is just the y value. So what's the slope between that point and that point? |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | When x is equal to 1, you get y is equal to m times 1, or it's equal to m plus b. We also know that the point 1, m plus b is also on the line. This is just the y value. So what's the slope between that point and that point? Let's take this as the end point. So you have m plus b, our change in y, m plus b minus b over our change in x. Over 1 minus 0. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So what's the slope between that point and that point? Let's take this as the end point. So you have m plus b, our change in y, m plus b minus b over our change in x. Over 1 minus 0. This is our change in y over change in x. We're using two points. That's our end point, that's our starting point. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Over 1 minus 0. This is our change in y over change in x. We're using two points. That's our end point, that's our starting point. So if you simplify this, b minus b is 0, 1 minus 0 is 1. So you get m over 1, or it's equal to m. So hopefully you're satisfied, and hopefully I didn't confuse you by staying in the abstract with all of these variables here. But this is definitely going to be the slope, and this is definitely going to be the y-intercept. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | That's our end point, that's our starting point. So if you simplify this, b minus b is 0, 1 minus 0 is 1. So you get m over 1, or it's equal to m. So hopefully you're satisfied, and hopefully I didn't confuse you by staying in the abstract with all of these variables here. But this is definitely going to be the slope, and this is definitely going to be the y-intercept. Now given that, what I want to do in this exercise is look at these graphs, and then use the already drawn graphs to figure out the equation. So we're going to look at these, figure out the slopes, figure out the y-intercepts, and then know the equation. So let's do this line A first. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | But this is definitely going to be the slope, and this is definitely going to be the y-intercept. Now given that, what I want to do in this exercise is look at these graphs, and then use the already drawn graphs to figure out the equation. So we're going to look at these, figure out the slopes, figure out the y-intercepts, and then know the equation. So let's do this line A first. So what is A's slope? So let's start at some arbitrary point. Let's start right over there. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So let's do this line A first. So what is A's slope? So let's start at some arbitrary point. Let's start right over there. And then let us see, and we want to get to even numbers. So let's see, if we run 1, 2, 3. So if our delta x is equal to 3, our delta y is equal to, we go down by 2, it's equal to negative 2. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Let's start right over there. And then let us see, and we want to get to even numbers. So let's see, if we run 1, 2, 3. So if our delta x is equal to 3, our delta y is equal to, we go down by 2, it's equal to negative 2. So for A, change in y for a change in x. When our change in x is 3, our change in y is negative 2. So our slope is negative 2 thirds. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So if our delta x is equal to 3, our delta y is equal to, we go down by 2, it's equal to negative 2. So for A, change in y for a change in x. When our change in x is 3, our change in y is negative 2. So our slope is negative 2 thirds. When we go over by 3, we're going to go down by 2. Or if we go over by 1, we're going to go down by 2 thirds. You can't exactly see it there, but you definitely see it when you go over by 3. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So our slope is negative 2 thirds. When we go over by 3, we're going to go down by 2. Or if we go over by 1, we're going to go down by 2 thirds. You can't exactly see it there, but you definitely see it when you go over by 3. So that's our slope. We've essentially done half of that problem. Now we have to figure out the y-intercept. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | You can't exactly see it there, but you definitely see it when you go over by 3. So that's our slope. We've essentially done half of that problem. Now we have to figure out the y-intercept. So that right there is our m. Now what is our b, our y-intercept? Well, where does this intersect the y-axis? Well, we already said the slope is 2 thirds, so this is the point y is equal to 2. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Now we have to figure out the y-intercept. So that right there is our m. Now what is our b, our y-intercept? Well, where does this intersect the y-axis? Well, we already said the slope is 2 thirds, so this is the point y is equal to 2. When we go over by 1 to the right, we would have gone down by 2 thirds. So this right here must be the point 1 and 1 third. Or another way to say it is 4 thirds. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Well, we already said the slope is 2 thirds, so this is the point y is equal to 2. When we go over by 1 to the right, we would have gone down by 2 thirds. So this right here must be the point 1 and 1 third. Or another way to say it is 4 thirds. That's the point y is equal to 4 thirds right there. A little bit more than 1, but 1 and 1 third. So we can say b is equal to 4 thirds. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Or another way to say it is 4 thirds. That's the point y is equal to 4 thirds right there. A little bit more than 1, but 1 and 1 third. So we can say b is equal to 4 thirds. And so we'll know that the equation is y is equal to m, negative 2 thirds, x, plus b, plus 4 thirds. That's equation A. Let's do equation B. Hopefully we won't have to deal with as many fractions here. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So we can say b is equal to 4 thirds. And so we'll know that the equation is y is equal to m, negative 2 thirds, x, plus b, plus 4 thirds. That's equation A. Let's do equation B. Hopefully we won't have to deal with as many fractions here. Equation B, let's figure out its slope first. Let's start at some reasonable point. Let's see. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Let's do equation B. Hopefully we won't have to deal with as many fractions here. Equation B, let's figure out its slope first. Let's start at some reasonable point. Let's see. We could start at that point. And then if we, let me do it right here. B, equation B. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Let's see. We could start at that point. And then if we, let me do it right here. B, equation B. When our delta x is equal to, let me write it this way. Delta x. So our delta x could be 1. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | B, equation B. When our delta x is equal to, let me write it this way. Delta x. So our delta x could be 1. When we move over 1 to the right, what happens to our delta y? We go up by 3. Delta x, delta y. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So our delta x could be 1. When we move over 1 to the right, what happens to our delta y? We go up by 3. Delta x, delta y. Our change in y is 3. So delta y over delta x, when we go to the right, our change in x is 1. Our change in x is 1, our change in y is positive 3. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Delta x, delta y. Our change in y is 3. So delta y over delta x, when we go to the right, our change in x is 1. Our change in x is 1, our change in y is positive 3. So our slope is equal to 3. What is our y intercept? Well when x is equal to 0, y is equal to 1. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Our change in x is 1, our change in y is positive 3. So our slope is equal to 3. What is our y intercept? Well when x is equal to 0, y is equal to 1. So b is equal to 1. So this was a lot easier. Here the equation is y is equal to 3x plus 1. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Well when x is equal to 0, y is equal to 1. So b is equal to 1. So this was a lot easier. Here the equation is y is equal to 3x plus 1. Let's do that last line there. Line C. Line C. Alright. So let's do the y intercept first. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Here the equation is y is equal to 3x plus 1. Let's do that last line there. Line C. Line C. Alright. So let's do the y intercept first. You see immediately the y intercept when x is equal to 0, y is negative 2. So b is equal to negative 2. And then what is the slope? |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So let's do the y intercept first. You see immediately the y intercept when x is equal to 0, y is negative 2. So b is equal to negative 2. And then what is the slope? M is equal to change in y over change in x. So let's see. Let's start at that y intercept. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | And then what is the slope? M is equal to change in y over change in x. So let's see. Let's start at that y intercept. And if we go over to the right by 1, 2, 3, 4. So our change in x is equal to 4. What is our change in y? |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Let's start at that y intercept. And if we go over to the right by 1, 2, 3, 4. So our change in x is equal to 4. What is our change in y? Our change in y is positive 2. Change in y is equal to 2. So change in y is 2 when change in x is 4. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | What is our change in y? Our change in y is positive 2. Change in y is equal to 2. So change in y is 2 when change in x is 4. So the slope is equal to 1 half. 2 over 4. So the equation here is y is equal to 1 half x. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So change in y is 2 when change in x is 4. So the slope is equal to 1 half. 2 over 4. So the equation here is y is equal to 1 half x. That's our slope. Minus 2. And we're done. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So the equation here is y is equal to 1 half x. That's our slope. Minus 2. And we're done. Now let's go the other way. Let's look at some equations of lines knowing that this is the slope and this is the y intercept. That's the m, that's the b. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | And we're done. Now let's go the other way. Let's look at some equations of lines knowing that this is the slope and this is the y intercept. That's the m, that's the b. And actually graph them. So let's do this first line. I already started circling it in orange. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | That's the m, that's the b. And actually graph them. So let's do this first line. I already started circling it in orange. The y intercept is 5. When x is equal to 0, y is equal to 5. You can verify that on the equation. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | I already started circling it in orange. The y intercept is 5. When x is equal to 0, y is equal to 5. You can verify that on the equation. So when x is equal to 0, y is equal to 1, 2, 3, 4, 5. That's the y intercept. And then the slope is 2. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | You can verify that on the equation. So when x is equal to 0, y is equal to 1, 2, 3, 4, 5. That's the y intercept. And then the slope is 2. That means when I move 1 in the x direction, I move up 2 in the y direction. So if I move 1 in the x direction, I move up 2 in the y direction. If I move 1 in the x direction, I move up 2 in the y direction. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | And then the slope is 2. That means when I move 1 in the x direction, I move up 2 in the y direction. So if I move 1 in the x direction, I move up 2 in the y direction. If I move 1 in the x direction, I move up 2 in the y direction. If I move back 1 in the x direction, I move down two in the y direction. If I move back 1 in the x direction, I move down 2 in the y direction. I keep doing that. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | If I move 1 in the x direction, I move up 2 in the y direction. If I move back 1 in the x direction, I move down two in the y direction. If I move back 1 in the x direction, I move down 2 in the y direction. I keep doing that. So this line is going to look. I can't draw lines too neatly but this is going to be my best shot. It's going to look something like that. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | I keep doing that. So this line is going to look. I can't draw lines too neatly but this is going to be my best shot. It's going to look something like that. And it'll just keep going on, on and on and on. So that's our first line. I could just keep going down like that. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | It's going to look something like that. And it'll just keep going on, on and on and on. So that's our first line. I could just keep going down like that. Now let's do this second line. y is equal to negative 0.2x plus 7. So let me write that. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | I could just keep going down like that. Now let's do this second line. y is equal to negative 0.2x plus 7. So let me write that. y is equal to negative 0.2x plus 7. So it's always easier to think in fractions. So 0.2 is the same thing as 1 5th. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So let me write that. y is equal to negative 0.2x plus 7. So it's always easier to think in fractions. So 0.2 is the same thing as 1 5th. So we could write y is equal to negative 1 5th x plus 7. So we know it's y-intercept. It's 7. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So 0.2 is the same thing as 1 5th. So we could write y is equal to negative 1 5th x plus 7. So we know it's y-intercept. It's 7. So this is 1, 2, 3, 4, 5, 6. That's our y-intercept when x is equal to 0. And this tells us that for every 5 we move to the right, we move down 1. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | It's 7. So this is 1, 2, 3, 4, 5, 6. That's our y-intercept when x is equal to 0. And this tells us that for every 5 we move to the right, we move down 1. So we could view this as negative 1 over 5. That delta y over delta x is equal to negative 1 over 5. So for every 5 we move to the right, we move down 1. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | And this tells us that for every 5 we move to the right, we move down 1. So we could view this as negative 1 over 5. That delta y over delta x is equal to negative 1 over 5. So for every 5 we move to the right, we move down 1. So every 5, 1, 2, 3, 4, 5, we moved 5 to the right. That means we must move down 1. We move 5 to the right, 1, 2, 3, 4, 5, we must move down 1. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So for every 5 we move to the right, we move down 1. So every 5, 1, 2, 3, 4, 5, we moved 5 to the right. That means we must move down 1. We move 5 to the right, 1, 2, 3, 4, 5, we must move down 1. If you go backwards, if you move 5 backwards, so if you view instead of this, you view this as 1 over negative 5. These are obviously equivalent numbers. So if you go back 5, that's negative 5, 1, 2, 3, 4, 5, then you move up 1. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | We move 5 to the right, 1, 2, 3, 4, 5, we must move down 1. If you go backwards, if you move 5 backwards, so if you view instead of this, you view this as 1 over negative 5. These are obviously equivalent numbers. So if you go back 5, that's negative 5, 1, 2, 3, 4, 5, then you move up 1. You go back 5, 1, 2, 3, 4, 5, you move up 1. So the line is going to look like this. I have to just connect the dots. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So if you go back 5, that's negative 5, 1, 2, 3, 4, 5, then you move up 1. You go back 5, 1, 2, 3, 4, 5, you move up 1. So the line is going to look like this. I have to just connect the dots. I think you get the idea. I just have to connect those dots. I could have drawn it a little bit straighter. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | I have to just connect the dots. I think you get the idea. I just have to connect those dots. I could have drawn it a little bit straighter. Now let's do this one. y is equal to negative x. y is equal to negative x. Where is the b term? |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | I could have drawn it a little bit straighter. Now let's do this one. y is equal to negative x. y is equal to negative x. Where is the b term? I don't see any b term. You remember we're saying y is equal to mx plus b. Where is the b? |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Where is the b term? I don't see any b term. You remember we're saying y is equal to mx plus b. Where is the b? Well, the b is 0. You could view this as plus 0. So here is b is 0. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | Where is the b? Well, the b is 0. You could view this as plus 0. So here is b is 0. When x is 0, y is 0. So that's our y-intercept right there at the origin. And then the slope, once again, you just see a negative sign. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | So here is b is 0. When x is 0, y is 0. So that's our y-intercept right there at the origin. And then the slope, once again, you just see a negative sign. You could view that as negative 1x plus 0. So slope is negative 1. When you move to the right by 1, when change in x is 1, change in y is negative 1. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | And then the slope, once again, you just see a negative sign. You could view that as negative 1x plus 0. So slope is negative 1. When you move to the right by 1, when change in x is 1, change in y is negative 1. When you move up by 1 in x, you go down by 1 in y. Or if you go down by 1 in x, you're going to go up by 1 in y. x and y are going to have opposite signs. They go in opposite directions. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | When you move to the right by 1, when change in x is 1, change in y is negative 1. When you move up by 1 in x, you go down by 1 in y. Or if you go down by 1 in x, you're going to go up by 1 in y. x and y are going to have opposite signs. They go in opposite directions. So the line is going to look like that. It's going to look like that. You can almost imagine it splitting the second and fourth quadrants. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | They go in opposite directions. So the line is going to look like that. It's going to look like that. You can almost imagine it splitting the second and fourth quadrants. Now I'll do one more. Let's do this last one right here. y is equal to 3.75. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | You can almost imagine it splitting the second and fourth quadrants. Now I'll do one more. Let's do this last one right here. y is equal to 3.75. So now you're saying, gee, we're looking for y is equal to mx plus b. Where is this x term? It's completely gone. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | y is equal to 3.75. So now you're saying, gee, we're looking for y is equal to mx plus b. Where is this x term? It's completely gone. Well, the reality here is this could be rewritten as y is equal to 0x plus 3.75. Now it makes sense. The slope is 0. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | It's completely gone. Well, the reality here is this could be rewritten as y is equal to 0x plus 3.75. Now it makes sense. The slope is 0. No matter how much we change our x, y does not change. Delta y over delta x is equal to 0. I don't care how much you change your x. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | The slope is 0. No matter how much we change our x, y does not change. Delta y over delta x is equal to 0. I don't care how much you change your x. So our y-intercept is 3.75. So 1, 2, 3.75 is right around there. Just want to get close. |
Slope-intercept equation from a graph examples Algebra I Khan Academy.mp3 | I don't care how much you change your x. So our y-intercept is 3.75. So 1, 2, 3.75 is right around there. Just want to get close. 3 and 3 fourths. And then as I change x, y will not change. y is always going to be 3.75. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | Everyone in the kingdom is very impressed with your ability to help with the party planning. Everyone except for this gentleman right over here. This is Arbegla, and he is the king's top advisor and also chief party planner. And he seems somewhat threatened by your ability to solve these otherwise unsolvable problems, or at least from his point of view, because he keeps over-ordering or under-ordering things like cupcakes. And so he says, king, that cupcake problem was easy. Ask them about the potato chip issue, because we can never get the potato chips right. And so the king says, Arbegla, that's a good idea. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | And he seems somewhat threatened by your ability to solve these otherwise unsolvable problems, or at least from his point of view, because he keeps over-ordering or under-ordering things like cupcakes. And so he says, king, that cupcake problem was easy. Ask them about the potato chip issue, because we can never get the potato chips right. And so the king says, Arbegla, that's a good idea. We need to get the potato chips right. So he comes to you and says, how do we figure out, on average, how many potato chips we need to order? And to do that, we have to figure out how much, on average, does each man eat and how much each woman eats. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | And so the king says, Arbegla, that's a good idea. We need to get the potato chips right. So he comes to you and says, how do we figure out, on average, how many potato chips we need to order? And to do that, we have to figure out how much, on average, does each man eat and how much each woman eats. And you say, well, what about the children? He say, well, in our kingdom, the king says, in our kingdom, we forbid potato chips for children. You say, oh, well, that's all in good. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | And to do that, we have to figure out how much, on average, does each man eat and how much each woman eats. And you say, well, what about the children? He say, well, in our kingdom, the king says, in our kingdom, we forbid potato chips for children. You say, oh, well, that's all in good. Well, tell me what happened at the previous parties. And so the king says, you might remember, at the last party, in fact, the last two parties, we had 500 adults. At the last party, 200 of them were men, 200 men, and 300 of them were women, 300 were women, and in total, they ate 1,200 bags of potato chips. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | You say, oh, well, that's all in good. Well, tell me what happened at the previous parties. And so the king says, you might remember, at the last party, in fact, the last two parties, we had 500 adults. At the last party, 200 of them were men, 200 men, and 300 of them were women, 300 were women, and in total, they ate 1,200 bags of potato chips. 1,200 bags of potato chips. And you say, what about the party before that? He says, that one, we had a bigger skew towards women. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | At the last party, 200 of them were men, 200 men, and 300 of them were women, 300 were women, and in total, they ate 1,200 bags of potato chips. 1,200 bags of potato chips. And you say, what about the party before that? He says, that one, we had a bigger skew towards women. We only had 100 men, 100 men, and we had 400 women, 400 women, and that time, we actually had fewer bags consumed, 1,100, 1,100 bags of potato chips. So you say, okay, king and our bigla, this seems like a fairly straightforward thing. Let me define some variables to represent our unknowns. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | He says, that one, we had a bigger skew towards women. We only had 100 men, 100 men, and we had 400 women, 400 women, and that time, we actually had fewer bags consumed, 1,100, 1,100 bags of potato chips. So you say, okay, king and our bigla, this seems like a fairly straightforward thing. Let me define some variables to represent our unknowns. So you go ahead and you say, well, let's let, let's let, let's let m equal the number of bags eaten, eaten by each man, by each man, and you could think of it on average, or maybe everyone, all the men in that kingdom are completely identical, or maybe this is the average, number of bags eaten by each man, and let's let w equal the number of bags eaten by each woman, each woman. And so with that, with these definitions of our variables, let's think about how we can represent this first piece of information, this piece of information in green. Well, let's think about the total number of bags that the men ate. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | Let me define some variables to represent our unknowns. So you go ahead and you say, well, let's let, let's let, let's let m equal the number of bags eaten, eaten by each man, by each man, and you could think of it on average, or maybe everyone, all the men in that kingdom are completely identical, or maybe this is the average, number of bags eaten by each man, and let's let w equal the number of bags eaten by each woman, each woman. And so with that, with these definitions of our variables, let's think about how we can represent this first piece of information, this piece of information in green. Well, let's think about the total number of bags that the men ate. You had 200, you had 200 men, 200, let me scroll over a little bit. You had 200 men, and they each ate m bags, m bags per man, so the men at this first party collectively ate 200 times m bags. If m is 10 bags per man, then this would be 2,000. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | Well, let's think about the total number of bags that the men ate. You had 200, you had 200 men, 200, let me scroll over a little bit. You had 200 men, and they each ate m bags, m bags per man, so the men at this first party collectively ate 200 times m bags. If m is 10 bags per man, then this would be 2,000. If m was five bags per man, then this would be 5,000. We don't know what m is, but 200 times m is the total eaten by the men. Same logic, total eaten by the women is 300, 300 women times the number of bags eaten by each woman, and so if you add the total eaten by the men and the women, you get the 1,200 bags. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | If m is 10 bags per man, then this would be 2,000. If m was five bags per man, then this would be 5,000. We don't know what m is, but 200 times m is the total eaten by the men. Same logic, total eaten by the women is 300, 300 women times the number of bags eaten by each woman, and so if you add the total eaten by the men and the women, you get the 1,200 bags. You get the 1,200 bags. So this is this information written algebraically given these variable definitions. Now let's do the same thing with the second party, the information that they gave us right over here. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | Same logic, total eaten by the women is 300, 300 women times the number of bags eaten by each woman, and so if you add the total eaten by the men and the women, you get the 1,200 bags. You get the 1,200 bags. So this is this information written algebraically given these variable definitions. Now let's do the same thing with the second party, the information that they gave us right over here. Let's think about how we can represent this algebraically. Well, similar logic. What was the total that the men ate at that party? |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | Now let's do the same thing with the second party, the information that they gave us right over here. Let's think about how we can represent this algebraically. Well, similar logic. What was the total that the men ate at that party? It was 100 men times m bags per man, and we're assuming that m is the same across parties, that men on average always eat the same number of bags. And how many did the women eat at that second party? Well, you had 400 women, and on average, they ate w bags per woman. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | What was the total that the men ate at that party? It was 100 men times m bags per man, and we're assuming that m is the same across parties, that men on average always eat the same number of bags. And how many did the women eat at that second party? Well, you had 400 women, and on average, they ate w bags per woman. So this is how 400 times w is the total number that the women ate. You add those two together, you have the total number that all the adults ate. So this is going to be 1,100 bags. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | Well, you had 400 women, and on average, they ate w bags per woman. So this is how 400 times w is the total number that the women ate. You add those two together, you have the total number that all the adults ate. So this is going to be 1,100 bags. So it looks pretty similar now. You have a system of two equations with two unknowns, and so you try your best to solve it. But when you solve it, you see something interesting. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | So this is going to be 1,100 bags. So it looks pretty similar now. You have a system of two equations with two unknowns, and so you try your best to solve it. But when you solve it, you see something interesting. Last time it was very convenient. You had a, I think it was a 500 here for 500 adults, and you had another 500, and so it seemed like it was pretty easy to cancel out one of the variables. Here it seems a little bit more difficult. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | But when you solve it, you see something interesting. Last time it was very convenient. You had a, I think it was a 500 here for 500 adults, and you had another 500, and so it seemed like it was pretty easy to cancel out one of the variables. Here it seems a little bit more difficult. What's multiplying by the m's, it's different here. The coefficient on the w is different over here. But you say, well, maybe I can change one of these equations so it makes it a little bit easier to cancel out with the other equation. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | Here it seems a little bit more difficult. What's multiplying by the m's, it's different here. The coefficient on the w is different over here. But you say, well, maybe I can change one of these equations so it makes it a little bit easier to cancel out with the other equation. So what if, for example, I were to take this blue equation right over here and multiply it by negative two? And you might say, well, Sal, why are we multiplying it by negative two? Well, if we were to multiply it by negative two, this 100m would become a negative 200m. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | But you say, well, maybe I can change one of these equations so it makes it a little bit easier to cancel out with the other equation. So what if, for example, I were to take this blue equation right over here and multiply it by negative two? And you might say, well, Sal, why are we multiplying it by negative two? Well, if we were to multiply it by negative two, this 100m would become a negative 200m. And if it was a negative 200m, then that would cancel out with a positive 200m when we add the two. So let's see what happens. So let's just multiply, let's multiply this blue equation by negative two. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | Well, if we were to multiply it by negative two, this 100m would become a negative 200m. And if it was a negative 200m, then that would cancel out with a positive 200m when we add the two. So let's see what happens. So let's just multiply, let's multiply this blue equation by negative two. We're gonna multiply by negative two. Let me scroll over to the left a little bit. So what happens? |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | So let's just multiply, let's multiply this blue equation by negative two. We're gonna multiply by negative two. Let me scroll over to the left a little bit. So what happens? Remember, when we multiply an equation, we can't just do one term. We have to do, and we can't just do one side of the equation. We have to do the entire equation in order for the equality to hold true. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | So what happens? Remember, when we multiply an equation, we can't just do one term. We have to do, and we can't just do one side of the equation. We have to do the entire equation in order for the equality to hold true. So negative two times 100m is negative 200m. Negative two times 400w, and there's a positive right over here, so it becomes negative 800w. And then negative two, now we did the left-hand side, but we also have to do the right-hand side. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | We have to do the entire equation in order for the equality to hold true. So negative two times 100m is negative 200m. Negative two times 400w, and there's a positive right over here, so it becomes negative 800w. And then negative two, now we did the left-hand side, but we also have to do the right-hand side. Negative two times 1,100 is negative 2,200. So just to be clear, this equation that I just wrote here essentially has the same information we just manipulated. We just changed this equation, multiplied both sides by negative two. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | And then negative two, now we did the left-hand side, but we also have to do the right-hand side. Negative two times 1,100 is negative 2,200. So just to be clear, this equation that I just wrote here essentially has the same information we just manipulated. We just changed this equation, multiplied both sides by negative two. But it's kind of the same constraint. What makes this interesting is now we can rewrite this green equation. Let me do it over here, this first one. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | We just changed this equation, multiplied both sides by negative two. But it's kind of the same constraint. What makes this interesting is now we can rewrite this green equation. Let me do it over here, this first one. 200m plus 300w is equal to 1,200. And the whole reason why I multiplied by negative two is so that if I were to add these two things, I might be able to get rid of that variable over there. And so let's do that. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | Let me do it over here, this first one. 200m plus 300w is equal to 1,200. And the whole reason why I multiplied by negative two is so that if I were to add these two things, I might be able to get rid of that variable over there. And so let's do that. Let's add the left-hand sides, and let's add the right-hand sides. And you could literally view it as, we're starting with this blue equation. We're adding this quantity, the left-hand side of the yellow equation to the left-hand side of the blue. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | And so let's do that. Let's add the left-hand sides, and let's add the right-hand sides. And you could literally view it as, we're starting with this blue equation. We're adding this quantity, the left-hand side of the yellow equation to the left-hand side of the blue. And then 1,200 is the exact same thing that we're adding to the right-hand side. We know that this is equal to this, so we can add this to the left-hand side and this to the right-hand side. So let's see what happens. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | We're adding this quantity, the left-hand side of the yellow equation to the left-hand side of the blue. And then 1,200 is the exact same thing that we're adding to the right-hand side. We know that this is equal to this, so we can add this to the left-hand side and this to the right-hand side. So let's see what happens. So the good thing is, the whole reason why we multiplied it by negative two is so that these two characters cancel out. You add those two together, you just get zero m, or just zero. You have negative 800w plus 300w. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | So let's see what happens. So the good thing is, the whole reason why we multiplied it by negative two is so that these two characters cancel out. You add those two together, you just get zero m, or just zero. You have negative 800w plus 300w. Well, that's negative 500w. And then on the right-hand side, you have negative 2,200 plus 1,200. So that's negative 1,000. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | You have negative 800w plus 300w. Well, that's negative 500w. And then on the right-hand side, you have negative 2,200 plus 1,200. So that's negative 1,000. And now this is pretty straightforward. One equation and one unknown, a fairly straightforward equation. We divide both sides by the coefficient of w, multiplying w, so divide by negative 500 on the left, divide by negative 500 on the right. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | So that's negative 1,000. And now this is pretty straightforward. One equation and one unknown, a fairly straightforward equation. We divide both sides by the coefficient of w, multiplying w, so divide by negative 500 on the left, divide by negative 500 on the right. And we are left with w is equal to, w is equal to two. On average, women ate two bags of potato chips at these parties. We're assuming that that's constant across the parties. |
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