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Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | So why don't we do that? Let's take three blocks from this side. But we can't just take it from that side if we want to keep it balanced. We have to do it to this side, too. We've got to take away three blocks. So we're subtracting three from that side and subtracting three from the right side. So on the left-hand side, we're going to be left with just these two blocks of mass y. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | We have to do it to this side, too. We've got to take away three blocks. So we're subtracting three from that side and subtracting three from the right side. So on the left-hand side, we're going to be left with just these two blocks of mass y. So our total mass is now going to be 2y. These 3 minus 3 is 0. And you see that here. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | So on the left-hand side, we're going to be left with just these two blocks of mass y. So our total mass is now going to be 2y. These 3 minus 3 is 0. And you see that here. We're just left with two y's right over here. And on the right-hand side, we got rid of three of the blocks. So we only have four of them left. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | And you see that here. We're just left with two y's right over here. And on the right-hand side, we got rid of three of the blocks. So we only have four of them left. So you have two of these y masses is equal to 4 kilograms. Because we did the same thing to both sides, the scale is still balanced. And now, how do we solve this? |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | So we only have four of them left. So you have two of these y masses is equal to 4 kilograms. Because we did the same thing to both sides, the scale is still balanced. And now, how do we solve this? And you might be able to solve this in your head. I have 2 times something is equal to 4. You could kind of think about what that is. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | And now, how do we solve this? And you might be able to solve this in your head. I have 2 times something is equal to 4. You could kind of think about what that is. But if we want to stay true to what we've been doing before, let's think about it. I have 2 of something is equal to something else. What if I multiplied both sides by 2? |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | You could kind of think about what that is. But if we want to stay true to what we've been doing before, let's think about it. I have 2 of something is equal to something else. What if I multiplied both sides by 2? Sorry, what if I multiplied both sides by 1 half? Or in other ways, dividing both sides by 2. If I multiply this side by 1 half, if I essentially take away half of the mass, or I only leave half of the mass, then I'm only going to have one block here. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | What if I multiplied both sides by 2? Sorry, what if I multiplied both sides by 1 half? Or in other ways, dividing both sides by 2. If I multiply this side by 1 half, if I essentially take away half of the mass, or I only leave half of the mass, then I'm only going to have one block here. And if I take away half of the mass over here, I'm going to have to take away two of these blocks right over there. And what I just did, you could say I multiplied both sides by 1 half. Or just for the sake of a little change, you could say I divided both sides by 2. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | If I multiply this side by 1 half, if I essentially take away half of the mass, or I only leave half of the mass, then I'm only going to have one block here. And if I take away half of the mass over here, I'm going to have to take away two of these blocks right over there. And what I just did, you could say I multiplied both sides by 1 half. Or just for the sake of a little change, you could say I divided both sides by 2. And on the left-hand side, I'm left with a mass of y. And on the right-hand side, I'm left with a mass of 4 divided by 2 is 2. And once again, I can still write this equal sign because the scale is balanced. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | Or just for the sake of a little change, you could say I divided both sides by 2. And on the left-hand side, I'm left with a mass of y. And on the right-hand side, I'm left with a mass of 4 divided by 2 is 2. And once again, I can still write this equal sign because the scale is balanced. I did the exact same thing to both sides. I left half of what was on the left-hand side and half of what was on the right-hand side. It was balanced before, half of each side, so it's going to be balanced again. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | And once again, I can still write this equal sign because the scale is balanced. I did the exact same thing to both sides. I left half of what was on the left-hand side and half of what was on the right-hand side. It was balanced before, half of each side, so it's going to be balanced again. But there we've done it. We've solved something that's actually not so easy to solve, or might not look so easy at first. We figured out that our mystery mass, y, is 2 kilograms. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | It was balanced before, half of each side, so it's going to be balanced again. But there we've done it. We've solved something that's actually not so easy to solve, or might not look so easy at first. We figured out that our mystery mass, y, is 2 kilograms. And you can verify this. This is the really fun thing about algebra, is that once you get to this point, you can go back and think about whether the original problem we saw made sense. Let's do that. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | We figured out that our mystery mass, y, is 2 kilograms. And you can verify this. This is the really fun thing about algebra, is that once you get to this point, you can go back and think about whether the original problem we saw made sense. Let's do that. Let's think about whether the original problem made sense. And to do that, I want you to calculate, now that we know what the mass, y, is, is 2 kilograms, what was the total mass on each side? Well, let's calculate it. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | Let's do that. Let's think about whether the original problem made sense. And to do that, I want you to calculate, now that we know what the mass, y, is, is 2 kilograms, what was the total mass on each side? Well, let's calculate it. You have 2, I'll write it right over here. This is 2 kilograms. I'll do it in purple color. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | Well, let's calculate it. You have 2, I'll write it right over here. This is 2 kilograms. I'll do it in purple color. So this is a 2. So we had 6 kilograms plus these 3. You had 9 kilograms on the left-hand side. |
Why we do the same thing to both sides Multi-step equations Algebra I Khan Academy.mp3 | I'll do it in purple color. So this is a 2. So we had 6 kilograms plus these 3. You had 9 kilograms on the left-hand side. And on the right-hand side, I had these 7 plus 2 here. 7 plus 2 is 9 kilograms. That's why it was balanced, our mystery mass. |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | So let's think about what four to the negative three times four to the fifth power is going to be equal to. And I encourage you to pause the video and think about it on your own. Well, there's a couple of ways to do this. One, you say, oh look, I'm multiplying two things that have the same base, so this is going to be that base, four, and then I add the exponents. Four to the negative three plus five power, which is equal to four to the second power. And that's just a straightforward exponent property, but you can also think about why does that actually make sense? Four to the negative three power, that is one over four to the third power. |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | One, you say, oh look, I'm multiplying two things that have the same base, so this is going to be that base, four, and then I add the exponents. Four to the negative three plus five power, which is equal to four to the second power. And that's just a straightforward exponent property, but you can also think about why does that actually make sense? Four to the negative three power, that is one over four to the third power. Or you could view that as one over four times four times four and then four to the fifth, that's five fours being multiplied together, so it's times four times four times four times four times four. And so notice, when you multiply this out, you're gonna have five fours in the numerator and three fours in the denominator. And so three of these in the denominator are gonna cancel out with three of these in the numerator. |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | Four to the negative three power, that is one over four to the third power. Or you could view that as one over four times four times four and then four to the fifth, that's five fours being multiplied together, so it's times four times four times four times four times four. And so notice, when you multiply this out, you're gonna have five fours in the numerator and three fours in the denominator. And so three of these in the denominator are gonna cancel out with three of these in the numerator. And so you're gonna be left with five minus three, or negative three plus five fours. So this four times four is the same thing as four squared. Now let's do one with variables. |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | And so three of these in the denominator are gonna cancel out with three of these in the numerator. And so you're gonna be left with five minus three, or negative three plus five fours. So this four times four is the same thing as four squared. Now let's do one with variables. So let's say that you have a to the negative four power times a to the, let's say a squared. What is that going to be? Well, once again, you have the same base. |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | Now let's do one with variables. So let's say that you have a to the negative four power times a to the, let's say a squared. What is that going to be? Well, once again, you have the same base. In this case, it's a. And so, and since I'm multiplying them, you can just add the exponents. So it's gonna be a to the negative four plus two power, which is equal to a to the negative two power. |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | Well, once again, you have the same base. In this case, it's a. And so, and since I'm multiplying them, you can just add the exponents. So it's gonna be a to the negative four plus two power, which is equal to a to the negative two power. And once again, it should make sense. This right over here, that is one over a times a times a times a. And then this is times a times a. |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | So it's gonna be a to the negative four plus two power, which is equal to a to the negative two power. And once again, it should make sense. This right over here, that is one over a times a times a times a. And then this is times a times a. So that cancels with that, that cancels with that, and you're still left with one over a times a, which is the same thing as a to the negative two power. Now let's do it with some quotients. So what if I were to ask you, what is 12 to the negative seven divided by 12 to the negative five power? |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | And then this is times a times a. So that cancels with that, that cancels with that, and you're still left with one over a times a, which is the same thing as a to the negative two power. Now let's do it with some quotients. So what if I were to ask you, what is 12 to the negative seven divided by 12 to the negative five power? Well, when you're dividing, you subtract exponents if you have the same base. So this is going to be equal to 12 to the negative seven minus negative five power. You're subtracting the bottom exponent. |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | So what if I were to ask you, what is 12 to the negative seven divided by 12 to the negative five power? Well, when you're dividing, you subtract exponents if you have the same base. So this is going to be equal to 12 to the negative seven minus negative five power. You're subtracting the bottom exponent. And so this is going to be equal to 12 to the, well, subtracting a negative is the same thing as adding the positive, a 12 to the negative two power. And once again, we just have to think about why does this actually make sense? Well, you can actually rewrite this. |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | You're subtracting the bottom exponent. And so this is going to be equal to 12 to the, well, subtracting a negative is the same thing as adding the positive, a 12 to the negative two power. And once again, we just have to think about why does this actually make sense? Well, you can actually rewrite this. 12 to the negative seven divided by 12 to the negative five, that's the same thing as 12 to the negative seven times 12 to the fifth power. If we take the reciprocal of, if we take the reciprocal of this right over here, you would make the exponent positive. And then you get exactly what we were doing in those previous examples with products. |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | Well, you can actually rewrite this. 12 to the negative seven divided by 12 to the negative five, that's the same thing as 12 to the negative seven times 12 to the fifth power. If we take the reciprocal of, if we take the reciprocal of this right over here, you would make the exponent positive. And then you get exactly what we were doing in those previous examples with products. And so let's just do one more with variables for good measure. Let's say I have x to the negative 20th power divided by x to the fifth power. Well, once again, we have the same base and we're taking a quotient. |
Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 | And then you get exactly what we were doing in those previous examples with products. And so let's just do one more with variables for good measure. Let's say I have x to the negative 20th power divided by x to the fifth power. Well, once again, we have the same base and we're taking a quotient. So this is going to be x to the negative 20 minus five because we have this one right over here in the denominator. So this is going to be equal to x to the negative 25th power. And once again, you could view our original expression as x to the negative 20th. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | We're told to plot these values on a number line. And you see every one of these values have an absolute value sign, so let's take a little bit of a review of what absolute value even is. Absolute value. The way I think about it, there's two ways to think about it. The first way to think about it is how far is something from 0? So let me plot this negative 3 here. So let me do a number line. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | The way I think about it, there's two ways to think about it. The first way to think about it is how far is something from 0? So let me plot this negative 3 here. So let me do a number line. This isn't the number line for our actual answer to this command, plot these values on a number line. I'm just first going to plot the numbers inside the absolute value sign, and then we're going to take the absolute value and plot those, just like they're asking us to do. So on this number line, if this is 0, if we go to the negative, we're going to go to the left of 0. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | So let me do a number line. This isn't the number line for our actual answer to this command, plot these values on a number line. I'm just first going to plot the numbers inside the absolute value sign, and then we're going to take the absolute value and plot those, just like they're asking us to do. So on this number line, if this is 0, if we go to the negative, we're going to go to the left of 0. So this is negative 1, negative 2, negative 3. Negative 3 sits right over there. So this is negative 3 right there. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | So on this number line, if this is 0, if we go to the negative, we're going to go to the left of 0. So this is negative 1, negative 2, negative 3. Negative 3 sits right over there. So this is negative 3 right there. The absolute value of negative 3 is essentially saying how far are you away from 0? How far is negative 3 from 0? And you say, well, it's 1, 2, 3 away from 0. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | So this is negative 3 right there. The absolute value of negative 3 is essentially saying how far are you away from 0? How far is negative 3 from 0? And you say, well, it's 1, 2, 3 away from 0. So you'd say that the absolute value of negative 3 is equal to positive 3. Now, that's really the conceptual way to imagine absolute value. How far are you away from 0? |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | And you say, well, it's 1, 2, 3 away from 0. So you'd say that the absolute value of negative 3 is equal to positive 3. Now, that's really the conceptual way to imagine absolute value. How far are you away from 0? But the easy way to calculate absolute value signs, if you don't care too much about the concept, is whether it's negative or positive, the absolute value of it's always going to be positive. Absolute value of negative 3 is positive 3. Absolute value of positive 3 is still positive 3. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | How far are you away from 0? But the easy way to calculate absolute value signs, if you don't care too much about the concept, is whether it's negative or positive, the absolute value of it's always going to be positive. Absolute value of negative 3 is positive 3. Absolute value of positive 3 is still positive 3. So you're always going to get the positive version of the number, so to speak. But conceptually, you're just saying how far away are you from 0? So let's do what they asked. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | Absolute value of positive 3 is still positive 3. So you're always going to get the positive version of the number, so to speak. But conceptually, you're just saying how far away are you from 0? So let's do what they asked. So that first value on this number line, so all of these are absolute values. So they're all going to be positive values. So they're all going to be greater than 0. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | So let's do what they asked. So that first value on this number line, so all of these are absolute values. So they're all going to be positive values. So they're all going to be greater than 0. So let me draw my number line like this. I could do a straighter number line than that. Let's see. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | So they're all going to be greater than 0. So let me draw my number line like this. I could do a straighter number line than that. Let's see. Well, that's a little bit straighter. And let's say if this is 0, this would be negative 1, then you'd have 1, 2, 3, 4, 5, 6, 7. I think that'll do the trick. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | Let's see. Well, that's a little bit straighter. And let's say if this is 0, this would be negative 1, then you'd have 1, 2, 3, 4, 5, 6, 7. I think that'll do the trick. So this first quantity here, I'll do it in orange, the absolute value of negative 3 we just figured out. That is positive 3. So I'll plot it right over there. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | I think that'll do the trick. So this first quantity here, I'll do it in orange, the absolute value of negative 3 we just figured out. That is positive 3. So I'll plot it right over there. Positive 3. Then this next value right here, the absolute value of 7. If we look over here, 1, 2, 3, 4, 5, 6, 7, 7 is how far away from 0? |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | So I'll plot it right over there. Positive 3. Then this next value right here, the absolute value of 7. If we look over here, 1, 2, 3, 4, 5, 6, 7, 7 is how far away from 0? It is 7 away from 0. So the absolute value of 7 is equal to 7. So you already see the pattern there. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | If we look over here, 1, 2, 3, 4, 5, 6, 7, 7 is how far away from 0? It is 7 away from 0. So the absolute value of 7 is equal to 7. So you already see the pattern there. If it's negative, it just becomes positive. If it's already positive, it just equals itself. So plotting this value, we'll just place it right over there. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | So you already see the pattern there. If it's negative, it just becomes positive. If it's already positive, it just equals itself. So plotting this value, we'll just place it right over there. So the absolute value of 7 is 7. Absolute value of negative 3 is positive 3. Let me mark out the 0 a little bit better so you see relative to 0. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | So plotting this value, we'll just place it right over there. So the absolute value of 7 is 7. Absolute value of negative 3 is positive 3. Let me mark out the 0 a little bit better so you see relative to 0. Now we have the absolute value of 8 minus 12. The absolute value of 8 minus 12. Well, first of all, let's figure out what 8 minus 12 is. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | Let me mark out the 0 a little bit better so you see relative to 0. Now we have the absolute value of 8 minus 12. The absolute value of 8 minus 12. Well, first of all, let's figure out what 8 minus 12 is. So if you take 12 away from 8, you're at negative 4. 12 less than 8 is negative 4. And you can do that on the number line if this is a little bit, if you don't quite remember how to do this. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | Well, first of all, let's figure out what 8 minus 12 is. So if you take 12 away from 8, you're at negative 4. 12 less than 8 is negative 4. And you can do that on the number line if this is a little bit, if you don't quite remember how to do this. But if you take 8 away from 8, you're at 0. And then you take another one, you're at negative 1, then at negative 2, negative 3, all the way to negative 4. So this is equal to the absolute value of negative 4. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | And you can do that on the number line if this is a little bit, if you don't quite remember how to do this. But if you take 8 away from 8, you're at 0. And then you take another one, you're at negative 1, then at negative 2, negative 3, all the way to negative 4. So this is equal to the absolute value of negative 4. If we just plot negative 4, we go 1, 2, 3, negative 4 is right over there. But if we're taking its absolute value, we're saying how far is negative 4 from 0? Well, it's 4 away from 0. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | So this is equal to the absolute value of negative 4. If we just plot negative 4, we go 1, 2, 3, negative 4 is right over there. But if we're taking its absolute value, we're saying how far is negative 4 from 0? Well, it's 4 away from 0. 1, 2, 3, 4. So this is equal to positive 4. So we'll plot it right here. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | Well, it's 4 away from 0. 1, 2, 3, 4. So this is equal to positive 4. So we'll plot it right here. This number line is the answer to this command up here. So the absolute value of 8 minus 12, which is negative 4, is positive 4. Then we have the absolute value of 0. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | So we'll plot it right here. This number line is the answer to this command up here. So the absolute value of 8 minus 12, which is negative 4, is positive 4. Then we have the absolute value of 0. So how far is 0 from 0? Well, it's 0 away from 0. The absolute value of 0 is 0. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | Then we have the absolute value of 0. So how far is 0 from 0? Well, it's 0 away from 0. The absolute value of 0 is 0. So you can just plot it right over there. And we have one left. Let me pick a suitable color here. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | The absolute value of 0 is 0. So you can just plot it right over there. And we have one left. Let me pick a suitable color here. The absolute value of 7 minus 2. The absolute value of 7 minus 2. Well, 7 minus 2 is 5. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | Let me pick a suitable color here. The absolute value of 7 minus 2. The absolute value of 7 minus 2. Well, 7 minus 2 is 5. So this is the same thing as the absolute value of 5. How far is 5 away from 0? Well, it's just 5 away. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | Well, 7 minus 2 is 5. So this is the same thing as the absolute value of 5. How far is 5 away from 0? Well, it's just 5 away. It's almost too easy. That's what makes it confusing. If I were to plot 5, it's 1, 2, 3, 4, 5. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | Well, it's just 5 away. It's almost too easy. That's what makes it confusing. If I were to plot 5, it's 1, 2, 3, 4, 5. It is 1, 2, 3, 4, 5 spaces from 0. So the absolute value of 5 is 5. So you plot it just like that. |
Absolute value and number lines Negative numbers and absolute value Pre-Algebra Khan Academy.mp3 | If I were to plot 5, it's 1, 2, 3, 4, 5. It is 1, 2, 3, 4, 5 spaces from 0. So the absolute value of 5 is 5. So you plot it just like that. So conceptually, it's how far you are away from 0. But if you think about it in kind of just very simple terms, if it's a negative number, it becomes a positive version of it. If it's a positive number already, it just equals itself when you take the absolute value. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | Well, we've already talked about when you divide by something, it's the exact same thing as multiplying by its reciprocal. So this is going to be the exact same thing as negative 5 6 times the reciprocal of 3 4ths, which is 4 over 3. I'm just swapping the numerator and the denominator. So it's going to be 4 over 3. And we've already seen lots of examples multiplying fractions. This is going to be the numerators times each other. So we're going to multiply negative 5 times 4. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | So it's going to be 4 over 3. And we've already seen lots of examples multiplying fractions. This is going to be the numerators times each other. So we're going to multiply negative 5 times 4. I'll give the negative sign to the 5 there. So negative 5 times 4. Let me do 4 in that yellow color. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | So we're going to multiply negative 5 times 4. I'll give the negative sign to the 5 there. So negative 5 times 4. Let me do 4 in that yellow color. And then the denominator is 6 times 3. 6 times 3. Now, in the numerator here, you see we have a negative number. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | Let me do 4 in that yellow color. And then the denominator is 6 times 3. 6 times 3. Now, in the numerator here, you see we have a negative number. You might already know that 5 times 4 is 20. And you just have to remember that, look, we're multiplying a negative times a positive. We're essentially going to have negative 5 four times. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | Now, in the numerator here, you see we have a negative number. You might already know that 5 times 4 is 20. And you just have to remember that, look, we're multiplying a negative times a positive. We're essentially going to have negative 5 four times. So negative 5 plus negative 5 plus negative 5 plus negative 5 is negative 20. So the numerator here is negative 20. And the denominator here is 18. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | We're essentially going to have negative 5 four times. So negative 5 plus negative 5 plus negative 5 plus negative 5 is negative 20. So the numerator here is negative 20. And the denominator here is 18. So we get 20 over 18. But we can simplify this. Both the numerator and the denominator, they're both divisible by 2. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | And the denominator here is 18. So we get 20 over 18. But we can simplify this. Both the numerator and the denominator, they're both divisible by 2. So let's divide them both by 2. Let me give myself a little more space. So if we divide both the numerator and the denominator by 2, just to simplify this. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | Both the numerator and the denominator, they're both divisible by 2. So let's divide them both by 2. Let me give myself a little more space. So if we divide both the numerator and the denominator by 2, just to simplify this. And I picked 2 because that's the largest number that goes into both of these. That's the greatest common divisor of 20 and 18. 20 divided by 2 is 10. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | So if we divide both the numerator and the denominator by 2, just to simplify this. And I picked 2 because that's the largest number that goes into both of these. That's the greatest common divisor of 20 and 18. 20 divided by 2 is 10. And 18 divided by 2 is 9. So negative 5 6 divided by 3 4ths is, oh, I have to be very careful here. It's negative 10 9ths. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | 20 divided by 2 is 10. And 18 divided by 2 is 9. So negative 5 6 divided by 3 4ths is, oh, I have to be very careful here. It's negative 10 9ths. Just how we always learned. If you have a negative divided by a positive, if the signs are different, then you're going to get a negative value. Let's do another example. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | It's negative 10 9ths. Just how we always learned. If you have a negative divided by a positive, if the signs are different, then you're going to get a negative value. Let's do another example. Let's say that I have negative 4 divided by negative 1 half. So using the exact logic that we just said, we said, hey, look, dividing by something is equivalent to multiplying by its reciprocal. So this is going to be equal to negative 4. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | Let's do another example. Let's say that I have negative 4 divided by negative 1 half. So using the exact logic that we just said, we said, hey, look, dividing by something is equivalent to multiplying by its reciprocal. So this is going to be equal to negative 4. Instead of writing it as negative 4, let me just write it as a fraction so that we are clear what its numerator is and what its denominator is. So negative 4 is the exact same thing as negative 4 over 1. And we're going to multiply that times the reciprocal of negative 1 half. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | So this is going to be equal to negative 4. Instead of writing it as negative 4, let me just write it as a fraction so that we are clear what its numerator is and what its denominator is. So negative 4 is the exact same thing as negative 4 over 1. And we're going to multiply that times the reciprocal of negative 1 half. The reciprocal of negative 1 half is negative 2 over 1. You could view it as negative 2 over 1, or you could use it as positive 2 over negative 1, or you could use it as negative 2. Either way, these are all the same value. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | And we're going to multiply that times the reciprocal of negative 1 half. The reciprocal of negative 1 half is negative 2 over 1. You could view it as negative 2 over 1, or you could use it as positive 2 over negative 1, or you could use it as negative 2. Either way, these are all the same value. And now we're ready to multiply. Notice, all I did here, I rewrote the negative 4 just as negative 4 over 1. Negative 4 divided by 1 is negative 4. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | Either way, these are all the same value. And now we're ready to multiply. Notice, all I did here, I rewrote the negative 4 just as negative 4 over 1. Negative 4 divided by 1 is negative 4. And here, for the negative 1 half, since I'm multiplying now, I'm multiplying by its reciprocal. I've swapped the denominator and the numerator. Or I swapped the denominator and the numerator. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | Negative 4 divided by 1 is negative 4. And here, for the negative 1 half, since I'm multiplying now, I'm multiplying by its reciprocal. I've swapped the denominator and the numerator. Or I swapped the denominator and the numerator. What was the denominator is now the numerator. What was the numerator is now the denominator. And I'm ready to multiply. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | Or I swapped the denominator and the numerator. What was the denominator is now the numerator. What was the numerator is now the denominator. And I'm ready to multiply. This is going to be equal to, I gave both the negative signs to the numerator, so it's going to be negative 4 times negative 2 in the numerator. And then in the denominator, it's going to be 1 times 1. Let me write that down. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | And I'm ready to multiply. This is going to be equal to, I gave both the negative signs to the numerator, so it's going to be negative 4 times negative 2 in the numerator. And then in the denominator, it's going to be 1 times 1. Let me write that down. 1 times 1. And so this gives us, so we have a negative 4 times a negative 2. So it's a negative times a negative. |
Dividing negative fractions Fractions Pre-Algebra Khan Academy.mp3 | Let me write that down. 1 times 1. And so this gives us, so we have a negative 4 times a negative 2. So it's a negative times a negative. So we're going to get a positive value here. And 4 times 2 is 8. So this is a positive 8 over 1. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | And so he storms out of the room, and then a few seconds later, he storms back in. He says, my fault, my apologies. I realize now, I realize now what the mistake was. There was a slight, I guess, typing error or writing error. The first week when they went to the market and bought two pounds of apples and one pounds of bananas, it wasn't a $3 cost. It was a $5 cost. It was a $5 cost. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | There was a slight, I guess, typing error or writing error. The first week when they went to the market and bought two pounds of apples and one pounds of bananas, it wasn't a $3 cost. It was a $5 cost. It was a $5 cost. Now surely, considering how smart you and this bird seem to be, you surely could figure out what is the per pound cost of apples and what is the per pound cost of bananas. So you think for a little bit. Is there now going to be a solution? |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | It was a $5 cost. Now surely, considering how smart you and this bird seem to be, you surely could figure out what is the per pound cost of apples and what is the per pound cost of bananas. So you think for a little bit. Is there now going to be a solution? So let's break it down using the exact same variables. You say, well, if a is the cost of apples per pound and b is the cost of bananas, this first constraint tells us that two pounds of apples, so two pounds of apples are going to cost 2a, because it's $a per pound. And one pound of bananas is going to cost b dollars, because it's one pound times b dollars per pound, is now going to cost $5. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | Is there now going to be a solution? So let's break it down using the exact same variables. You say, well, if a is the cost of apples per pound and b is the cost of bananas, this first constraint tells us that two pounds of apples, so two pounds of apples are going to cost 2a, because it's $a per pound. And one pound of bananas is going to cost b dollars, because it's one pound times b dollars per pound, is now going to cost $5. This is the corrected. This is the corrected number. And we saw from the last scenario, this information hasn't changed. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | And one pound of bananas is going to cost b dollars, because it's one pound times b dollars per pound, is now going to cost $5. This is the corrected. This is the corrected number. And we saw from the last scenario, this information hasn't changed. Six pounds of apples is going to cost 6a, six pounds times $a per pound. And three pounds of bananas is going to cost 3b, three pounds times b dollars per pound. The total cost of the apples and bananas in this trip we are given is $15. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | And we saw from the last scenario, this information hasn't changed. Six pounds of apples is going to cost 6a, six pounds times $a per pound. And three pounds of bananas is going to cost 3b, three pounds times b dollars per pound. The total cost of the apples and bananas in this trip we are given is $15. So once again, you say, well, let me try to solve this maybe through elimination. And once again, you say, well, let me cancel out the a's. I have 2a here. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | The total cost of the apples and bananas in this trip we are given is $15. So once again, you say, well, let me try to solve this maybe through elimination. And once again, you say, well, let me cancel out the a's. I have 2a here. I have 6a here. If I multiply the 2a here by negative 3, then this will become a negative 6a, and it might be able to cancel out with all of this business. So you do that. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | I have 2a here. I have 6a here. If I multiply the 2a here by negative 3, then this will become a negative 6a, and it might be able to cancel out with all of this business. So you do that. You multiply this entire equation. You can't just multiply one term. You have to multiply the entire equation times negative 3 if you want the equation to still hold. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | So you do that. You multiply this entire equation. You can't just multiply one term. You have to multiply the entire equation times negative 3 if you want the equation to still hold. And so we're multiplying by negative 3. So 2a times negative 3 is negative 6a. b times negative 3 is negative 3b. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | You have to multiply the entire equation times negative 3 if you want the equation to still hold. And so we're multiplying by negative 3. So 2a times negative 3 is negative 6a. b times negative 3 is negative 3b. And then 5 times negative 3 is negative 15. And now something fishy starts to look like it's about to happen. Because when you add the left-hand side of this blue equation or this purplish equation to the green one, you get 0. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | b times negative 3 is negative 3b. And then 5 times negative 3 is negative 15. And now something fishy starts to look like it's about to happen. Because when you add the left-hand side of this blue equation or this purplish equation to the green one, you get 0. All of these things right over here just cancel out. And on the right-hand side, 15 minus 15, that is also equal to 0. And you get 0 equals 0, which seems a little bit better than the last time you worked through it. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | Because when you add the left-hand side of this blue equation or this purplish equation to the green one, you get 0. All of these things right over here just cancel out. And on the right-hand side, 15 minus 15, that is also equal to 0. And you get 0 equals 0, which seems a little bit better than the last time you worked through it. Last time we got 0 equals 6. But 0 equals 0 doesn't really tell you anything about the x's and y's. This is true. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | And you get 0 equals 0, which seems a little bit better than the last time you worked through it. Last time we got 0 equals 6. But 0 equals 0 doesn't really tell you anything about the x's and y's. This is true. This is absolutely true that 0 does definitely equal 0. But it doesn't tell you any information about x and y. And so then the bird whispers in the king's ear. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | This is true. This is absolutely true that 0 does definitely equal 0. But it doesn't tell you any information about x and y. And so then the bird whispers in the king's ear. And then the king says, well, the bird says you should graph it to figure out what's actually going on. And so you've learned that listening to the bird actually makes a lot of sense. So you try to graph these two constraints. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | And so then the bird whispers in the king's ear. And then the king says, well, the bird says you should graph it to figure out what's actually going on. And so you've learned that listening to the bird actually makes a lot of sense. So you try to graph these two constraints. So let's do it the same way. We'll have a b-axis. That's our b-axis. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | So you try to graph these two constraints. So let's do it the same way. We'll have a b-axis. That's our b-axis. And we will have our a-axis. Let me mark off some markers here. 1, 2, 3, 4, 5, and 1, 2, 3, 4, 5. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | That's our b-axis. And we will have our a-axis. Let me mark off some markers here. 1, 2, 3, 4, 5, and 1, 2, 3, 4, 5. So this first equation right over here, if we subtract 2a from both sides, I'm just going to put it into slope-intercept form, you get b is equal to negative 2a plus 5. All I did is subtract 2a from both sides. And if we were to graph that, our b-intercept, when a is equal to 0, b is equal to 5. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | 1, 2, 3, 4, 5, and 1, 2, 3, 4, 5. So this first equation right over here, if we subtract 2a from both sides, I'm just going to put it into slope-intercept form, you get b is equal to negative 2a plus 5. All I did is subtract 2a from both sides. And if we were to graph that, our b-intercept, when a is equal to 0, b is equal to 5. So that's right over here. And our slope is negative 2. Every time you add 1 to a, so if a goes from 0 to 1, b is going to go down by 2. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | And if we were to graph that, our b-intercept, when a is equal to 0, b is equal to 5. So that's right over here. And our slope is negative 2. Every time you add 1 to a, so if a goes from 0 to 1, b is going to go down by 2. So go down by 2, go down by 2. So this first white equation looks like this if we graph the solution set. These are all of the prices for bananas and apples that meet this constraint. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | Every time you add 1 to a, so if a goes from 0 to 1, b is going to go down by 2. So go down by 2, go down by 2. So this first white equation looks like this if we graph the solution set. These are all of the prices for bananas and apples that meet this constraint. Now let's graph the second equation. If we subtract 6a from both sides, we get 3b is equal to negative 6a plus 15. And now we can divide both sides by 3. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | These are all of the prices for bananas and apples that meet this constraint. Now let's graph the second equation. If we subtract 6a from both sides, we get 3b is equal to negative 6a plus 15. And now we can divide both sides by 3. Divide everything by 3. We are left with b is equal to negative 2a plus 5. Well, this is interesting. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | And now we can divide both sides by 3. Divide everything by 3. We are left with b is equal to negative 2a plus 5. Well, this is interesting. This looks very similar, or it looks exactly the same. Our b-intercept is 5, and our slope is negative 2a. So this is essentially the same line. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | Well, this is interesting. This looks very similar, or it looks exactly the same. Our b-intercept is 5, and our slope is negative 2a. So this is essentially the same line. So these are essentially the same constraints. And so you start to look at it a little bit confused. And you say, OK, I see why we got 0 equals 0. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | So this is essentially the same line. So these are essentially the same constraints. And so you start to look at it a little bit confused. And you say, OK, I see why we got 0 equals 0. There's actually an infinite number of solutions. You pick any x, and then the corresponding y for each of these could be a solution for either of these things. So there's an infinite number of solutions. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | And you say, OK, I see why we got 0 equals 0. There's actually an infinite number of solutions. You pick any x, and then the corresponding y for each of these could be a solution for either of these things. So there's an infinite number of solutions. But you start to wonder, why is this happening? And so the bird whispers again into the king's ear. And the king says, well, the bird says this is because in both trips to the market, the same ratio of apples and bananas was bought. |