video_title stringlengths 25 104 | Sentence stringlengths 91 1.69k |
|---|---|
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | So let's just make this over here our starting point and make that our ending point. So what is our change in y? So our change in y, we started at y is equal to 6. We started at y is equal to 6. And we go down all the way to y is equal to negative 4. So this is right here. That is our change in y. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | We started at y is equal to 6. And we go down all the way to y is equal to negative 4. So this is right here. That is our change in y. You could look at the graph. You say, well, if I start at 6 and I go to negative 4, I went down 10. Or if you just want to use this formula here, it'll give you the same thing. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | That is our change in y. You could look at the graph. You say, well, if I start at 6 and I go to negative 4, I went down 10. Or if you just want to use this formula here, it'll give you the same thing. We finished at negative 4. We finished at negative 4. And from that, we want to subtract 6. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | Or if you just want to use this formula here, it'll give you the same thing. We finished at negative 4. We finished at negative 4. And from that, we want to subtract 6. This right here is y2. This is our ending y. And this is our beginning y. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | And from that, we want to subtract 6. This right here is y2. This is our ending y. And this is our beginning y. This is y1. So y2, negative 4, minus y1. Negative 6. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | And this is our beginning y. This is y1. So y2, negative 4, minus y1. Negative 6. So negative 4 minus 6. That is equal to negative 10. And all that's doing is telling us the change in y. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | Negative 6. So negative 4 minus 6. That is equal to negative 10. And all that's doing is telling us the change in y. To go from this point to that point, we had to go down. Our rise was negative. We had to go down 10. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | And all that's doing is telling us the change in y. To go from this point to that point, we had to go down. Our rise was negative. We had to go down 10. That's where the negative 10 comes from. Now we just have to find our change in x. So we can look at this graph over here. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | We had to go down 10. That's where the negative 10 comes from. Now we just have to find our change in x. So we can look at this graph over here. We started at x is equal to negative 1. And we go all the way to x is equal to 5. So we start at x is equal to negative 1. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | So we can look at this graph over here. We started at x is equal to negative 1. And we go all the way to x is equal to 5. So we start at x is equal to negative 1. And we go all the way to x is equal to 5. So it takes us 1 to get to 0, and then 5 more. So our change in x is 6. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | So we start at x is equal to negative 1. And we go all the way to x is equal to 5. So it takes us 1 to get to 0, and then 5 more. So our change in x is 6. You can look at it visually there. Or you could use this formula. Same exact idea. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | So our change in x is 6. You can look at it visually there. Or you could use this formula. Same exact idea. Our ending x value is 5. And our starting x value is negative 1. 5 minus negative 1. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | Same exact idea. Our ending x value is 5. And our starting x value is negative 1. 5 minus negative 1. 5 minus negative 1 is the same thing as 5 plus 1. So it is 6. So our slope here is negative 10 over 6, which is the exact same thing as negative 5 thirds. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | 5 minus negative 1. 5 minus negative 1 is the same thing as 5 plus 1. So it is 6. So our slope here is negative 10 over 6, which is the exact same thing as negative 5 thirds. As negative 5 over 3. Divide the numerator and denominator by 2. So we now know our equation will be y is equal to negative 5 thirds, that's our slope, x plus b. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | So our slope here is negative 10 over 6, which is the exact same thing as negative 5 thirds. As negative 5 over 3. Divide the numerator and denominator by 2. So we now know our equation will be y is equal to negative 5 thirds, that's our slope, x plus b. So we still need to solve for our y-intercept to get our equation. And to do that, we can use the information that we know. Or we have several points of information. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | So we now know our equation will be y is equal to negative 5 thirds, that's our slope, x plus b. So we still need to solve for our y-intercept to get our equation. And to do that, we can use the information that we know. Or we have several points of information. But we can use the fact that the line goes through the point negative 1, 6. We could use the other point as well. But we know that when x is equal to negative 1, so y is equal to 6. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | Or we have several points of information. But we can use the fact that the line goes through the point negative 1, 6. We could use the other point as well. But we know that when x is equal to negative 1, so y is equal to 6. So y is equal to 6 when x is equal to negative 1. So negative 5 thirds times x. When x is equal to negative 1, y is equal to 6. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | But we know that when x is equal to negative 1, so y is equal to 6. So y is equal to 6 when x is equal to negative 1. So negative 5 thirds times x. When x is equal to negative 1, y is equal to 6. So we literally just substitute this x and y value back into this. And now we can solve for b. So let's see. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | When x is equal to negative 1, y is equal to 6. So we literally just substitute this x and y value back into this. And now we can solve for b. So let's see. This negative 1 times negative 5 thirds. So we get 6 is equal to positive 5 thirds plus b. And now we can subtract 5 thirds from both sides of this equation. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | So let's see. This negative 1 times negative 5 thirds. So we get 6 is equal to positive 5 thirds plus b. And now we can subtract 5 thirds from both sides of this equation. So we have subtract the left-hand side from the left-hand side. And subtract from the right-hand side. And then we get, what's 6 minus 5 thirds? |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | And now we can subtract 5 thirds from both sides of this equation. So we have subtract the left-hand side from the left-hand side. And subtract from the right-hand side. And then we get, what's 6 minus 5 thirds? So that's going to be, let me do it over here. We could take a common denominator. So 6 is the same thing as, let me just do it over here. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | And then we get, what's 6 minus 5 thirds? So that's going to be, let me do it over here. We could take a common denominator. So 6 is the same thing as, let me just do it over here. So 6 minus 5 over 3 is the same thing as 6 is 18 over 3 minus 5 over 3. That's 6 is 18 over 3. And this is just 13 over 3. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | So 6 is the same thing as, let me just do it over here. So 6 minus 5 over 3 is the same thing as 6 is 18 over 3 minus 5 over 3. That's 6 is 18 over 3. And this is just 13 over 3. So this is 13 over 3. And then, of course, these cancel out. So we get b is equal to 13 thirds. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | And this is just 13 over 3. So this is 13 over 3. And then, of course, these cancel out. So we get b is equal to 13 thirds. So we're done. We know the slope and we know the y-intercept. The equation of our line is y is equal to negative 5 thirds x plus our y-intercept, which is 13 over 3. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | So we get b is equal to 13 thirds. So we're done. We know the slope and we know the y-intercept. The equation of our line is y is equal to negative 5 thirds x plus our y-intercept, which is 13 over 3. And we could write these as mixed numbers if it's easier to visualize. 13 over 3 is 4 and 1 third. So this y-intercept right over here, that's 0 comma 13 over 3 or 0 comma 4 and 1 third. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | The equation of our line is y is equal to negative 5 thirds x plus our y-intercept, which is 13 over 3. And we could write these as mixed numbers if it's easier to visualize. 13 over 3 is 4 and 1 third. So this y-intercept right over here, that's 0 comma 13 over 3 or 0 comma 4 and 1 third. And even with my very roughly drawn diagram, it does look like this. And this slope, negative 5 thirds, that's the same thing as negative 1 and 2 thirds. And you can see here, the slope is downward sloping. |
Slope-intercept equation from two solutions example Algebra I Khan Academy.mp3 | So this y-intercept right over here, that's 0 comma 13 over 3 or 0 comma 4 and 1 third. And even with my very roughly drawn diagram, it does look like this. And this slope, negative 5 thirds, that's the same thing as negative 1 and 2 thirds. And you can see here, the slope is downward sloping. It's negative. And it's a little bit steeper than a slope of 1. It's not quite a negative 2. |
Subtracting decimals example 1 Decimals Pre-Algebra Khan Academy.mp3 | Let's try to subtract 9.57 minus 8.09. So try to pause this video and figure this out first before we work through it together. All right, well let's just rewrite it. Let's rewrite it. And when I rewrite it, I like to line up the decimals. This one is a little intuitive. We have 8.09, just like that. |
Subtracting decimals example 1 Decimals Pre-Algebra Khan Academy.mp3 | Let's rewrite it. And when I rewrite it, I like to line up the decimals. This one is a little intuitive. We have 8.09, just like that. And now we're ready to subtract. And we want to subtract 9 hundredths from 7 hundredths. Well, we don't have enough hundredths up here, so let's move over here. |
Subtracting decimals example 1 Decimals Pre-Algebra Khan Academy.mp3 | We have 8.09, just like that. And now we're ready to subtract. And we want to subtract 9 hundredths from 7 hundredths. Well, we don't have enough hundredths up here, so let's move over here. Let's see if we can do some regrouping so we always have a higher number on top. So over here, we want to subtract 0 tenths from 5 tenths. We have enough tenths over here, so let's regroup. |
Subtracting decimals example 1 Decimals Pre-Algebra Khan Academy.mp3 | Well, we don't have enough hundredths up here, so let's move over here. Let's see if we can do some regrouping so we always have a higher number on top. So over here, we want to subtract 0 tenths from 5 tenths. We have enough tenths over here, so let's regroup. So instead of 5 tenths, I'm going to have 4 tenths. And then I'm going to give that other tenth, which is the same thing as 10 hundredths over here. So this becomes 17 hundredths. |
Subtracting decimals example 1 Decimals Pre-Algebra Khan Academy.mp3 | We have enough tenths over here, so let's regroup. So instead of 5 tenths, I'm going to have 4 tenths. And then I'm going to give that other tenth, which is the same thing as 10 hundredths over here. So this becomes 17 hundredths. 17 minus 9 is 8. 4 minus 0 is 4. And then I have 9 minus 8 is 1. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | And try pausing the video and solving it on your own before I work through it. So there's a couple of ways you could think about it. We could just write it as 30.24 divided by 0.42. But what do you do now? Well, the important realization is when you're doing a division problem like this, you will get the same answer as long as you multiply or divide both numbers by the same thing. And to understand that, we could rewrite this division as 30.42 over 0.42. We could write it really as a fraction. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | But what do you do now? Well, the important realization is when you're doing a division problem like this, you will get the same answer as long as you multiply or divide both numbers by the same thing. And to understand that, we could rewrite this division as 30.42 over 0.42. We could write it really as a fraction. And we know that when we have a fraction like this, we're not changing the value of the fraction. If we multiply the numerator and the denominator by the same quantity. And so what could we multiply this denominator by to make it a whole number? |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | We could write it really as a fraction. And we know that when we have a fraction like this, we're not changing the value of the fraction. If we multiply the numerator and the denominator by the same quantity. And so what could we multiply this denominator by to make it a whole number? Well, we could multiply it by 10 and then another 10. So we could multiply it by 100. So let's do that. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | And so what could we multiply this denominator by to make it a whole number? Well, we could multiply it by 10 and then another 10. So we could multiply it by 100. So let's do that. If we multiply the denominator by 100, in order to not change the value of this, we also need to multiply the numerator by 100. We're essentially multiplying by 100 over 100, which is just 1. So we're not changing the value of this fraction or of this division problem. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | So let's do that. If we multiply the denominator by 100, in order to not change the value of this, we also need to multiply the numerator by 100. We're essentially multiplying by 100 over 100, which is just 1. So we're not changing the value of this fraction or of this division problem. So this is going to be 30.42 times 100. Move the decimal two places to the right. Gets you 3,042. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | So we're not changing the value of this fraction or of this division problem. So this is going to be 30.42 times 100. Move the decimal two places to the right. Gets you 3,042. The decimal is now there if you care about it. And 0.42 times 100. Once again, move the decimal one, two places to the right. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | Gets you 3,042. The decimal is now there if you care about it. And 0.42 times 100. Once again, move the decimal one, two places to the right. It is now 42. So this is going to be the exact same thing as 3,042 divided by 42. So once again, we can move the decimal here two to the right. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | Once again, move the decimal one, two places to the right. It is now 42. So this is going to be the exact same thing as 3,042 divided by 42. So once again, we can move the decimal here two to the right. And if we move that two to the right, then we can move this two to the right. Or we need to move this two to the right. And so this is where now the decimal place is. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | So once again, we can move the decimal here two to the right. And if we move that two to the right, then we can move this two to the right. Or we need to move this two to the right. And so this is where now the decimal place is. You could view this as 3,024. Let me clear that. 3,024 divided by 42. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | And so this is where now the decimal place is. You could view this as 3,024. Let me clear that. 3,024 divided by 42. Let me clear that. And we already know how to tackle that already, so let's do it step by step. How many times does 42 go into 3? |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | 3,024 divided by 42. Let me clear that. And we already know how to tackle that already, so let's do it step by step. How many times does 42 go into 3? Well, it doesn't go at all, so we can move on to 30. How many times does 42 go into 30? Well, it doesn't go into 30, so we can move on to 302. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | How many times does 42 go into 3? Well, it doesn't go at all, so we can move on to 30. How many times does 42 go into 30? Well, it doesn't go into 30, so we can move on to 302. How many times does 42 go into 302? And like always, this is a bit of an art when you're dividing by a two-digit or a multi-digit number, I should say. So let's think about it a little bit. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | Well, it doesn't go into 30, so we can move on to 302. How many times does 42 go into 302? And like always, this is a bit of an art when you're dividing by a two-digit or a multi-digit number, I should say. So let's think about it a little bit. So this is roughly 40. This is roughly 300. So how many times does 40 go into 300? |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | So let's think about it a little bit. So this is roughly 40. This is roughly 300. So how many times does 40 go into 300? Well, how many times does 4 go into 30? It looks like it's about 7 times, so I'm going to try out a 7, see if it works out. 7 times 2 is 14. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | So how many times does 40 go into 300? Well, how many times does 4 go into 30? It looks like it's about 7 times, so I'm going to try out a 7, see if it works out. 7 times 2 is 14. 7 times 4 is 28, plus 1 is 29. And now I can subtract to do a little bit of regrouping here. So let's see. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | 7 times 2 is 14. 7 times 4 is 28, plus 1 is 29. And now I can subtract to do a little bit of regrouping here. So let's see. If I regroup, I take 100 from the 300. That becomes a 200. Then our 0 tens, now I have 10 tens. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | So let's see. If I regroup, I take 100 from the 300. That becomes a 200. Then our 0 tens, now I have 10 tens. But I'm going to need one of those 10 tens, so that's going to be 9 tens. And I'm going to give it over here. So this is going to be a 12. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | Then our 0 tens, now I have 10 tens. But I'm going to need one of those 10 tens, so that's going to be 9 tens. And I'm going to give it over here. So this is going to be a 12. 12 minus 4 is 8. 9 minus 9 is 0. 2 minus 2 is 0. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | So this is going to be a 12. 12 minus 4 is 8. 9 minus 9 is 0. 2 minus 2 is 0. So what I got left over is less than 42, so I know that 7 is the right number. I want to go as many times as possible into 302 without going over. So now let's bring down the next digit. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | 2 minus 2 is 0. So what I got left over is less than 42, so I know that 7 is the right number. I want to go as many times as possible into 302 without going over. So now let's bring down the next digit. Let's bring down this 4 over here. How many times does 42 go into 84? Well, that jumps out at you. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | So now let's bring down the next digit. Let's bring down this 4 over here. How many times does 42 go into 84? Well, that jumps out at you. Or hopefully it jumps. It starts to. It goes 2 times. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | Well, that jumps out at you. Or hopefully it jumps. It starts to. It goes 2 times. 2 times 2 is 4. 2 times 4 is 8. You subtract, and we have no remainder. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | It goes 2 times. 2 times 2 is 4. 2 times 4 is 8. You subtract, and we have no remainder. So 3,042 divided by 42 is the same thing as 30.42 divided by 0.42. And it's going to be equal to 72. It's going to be equal to 72. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | You subtract, and we have no remainder. So 3,042 divided by 42 is the same thing as 30.42 divided by 0.42. And it's going to be equal to 72. It's going to be equal to 72. Actually, I didn't have to copy and paste that. I'll just write this. This is equal to 72. |
Dividing decimals with hundredths Arithmetic operations 5th grade Khan Academy.mp3 | It's going to be equal to 72. Actually, I didn't have to copy and paste that. I'll just write this. This is equal to 72. This is equal to 72. 72. Just like that. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | So for example, the equation y is equal to two x minus three. This is a linear equation. Now why do we call it a linear equation? Well if you were to take the set of all of the xy pairs that satisfy this equation, and if you were to graph them on the coordinate plane, you would actually get a line. That's why it's called a linear equation. Let's actually feel good about that statement. Let's see if, let's plot some of the xy pairs that satisfy this equation, and then feel good that it does indeed generate a line. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Well if you were to take the set of all of the xy pairs that satisfy this equation, and if you were to graph them on the coordinate plane, you would actually get a line. That's why it's called a linear equation. Let's actually feel good about that statement. Let's see if, let's plot some of the xy pairs that satisfy this equation, and then feel good that it does indeed generate a line. So I'm just gonna pick some x values that make it easy to calculate the corresponding y values. So if x is equal to zero, y is gonna be two times zero minus three, which is negative three. And on our coordinate plane here, that's, we're gonna move zero in the x direction, zero in the horizontal direction, and we're gonna go down three in the vertical direction, in the y direction. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Let's see if, let's plot some of the xy pairs that satisfy this equation, and then feel good that it does indeed generate a line. So I'm just gonna pick some x values that make it easy to calculate the corresponding y values. So if x is equal to zero, y is gonna be two times zero minus three, which is negative three. And on our coordinate plane here, that's, we're gonna move zero in the x direction, zero in the horizontal direction, and we're gonna go down three in the vertical direction, in the y direction. So that's that point there. If x is equal to one, what is y equal to? Well two times one is two minus three is negative one. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | And on our coordinate plane here, that's, we're gonna move zero in the x direction, zero in the horizontal direction, and we're gonna go down three in the vertical direction, in the y direction. So that's that point there. If x is equal to one, what is y equal to? Well two times one is two minus three is negative one. So we move positive one in the x direction, and negative one, or down one, in the y direction. Now let's see, if x is equal to two, what is y? Two times two is four, minus three is one. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Well two times one is two minus three is negative one. So we move positive one in the x direction, and negative one, or down one, in the y direction. Now let's see, if x is equal to two, what is y? Two times two is four, minus three is one. When x is equal to two, y is equal to one. And hopefully you're seeing now, that if I were to keep going, and I encourage you to, if you want, pause the video, try x equals three, or x equals negative one, and keep going, you will see that this is going to generate a line. And in fact, let me connect these dots, and you will see, you will see the line that I'm talking about. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Two times two is four, minus three is one. When x is equal to two, y is equal to one. And hopefully you're seeing now, that if I were to keep going, and I encourage you to, if you want, pause the video, try x equals three, or x equals negative one, and keep going, you will see that this is going to generate a line. And in fact, let me connect these dots, and you will see, you will see the line that I'm talking about. So, let me see if I can draw, I'm gonna use the line tool here. Try to connect the dots as neatly as I can. There you go. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | And in fact, let me connect these dots, and you will see, you will see the line that I'm talking about. So, let me see if I can draw, I'm gonna use the line tool here. Try to connect the dots as neatly as I can. There you go. This line that I have just drawn, this is the graph, this is the graph of y is equal to two x minus three. So if you were to graph all of the xy pairs that satisfy this equation, you are going to get this line. And you might be saying, hey wait, wait, hold on Sal, you just tried some particular points, why don't I just get a bunch of points, how do I actually get a line? |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | There you go. This line that I have just drawn, this is the graph, this is the graph of y is equal to two x minus three. So if you were to graph all of the xy pairs that satisfy this equation, you are going to get this line. And you might be saying, hey wait, wait, hold on Sal, you just tried some particular points, why don't I just get a bunch of points, how do I actually get a line? Well I just tried, over here, I just tried integer values of x, but you could try any value in between here, all of these, it's actually a pretty neat concept. Any value of x that you input into this, you find the corresponding value for y, it will sit on this line. So for example, for example, if we were to take x is equal to, actually let's say x is equal to negative.5. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | And you might be saying, hey wait, wait, hold on Sal, you just tried some particular points, why don't I just get a bunch of points, how do I actually get a line? Well I just tried, over here, I just tried integer values of x, but you could try any value in between here, all of these, it's actually a pretty neat concept. Any value of x that you input into this, you find the corresponding value for y, it will sit on this line. So for example, for example, if we were to take x is equal to, actually let's say x is equal to negative.5. So if x is equal to negative.5, if we look at the line, when x is equal to negative.5, it looks like, it looks like y is equal to negative four. That looks like that sits on the line. Well let's verify that. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | So for example, for example, if we were to take x is equal to, actually let's say x is equal to negative.5. So if x is equal to negative.5, if we look at the line, when x is equal to negative.5, it looks like, it looks like y is equal to negative four. That looks like that sits on the line. Well let's verify that. If x is equal to negative, I'll write that as negative 1 1 2, then what is y equal to? Let's see, two times negative 1 1 2, I'll try it out, two times negative, two times negative 1 1 2 minus three. Well this is two times negative 1 1 2 is negative one minus three is indeed negative four. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Well let's verify that. If x is equal to negative, I'll write that as negative 1 1 2, then what is y equal to? Let's see, two times negative 1 1 2, I'll try it out, two times negative, two times negative 1 1 2 minus three. Well this is two times negative 1 1 2 is negative one minus three is indeed negative four. It is indeed negative four. So you can literally take any, for any x value that you put here and the corresponding y value, it is going to sit on the line. This point right over here represents a solution to this linear equation. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Well this is two times negative 1 1 2 is negative one minus three is indeed negative four. It is indeed negative four. So you can literally take any, for any x value that you put here and the corresponding y value, it is going to sit on the line. This point right over here represents a solution to this linear equation. Let me do this in a color you can see. So this point represents a solution to a linear equation. This point represents a solution to a linear equation. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | This point right over here represents a solution to this linear equation. Let me do this in a color you can see. So this point represents a solution to a linear equation. This point represents a solution to a linear equation. This point is not a solution to a linear equation. So if x is equal to five, then y is not going to be equal to three. If x is going to be equal to five, you go to the line to see what the solution to the linear equation is. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | This point represents a solution to a linear equation. This point is not a solution to a linear equation. So if x is equal to five, then y is not going to be equal to three. If x is going to be equal to five, you go to the line to see what the solution to the linear equation is. If x is five, this shows us that y is going to be seven. And it's indeed, that's indeed the case. Two times five is 10 minus three is seven. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | If x is going to be equal to five, you go to the line to see what the solution to the linear equation is. If x is five, this shows us that y is going to be seven. And it's indeed, that's indeed the case. Two times five is 10 minus three is seven. The point, the point five comma seven is on, or it satisfies this linear equation. So if you take all of the xy pairs that satisfy it, you get a line. That is why it's called a linear equation. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Two times five is 10 minus three is seven. The point, the point five comma seven is on, or it satisfies this linear equation. So if you take all of the xy pairs that satisfy it, you get a line. That is why it's called a linear equation. Now this isn't the only way that we could write a linear equation. You could write a linear equation like, let me do this in a new color. You could write a linear equation like this. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | That is why it's called a linear equation. Now this isn't the only way that we could write a linear equation. You could write a linear equation like, let me do this in a new color. You could write a linear equation like this. Four x minus three y is equal to 12. This also is a linear equation. And we can see that if we were to graph the xy pairs that satisfy this, we would once again get a line, x and y. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | You could write a linear equation like this. Four x minus three y is equal to 12. This also is a linear equation. And we can see that if we were to graph the xy pairs that satisfy this, we would once again get a line, x and y. If x is equal to zero, then this goes away. You have negative three y is equal to 12. See if negative three y equals 12, then y would be equal to negative four. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | And we can see that if we were to graph the xy pairs that satisfy this, we would once again get a line, x and y. If x is equal to zero, then this goes away. You have negative three y is equal to 12. See if negative three y equals 12, then y would be equal to negative four. Negative zero comma negative four. You can verify that. Four times zero minus three times negative four, well that's going to be equal to positive 12. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | See if negative three y equals 12, then y would be equal to negative four. Negative zero comma negative four. You can verify that. Four times zero minus three times negative four, well that's going to be equal to positive 12. And let's see, if y were to equal zero, if y were to equal zero, then this is going to be four times x is equal to 12. Well then x is equal to three. And so you have the point zero comma negative four. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Four times zero minus three times negative four, well that's going to be equal to positive 12. And let's see, if y were to equal zero, if y were to equal zero, then this is going to be four times x is equal to 12. Well then x is equal to three. And so you have the point zero comma negative four. Zero comma negative four on this line. And you have the point three comma zero on this line. Three comma zero. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | And so you have the point zero comma negative four. Zero comma negative four on this line. And you have the point three comma zero on this line. Three comma zero. Did I do that right? Yep. So zero comma negative four, and then three comma zero. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Three comma zero. Did I do that right? Yep. So zero comma negative four, and then three comma zero. These are both going to be on this line. Three comma zero. Is also on this line. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | So zero comma negative four, and then three comma zero. These are both going to be on this line. Three comma zero. Is also on this line. So this line is going to look something like, I'll just try to hand draw it, something like that. So once again, all of the xy pairs that satisfy this, if you were to plot them out, it forms a line. Now what are some examples of, maybe you're saying, well isn't any equation a linear equation? |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Is also on this line. So this line is going to look something like, I'll just try to hand draw it, something like that. So once again, all of the xy pairs that satisfy this, if you were to plot them out, it forms a line. Now what are some examples of, maybe you're saying, well isn't any equation a linear equation? And the simple answer is no. Not any equation is a linear equation. I'll give you some examples of nonlinear equations. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Now what are some examples of, maybe you're saying, well isn't any equation a linear equation? And the simple answer is no. Not any equation is a linear equation. I'll give you some examples of nonlinear equations. So nonlinear, whoops, let me write a little bit neater than that. Nonlinear equations. Well, those could include something like y is equal to x squared. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | I'll give you some examples of nonlinear equations. So nonlinear, whoops, let me write a little bit neater than that. Nonlinear equations. Well, those could include something like y is equal to x squared. If you graph this, you'll see that this is going to be a curve. It could be something like x times y is equal to 12. This is also not going to be a line. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Well, those could include something like y is equal to x squared. If you graph this, you'll see that this is going to be a curve. It could be something like x times y is equal to 12. This is also not going to be a line. Or it could be something like five over x plus y is equal to 10. This also is not going to be a line. So now, and at some point, I encourage you to try to graph these things. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | This is also not going to be a line. Or it could be something like five over x plus y is equal to 10. This also is not going to be a line. So now, and at some point, I encourage you to try to graph these things. These are actually quite interesting. But given that we've now seen examples of linear equations and nonlinear equations, let's see if we can come up with a definition for linear equations. One way to think about it is it's an equation that if you were to graph all the x and y pairs that satisfy this equation, you'll get a line. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | So now, and at some point, I encourage you to try to graph these things. These are actually quite interesting. But given that we've now seen examples of linear equations and nonlinear equations, let's see if we can come up with a definition for linear equations. One way to think about it is it's an equation that if you were to graph all the x and y pairs that satisfy this equation, you'll get a line. And that's actually literally where the word linear equation comes from. But another way to think about it is it's going to be an equation where every term is either going to be a constant. So for example, 12 is a constant. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | One way to think about it is it's an equation that if you were to graph all the x and y pairs that satisfy this equation, you'll get a line. And that's actually literally where the word linear equation comes from. But another way to think about it is it's going to be an equation where every term is either going to be a constant. So for example, 12 is a constant. It's not going to change based on the value of some variable. 12 is 12. Or negative three is negative three. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | So for example, 12 is a constant. It's not going to change based on the value of some variable. 12 is 12. Or negative three is negative three. So every term is either going to be a constant or it's going to be a constant times a variable raised to the first power. So this is a constant two times x to the first power. This is the variable y raised to the first power. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | Or negative three is negative three. So every term is either going to be a constant or it's going to be a constant times a variable raised to the first power. So this is a constant two times x to the first power. This is the variable y raised to the first power. You could say this is just one y. We're not dividing by x or y. We're not multiplying, or we don't have a term that has x to the second power or x to the third power or y to the fifth power. |
Two-variable linear equations and their graphs Algebra I Khan Academy.mp3 | This is the variable y raised to the first power. You could say this is just one y. We're not dividing by x or y. We're not multiplying, or we don't have a term that has x to the second power or x to the third power or y to the fifth power. We just have y to the first power. We have x to the first power. We're not multiplying x and y together like we did over here. |
Factoring polynomials common factor area model Mathematics II High School Math Khan Academy.mp3 | And we can see the area right over here, and they broke it up. This green area is 12x to the fourth, this purple area is 6x to the third, this blue area is 15x squared. You add them all together, you get this entire rectangle, which would be the combined areas, 12x to the fourth plus 6x to the third plus 15x squared. The length of the rectangle in meters, so this is the length right over here that we're talking about, we're talking about this distance. The length of the rectangle in meters is equal to the greatest common monomial factor of 12x to the fourth, 6x to the third, and 15x squared. What is the length and width of the rectangle? I encourage you to pause the video and try to work through it on your own. |
Factoring polynomials common factor area model Mathematics II High School Math Khan Academy.mp3 | The length of the rectangle in meters, so this is the length right over here that we're talking about, we're talking about this distance. The length of the rectangle in meters is equal to the greatest common monomial factor of 12x to the fourth, 6x to the third, and 15x squared. What is the length and width of the rectangle? I encourage you to pause the video and try to work through it on your own. Well, the key realization here is that the length times the width, the length times the width, is going to be equal to this area. If the length is the greatest common monomial factor of these terms, of 12x to the fourth, 6x to the third, and 15x squared, well then we can factor that out, and then what we have left over is going to be the width. So let's figure out what is the greatest common monomial factor of these three terms. |
Factoring polynomials common factor area model Mathematics II High School Math Khan Academy.mp3 | I encourage you to pause the video and try to work through it on your own. Well, the key realization here is that the length times the width, the length times the width, is going to be equal to this area. If the length is the greatest common monomial factor of these terms, of 12x to the fourth, 6x to the third, and 15x squared, well then we can factor that out, and then what we have left over is going to be the width. So let's figure out what is the greatest common monomial factor of these three terms. The first thing we can look at is let's look at the coefficients. Let's figure out what's the greatest common factor of 12, 6, and 15. And there's a couple of ways you could do it. |
Factoring polynomials common factor area model Mathematics II High School Math Khan Academy.mp3 | So let's figure out what is the greatest common monomial factor of these three terms. The first thing we can look at is let's look at the coefficients. Let's figure out what's the greatest common factor of 12, 6, and 15. And there's a couple of ways you could do it. You could do it by looking at a prime factorization. You could say, all right, well 12 is two times six, which is two times three. That's the prime factorization of 12. |
Factoring polynomials common factor area model Mathematics II High School Math Khan Academy.mp3 | And there's a couple of ways you could do it. You could do it by looking at a prime factorization. You could say, all right, well 12 is two times six, which is two times three. That's the prime factorization of 12. Prime factorization of six is just two times three. Prime factorization of 15 is three times five. And so the greatest common factor, the largest factor that's divisible into all of them, so let's see, we can throw a three in there. |
Factoring polynomials common factor area model Mathematics II High School Math Khan Academy.mp3 | That's the prime factorization of 12. Prime factorization of six is just two times three. Prime factorization of 15 is three times five. And so the greatest common factor, the largest factor that's divisible into all of them, so let's see, we can throw a three in there. Three is divisible into all of them. And that's it, because we can't say a three and a two. A three and a two would be divisible into 12 and six, but there's no two that's divisible into 15. |
Factoring polynomials common factor area model Mathematics II High School Math Khan Academy.mp3 | And so the greatest common factor, the largest factor that's divisible into all of them, so let's see, we can throw a three in there. Three is divisible into all of them. And that's it, because we can't say a three and a two. A three and a two would be divisible into 12 and six, but there's no two that's divisible into 15. We can't say a three and a five, because five isn't divisible into 12 or six. So the greatest common factor is going to be three. Another way we could have done this is we could have said, what are the non-prime factors of each of these numbers? |
Factoring polynomials common factor area model Mathematics II High School Math Khan Academy.mp3 | A three and a two would be divisible into 12 and six, but there's no two that's divisible into 15. We can't say a three and a five, because five isn't divisible into 12 or six. So the greatest common factor is going to be three. Another way we could have done this is we could have said, what are the non-prime factors of each of these numbers? 12, you could have said, okay, I can get 12 by saying one times 12, or two times six, or three times four. Six, you could have said, let's see, that could be one times six, or two times three. So those are the factors of six. |
Factoring polynomials common factor area model Mathematics II High School Math Khan Academy.mp3 | Another way we could have done this is we could have said, what are the non-prime factors of each of these numbers? 12, you could have said, okay, I can get 12 by saying one times 12, or two times six, or three times four. Six, you could have said, let's see, that could be one times six, or two times three. So those are the factors of six. And then 15, you could have said, well, one times 15, or three times five. And so you say the greatest common factor, well, three is the largest number that I've listed here that is common to all three of these factors. So once again, the greatest common factor of 12, six, and 15 is three. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.