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Finding intercepts from an equation Algebra I Khan Academy.mp3 | So the y-intercept occurs when x is equal to 0. So to figure out the intercepts, let's set y equal to 0 in this equation and solve for x. And then let's set x is equal to 0 and then solve for y. So when y is equal to 0, what does this equation become? I'll do it in orange. You get negative 5x plus 4y. Well, we're saying y is 0, so 4 times 0 is equal to 20. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | So when y is equal to 0, what does this equation become? I'll do it in orange. You get negative 5x plus 4y. Well, we're saying y is 0, so 4 times 0 is equal to 20. 4 times 0 is just 0, so we can just not write that. So let me just rewrite it. So we have negative 5x is equal to 20. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | Well, we're saying y is 0, so 4 times 0 is equal to 20. 4 times 0 is just 0, so we can just not write that. So let me just rewrite it. So we have negative 5x is equal to 20. We can divide both sides of this equation by negative 5. The negative 5's cancel out. That was the whole point behind dividing by negative 5. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | So we have negative 5x is equal to 20. We can divide both sides of this equation by negative 5. The negative 5's cancel out. That was the whole point behind dividing by negative 5. And we get x is equal to 20 divided by negative 5 is negative 4. So when y is equal to 0, we saw that right there, x is equal to negative 4. Or if we wanted to plot that point, we always put the x-coordinate for it first. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | That was the whole point behind dividing by negative 5. And we get x is equal to 20 divided by negative 5 is negative 4. So when y is equal to 0, we saw that right there, x is equal to negative 4. Or if we wanted to plot that point, we always put the x-coordinate for it first. So that would be the point negative 4 comma 0. So let me graph that. So if we go 1, 2, 3, 4, that's negative 4. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | Or if we wanted to plot that point, we always put the x-coordinate for it first. So that would be the point negative 4 comma 0. So let me graph that. So if we go 1, 2, 3, 4, that's negative 4. And then the y value is just 0, so that point is right over there. That is the x-intercept. y is 0, x is negative 4. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | So if we go 1, 2, 3, 4, that's negative 4. And then the y value is just 0, so that point is right over there. That is the x-intercept. y is 0, x is negative 4. Notice we're intersecting the x-axis. Now let's do the exact same thing for the y-intercept. Let's set x equal to 0. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | y is 0, x is negative 4. Notice we're intersecting the x-axis. Now let's do the exact same thing for the y-intercept. Let's set x equal to 0. So if we set x is equal to 0, we have negative 5 times 0 plus 4y is equal to 20. Well, anything times 0 is 0, so we can just put that out of the way. And remember, this was setting x is equal to 0. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | Let's set x equal to 0. So if we set x is equal to 0, we have negative 5 times 0 plus 4y is equal to 20. Well, anything times 0 is 0, so we can just put that out of the way. And remember, this was setting x is equal to 0. We're doing the y-intercept now. So this just simplifies to 4y is equal to 20. We can divide both sides of this equation by 4 to get rid of this 4 right there. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | And remember, this was setting x is equal to 0. We're doing the y-intercept now. So this just simplifies to 4y is equal to 20. We can divide both sides of this equation by 4 to get rid of this 4 right there. And you get y is equal to 20 over 4, which is 5. So when x is equal to 0, y is equal to 5. So the point 0, 5 is on the graph for this line. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | We can divide both sides of this equation by 4 to get rid of this 4 right there. And you get y is equal to 20 over 4, which is 5. So when x is equal to 0, y is equal to 5. So the point 0, 5 is on the graph for this line. So 0, 5, 0, x is 0, and y is 1, 2, 3, 4, 5 right over there. And notice, when x is 0, we're right on the y-axis. This is our y-intercept right over there. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | So the point 0, 5 is on the graph for this line. So 0, 5, 0, x is 0, and y is 1, 2, 3, 4, 5 right over there. And notice, when x is 0, we're right on the y-axis. This is our y-intercept right over there. And if we graph the line, all you need is two points to graph any line. So we just have to connect the dots. And that is our line. |
Finding intercepts from an equation Algebra I Khan Academy.mp3 | This is our y-intercept right over there. And if we graph the line, all you need is two points to graph any line. So we just have to connect the dots. And that is our line. So let me connect the dots, try my best to draw as straight of a line as I can. Well, I could do a better job than that. To draw as straight of a line as I can. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | So let's ask ourselves, what percent of, I don't know, let's say what percent of 16 is 4? And I encourage you to pause this video and to try it out yourself. So when you're saying what percent of 16 is 4, percent is another way of saying what fraction of 16 is 4. And we just need to write it as a percent, as per 100. So if you said what fraction of 16 is 4, you would say, well, look, this is the same thing as 4 16ths, which is the same thing as 1 4th. But this is saying what fraction 4 is of 16. You'd say, well, 4 is 1 4th of 16. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | And we just need to write it as a percent, as per 100. So if you said what fraction of 16 is 4, you would say, well, look, this is the same thing as 4 16ths, which is the same thing as 1 4th. But this is saying what fraction 4 is of 16. You'd say, well, 4 is 1 4th of 16. But it still doesn't answer our question. What percent? So in order to write this as a percent, we literally have to write it as something over 100. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | You'd say, well, 4 is 1 4th of 16. But it still doesn't answer our question. What percent? So in order to write this as a percent, we literally have to write it as something over 100. Percent literally means per cent. The word cent you know from cents and century. It relates to the number 100. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | So in order to write this as a percent, we literally have to write it as something over 100. Percent literally means per cent. The word cent you know from cents and century. It relates to the number 100. So it's per 100. So you could say, well, this is going to be equal to question mark over 100, the part of 100. And there's a bunch of ways that you could think about this. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | It relates to the number 100. So it's per 100. So you could say, well, this is going to be equal to question mark over 100, the part of 100. And there's a bunch of ways that you could think about this. You could say, well, look, if in the denominator to go from 4 to 100, I have to multiply by 25. In the numerator, I need to also multiply by 25 in order to have an equivalent fraction. So I'm also going to multiply by 25. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | And there's a bunch of ways that you could think about this. You could say, well, look, if in the denominator to go from 4 to 100, I have to multiply by 25. In the numerator, I need to also multiply by 25 in order to have an equivalent fraction. So I'm also going to multiply by 25. So 1 4th is the same thing as 25 over 100. And another way of saying 25 over 100 is this is 25 per 100, or 25 per cent. So this is equal to 25 per cent. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | So I'm also going to multiply by 25. So 1 4th is the same thing as 25 over 100. And another way of saying 25 over 100 is this is 25 per 100, or 25 per cent. So this is equal to 25 per cent. Now, there's a couple of other ways you could have thought about it. You could have said, well, 4 over 16, this is literally 4 divided by 16. Well, let me just do the division and convert to a decimal, which is very easy to convert to a percentage. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | So this is equal to 25 per cent. Now, there's a couple of other ways you could have thought about it. You could have said, well, 4 over 16, this is literally 4 divided by 16. Well, let me just do the division and convert to a decimal, which is very easy to convert to a percentage. So let's try to actually do this division right over here. So we're going to literally divide 4 by 16. Now, 16 goes into 4 0 times. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | Well, let me just do the division and convert to a decimal, which is very easy to convert to a percentage. So let's try to actually do this division right over here. So we're going to literally divide 4 by 16. Now, 16 goes into 4 0 times. 0 times 16 is 0. You subtract, and you get a 4. And we're not satisfied just having this remainder. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | Now, 16 goes into 4 0 times. 0 times 16 is 0. You subtract, and you get a 4. And we're not satisfied just having this remainder. We want to keep adding 0's to get a decimal answer right over here. So let's put a decimal right over here. We're going into the tenths place. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | And we're not satisfied just having this remainder. We want to keep adding 0's to get a decimal answer right over here. So let's put a decimal right over here. We're going into the tenths place. And let's throw some 0's right over here. The decimal will make sure we keep track of the fact that we are now in the tenths, and then the hundredths, and then the thousandths place if we have to go that far. But let's bring another 0 down. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | We're going into the tenths place. And let's throw some 0's right over here. The decimal will make sure we keep track of the fact that we are now in the tenths, and then the hundredths, and then the thousandths place if we have to go that far. But let's bring another 0 down. 16 goes into 40 2 times. 2 times 16 is 32. You subtract, you get 8. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | But let's bring another 0 down. 16 goes into 40 2 times. 2 times 16 is 32. You subtract, you get 8. And you could bring down another 0. And we have 16 goes into 80 5 times. 5 times 16 is 80. |
Finding a percentage Decimals Pre-Algebra Khan Academy.mp3 | You subtract, you get 8. And you could bring down another 0. And we have 16 goes into 80 5 times. 5 times 16 is 80. You subtract, you have no remainder, and you're done. 4 16ths is the same thing as 0.25. Now, 0.25 is the same thing as 25 hundredths. |
Finding the x-intercept of a line Algebra I Khan Academy.mp3 | So the x-intercept is the x value when y is equal to zero. Or it's the x value where our graph actually intersects the x-axis. Notice right over here our y value is exactly zero. We're sitting on the x-axis. So let's think about what this x value must be. Well just looking at it from, just trying to eyeball a little bit, it looks like it's a little over two. It's between two and three. |
Finding the x-intercept of a line Algebra I Khan Academy.mp3 | We're sitting on the x-axis. So let's think about what this x value must be. Well just looking at it from, just trying to eyeball a little bit, it looks like it's a little over two. It's between two and three. It looks like it's less than two and a half. But we don't know the exact value. So let's go turn to the equation to figure out the exact value. |
Finding the x-intercept of a line Algebra I Khan Academy.mp3 | It's between two and three. It looks like it's less than two and a half. But we don't know the exact value. So let's go turn to the equation to figure out the exact value. So we essentially have to figure out what x value when y is equal to zero will have this equation be true. So we could just say two times zero plus 3x is equal to seven. Well two times zero is just gonna be zero. |
Finding the x-intercept of a line Algebra I Khan Academy.mp3 | So let's go turn to the equation to figure out the exact value. So we essentially have to figure out what x value when y is equal to zero will have this equation be true. So we could just say two times zero plus 3x is equal to seven. Well two times zero is just gonna be zero. So we have 3x is equal to seven. And then we can divide both sides by three to solve for x. And we get x is equal to seven over three. |
Finding the x-intercept of a line Algebra I Khan Academy.mp3 | Well two times zero is just gonna be zero. So we have 3x is equal to seven. And then we can divide both sides by three to solve for x. And we get x is equal to seven over three. Now does that look like 7 3rds? Well we just have to remind ourselves that seven over three is the same thing as six over three plus one over three. And six over three is two. |
Finding the x-intercept of a line Algebra I Khan Academy.mp3 | And we get x is equal to seven over three. Now does that look like 7 3rds? Well we just have to remind ourselves that seven over three is the same thing as six over three plus one over three. And six over three is two. So this is the same thing as two and 1 3rd. Another way you could think about it is three goes into seven two times. And then you have a remainder of one. |
Finding the x-intercept of a line Algebra I Khan Academy.mp3 | And six over three is two. So this is the same thing as two and 1 3rd. Another way you could think about it is three goes into seven two times. And then you have a remainder of one. So you still gotta divide that one by three. So it's two full times and then a 1 3rd. So this looks like two and 1 3rd. |
Finding the x-intercept of a line Algebra I Khan Academy.mp3 | And then you have a remainder of one. So you still gotta divide that one by three. So it's two full times and then a 1 3rd. So this looks like two and 1 3rd. And so that's its x intercept. Seven 7 3rds. If I was doing this on the exercise on Khan Academy, it's always a little easier to type in the improper fraction. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | Now when you multiply decimals, you multiply them the exact same way you would multiply whole numbers. And then you count the number of spaces behind the decimal you have in your two numbers you're multiplying and you're going to have that many spaces in your product. And let me show you what I'm talking about. So let's just multiply these two characters. So we have 32.12 times 0.5. And when you write them out, you can just push both of them all the way to the right. You can almost ignore the decimal. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | So let's just multiply these two characters. So we have 32.12 times 0.5. And when you write them out, you can just push both of them all the way to the right. You can almost ignore the decimal. Right now, you should write the decimal where they belong. But you can almost pretend that this is 3,212 times 5. And then we'll worry about the decimals in a second. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | You can almost ignore the decimal. Right now, you should write the decimal where they belong. But you can almost pretend that this is 3,212 times 5. And then we'll worry about the decimals in a second. So let's get started. So if we were just multiplying 5 times 3,212, we would say, well, 5 times 2 is 10. Regroup the 1. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | And then we'll worry about the decimals in a second. So let's get started. So if we were just multiplying 5 times 3,212, we would say, well, 5 times 2 is 10. Regroup the 1. 5 times 1 is 5. Plus 1 is 6. 5 times 2 is 10. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | Regroup the 1. 5 times 1 is 5. Plus 1 is 6. 5 times 2 is 10. Regroup the 1. And then finally, you have 5 times 3 is 15. Plus 1 is 16. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | 5 times 2 is 10. Regroup the 1. And then finally, you have 5 times 3 is 15. Plus 1 is 16. And then we don't have any other places. This 0 really isn't. If we were just doing this as 0,5, we wouldn't multiply 0 times this whole thing. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | Plus 1 is 16. And then we don't have any other places. This 0 really isn't. If we were just doing this as 0,5, we wouldn't multiply 0 times this whole thing. We'd just get 0 anyway. So just 5 times 3,212 gives us this number. But now we want to care about the decimals. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | If we were just doing this as 0,5, we wouldn't multiply 0 times this whole thing. We'd just get 0 anyway. So just 5 times 3,212 gives us this number. But now we want to care about the decimals. We just have to count the total number of spaces or places we have behind the decimal points in the two numbers we're multiplying. So we have 1, 2, 3 spaces or 3 numbers to the right of the decimals in the two numbers that we're multiplying. So we need that many numbers to the right of the decimal in our answer. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | But now we want to care about the decimals. We just have to count the total number of spaces or places we have behind the decimal points in the two numbers we're multiplying. So we have 1, 2, 3 spaces or 3 numbers to the right of the decimals in the two numbers that we're multiplying. So we need that many numbers to the right of the decimal in our answer. So we go 1, 2, 3. Put the decimal right over there. So 32.12 times 0.5 is 16.060. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | So we need that many numbers to the right of the decimal in our answer. So we go 1, 2, 3. Put the decimal right over there. So 32.12 times 0.5 is 16.060. And this trailing 0 right here, we can ignore because it's really not adding any information there. So we can just write this as 16.06. Now the last thing you want to do is just kind of make sure that this makes sense. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | So 32.12 times 0.5 is 16.060. And this trailing 0 right here, we can ignore because it's really not adding any information there. So we can just write this as 16.06. Now the last thing you want to do is just kind of make sure that this makes sense. You have a number that's almost 32 and we're multiplying it by 0.5. Remember, 0.5 is the same thing as 5 over 10, which is the same thing as 1 half. So we're really multiplying 32.12 times 1 half. |
Multiplying decimals example Decimals Pre-Algebra Khan Academy.mp3 | Now the last thing you want to do is just kind of make sure that this makes sense. You have a number that's almost 32 and we're multiplying it by 0.5. Remember, 0.5 is the same thing as 5 over 10, which is the same thing as 1 half. So we're really multiplying 32.12 times 1 half. We're trying to figure out what 1 half of 32.12 is. And 1 half of 32 is 16 and then 1 half of 0.12 is 0.06. So this makes complete sense. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | We actually have two of them. So now we're going to try to figure out what x is. But before we even do that, what I want you to think about is a mathematical equation that can represent what is going on right here, that equates what we have on the left hand here to what we have on the right side of the scale right over there. And I'll give you a few seconds to think about it. So let's think about what we have on the left side here. We have three masses with mass x. So you could say that we have three x over here. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | And I'll give you a few seconds to think about it. So let's think about what we have on the left side here. We have three masses with mass x. So you could say that we have three x over here. We have three x's. And then we have two masses of one kilogram. So in total we have two kilograms, so plus two. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | So you could say that we have three x over here. We have three x's. And then we have two masses of one kilogram. So in total we have two kilograms, so plus two. So one way to think about it, the total mass on the left hand side is three x plus two. Three masses with mass x plus two kilograms. That's what we have on the left hand side. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | So in total we have two kilograms, so plus two. So one way to think about it, the total mass on the left hand side is three x plus two. Three masses with mass x plus two kilograms. That's what we have on the left hand side. Now let's think about what we have on the right hand side. We just have to count these. We have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | That's what we have on the left hand side. Now let's think about what we have on the right hand side. We just have to count these. We have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14. 14 blocks each have a mass of one kilogram. So the total mass right over here is going to be 14 kilograms. So we get, and we see that the scale is balanced. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | We have one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14. 14 blocks each have a mass of one kilogram. So the total mass right over here is going to be 14 kilograms. So we get, and we see that the scale is balanced. It's not tilting down or upwards. So the scale is balanced. So this mass, this total mass right over here, must be equal to this total mass. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | So we get, and we see that the scale is balanced. It's not tilting down or upwards. So the scale is balanced. So this mass, this total mass right over here, must be equal to this total mass. The scale is balanced. So we can write an equal sign. Maybe we'll do that in that white color. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | So this mass, this total mass right over here, must be equal to this total mass. The scale is balanced. So we can write an equal sign. Maybe we'll do that in that white color. I don't like that brown. We can do it in that white color. Now, what I want you to think about, and you can think about it either through the symbols or think about it through the scale, is how would you go about, let's think about a few things. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | Maybe we'll do that in that white color. I don't like that brown. We can do it in that white color. Now, what I want you to think about, and you can think about it either through the symbols or think about it through the scale, is how would you go about, let's think about a few things. How would you first go about at least getting rid of these little one kilogram blocks over here? And I'll give you a second to think about that. Well, the simplest thing is, well, you could take these one kilogram blocks off of the left-hand side. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | Now, what I want you to think about, and you can think about it either through the symbols or think about it through the scale, is how would you go about, let's think about a few things. How would you first go about at least getting rid of these little one kilogram blocks over here? And I'll give you a second to think about that. Well, the simplest thing is, well, you could take these one kilogram blocks off of the left-hand side. But remember, if you just took these one kilogram blocks off the left-hand side and it was balanced before, now the left-hand side will be lighter and it'll move up. But we want to keep it balanced so that we can keep saying equal, that this mass is equal to that mass. So if we're going to remove two blocks from the left-hand side, we need to remove two blocks from the right-hand side. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | Well, the simplest thing is, well, you could take these one kilogram blocks off of the left-hand side. But remember, if you just took these one kilogram blocks off the left-hand side and it was balanced before, now the left-hand side will be lighter and it'll move up. But we want to keep it balanced so that we can keep saying equal, that this mass is equal to that mass. So if we're going to remove two blocks from the left-hand side, we need to remove two blocks from the right-hand side. So we can remove two there, and then we can remove two right over there. And mathematically, what we're essentially doing is we're subtracting two kilograms from each side. So we're subtracting two from this side. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | So if we're going to remove two blocks from the left-hand side, we need to remove two blocks from the right-hand side. So we can remove two there, and then we can remove two right over there. And mathematically, what we're essentially doing is we're subtracting two kilograms from each side. So we're subtracting two from this side. So on the left-hand side, we now have 3x plus 2 minus 2. We're just left with 3x. And on the right-hand side, we had 14, and we took away 2. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | So we're subtracting two from this side. So on the left-hand side, we now have 3x plus 2 minus 2. We're just left with 3x. And on the right-hand side, we had 14, and we took away 2. Let me write this. We took away 2, so we're going to be left with 12. We're going to be left with 12 blocks. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | And on the right-hand side, we had 14, and we took away 2. Let me write this. We took away 2, so we're going to be left with 12. We're going to be left with 12 blocks. And you see that there. The ones that I haven't crossed out, there's 12 left. And here, you only have three of those x blocks. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | We're going to be left with 12 blocks. And you see that there. The ones that I haven't crossed out, there's 12 left. And here, you only have three of those x blocks. And since we removed the exact same amount from both sides, our scale is still balanced. And our equation, 3x is now equal to 12. And now this turns into a problem very similar to what we saw in the last video. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | And here, you only have three of those x blocks. And since we removed the exact same amount from both sides, our scale is still balanced. And our equation, 3x is now equal to 12. And now this turns into a problem very similar to what we saw in the last video. So I will ask you, what can we do to isolate 1x? To only have 1x on the scale while keeping the scale, or 1x on the left-hand side of the scale, while keeping the scale balanced? So the easiest way to think about it is, if I want 1x on this left-hand side, that's 1 third of the total x is here. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | And now this turns into a problem very similar to what we saw in the last video. So I will ask you, what can we do to isolate 1x? To only have 1x on the scale while keeping the scale, or 1x on the left-hand side of the scale, while keeping the scale balanced? So the easiest way to think about it is, if I want 1x on this left-hand side, that's 1 third of the total x is here. So what if I were to essentially multiply the left-hand side by 1 third? But if I want to keep the scale balanced, I have to multiply the right-hand side by 1 third. And so if we can do that mathematically, so right over here, I can multiply the left-hand side by 1 third. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | So the easiest way to think about it is, if I want 1x on this left-hand side, that's 1 third of the total x is here. So what if I were to essentially multiply the left-hand side by 1 third? But if I want to keep the scale balanced, I have to multiply the right-hand side by 1 third. And so if we can do that mathematically, so right over here, I can multiply the left-hand side by 1 third. But if I want to keep my scale balanced, I also have to multiply the right-hand side by 1 third. And multiplying it physically, that literally means just keeping 1 third of what we had here originally. So we would get rid of 2 of these. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | And so if we can do that mathematically, so right over here, I can multiply the left-hand side by 1 third. But if I want to keep my scale balanced, I also have to multiply the right-hand side by 1 third. And multiplying it physically, that literally means just keeping 1 third of what we had here originally. So we would get rid of 2 of these. And if we want to keep 1 third of what we had here originally, there are 12 blocks left over after removing those first 2. So 1 third of 12, we're only going to keep 4 of these little 1 kilogram boxes left. So let me remove all but 4. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | So we would get rid of 2 of these. And if we want to keep 1 third of what we had here originally, there are 12 blocks left over after removing those first 2. So 1 third of 12, we're only going to keep 4 of these little 1 kilogram boxes left. So let me remove all but 4. So we're going to move those and remove those. I have left 1, 2, 3, 4 here. And so what you're left with, the only thing you have left is this x. I'll shade it in to show this is the one that we actually have left. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | So let me remove all but 4. So we're going to move those and remove those. I have left 1, 2, 3, 4 here. And so what you're left with, the only thing you have left is this x. I'll shade it in to show this is the one that we actually have left. And then we have these boxes. We have these 1 kilogram boxes. And you see it mathematically right over here, 1 third times 3x. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | And so what you're left with, the only thing you have left is this x. I'll shade it in to show this is the one that we actually have left. And then we have these boxes. We have these 1 kilogram boxes. And you see it mathematically right over here, 1 third times 3x. Or you could have said 3x divided by 3. Either way, that gives us that. You could say that these 3's cancel out. |
Solving two-step equations Linear equations Algebra I Khan Academy.mp3 | And you see it mathematically right over here, 1 third times 3x. Or you could have said 3x divided by 3. Either way, that gives us that. You could say that these 3's cancel out. That'll give you just an x. And then on the right-hand side, 12 times 1 third, which is the same thing as 12 divided by 3, is equal to 4. And since we did the same thing to both sides, the scale is still balanced. |
Constructing and solving a one-step inequality Linear inequalities Algebra I Khan Academy.mp3 | A contractor is purchasing some stone tiles for a new patio. Each tile costs $3, and he wants to spend less than $1,000. And it's less than $1,000, not less than or equal to $1,000. The size of each tile is 1 square foot. Write an inequality that represents the number of tiles he can purchase with a $1,000 limit, and then figure out how large the stone patio can be. So let x be equal to the number of tiles purchased. And so the cost of purchasing x tiles, they're going to be $3 each, so it's going to be 3x. |
Constructing and solving a one-step inequality Linear inequalities Algebra I Khan Academy.mp3 | The size of each tile is 1 square foot. Write an inequality that represents the number of tiles he can purchase with a $1,000 limit, and then figure out how large the stone patio can be. So let x be equal to the number of tiles purchased. And so the cost of purchasing x tiles, they're going to be $3 each, so it's going to be 3x. So 3x is going to be the total cost of purchasing the tiles, and he wants to spend less than $1,000. 3x is how much he spends if he buys x tiles. It has to be less than $1,000. |
Constructing and solving a one-step inequality Linear inequalities Algebra I Khan Academy.mp3 | And so the cost of purchasing x tiles, they're going to be $3 each, so it's going to be 3x. So 3x is going to be the total cost of purchasing the tiles, and he wants to spend less than $1,000. 3x is how much he spends if he buys x tiles. It has to be less than $1,000. We say it right there. If it was less than or equal to, we'd have a little equal sign right there. So if we want to solve for x, how many tiles can he buy? |
Constructing and solving a one-step inequality Linear inequalities Algebra I Khan Academy.mp3 | It has to be less than $1,000. We say it right there. If it was less than or equal to, we'd have a little equal sign right there. So if we want to solve for x, how many tiles can he buy? We can divide both sides of this inequality by 3. And because we're dividing or multiplying, you can imagine we're multiplying by 1 third or dividing by 3, because this is a positive number, we do not have to swap the inequality sign. So we are left with x is less than 1,000 over 3, which is 333 and 1 third. |
Constructing and solving a one-step inequality Linear inequalities Algebra I Khan Academy.mp3 | So if we want to solve for x, how many tiles can he buy? We can divide both sides of this inequality by 3. And because we're dividing or multiplying, you can imagine we're multiplying by 1 third or dividing by 3, because this is a positive number, we do not have to swap the inequality sign. So we are left with x is less than 1,000 over 3, which is 333 and 1 third. So he has to buy less than 333 and 1 third tiles. That's how many tiles. And each tile is 1 square foot. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | You're traveling in some type of strange fantasy land, and you're trying to get to the castle up here to save the princess or the prince or whomever you are trying to save. But to get there, you have to cross this river. You can't swim across it. It's a very rough river. So you have to cross this bridge. And so as you approach the bridge, this troll shows up. That's the troll. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | It's a very rough river. So you have to cross this bridge. And so as you approach the bridge, this troll shows up. That's the troll. And he says, well, I'm a reasonable troll. You just have to pay $5. And when you look a little bit more carefully, you see that there actually was a sign there that says $5 toll to cross the bridge. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | That's the troll. And he says, well, I'm a reasonable troll. You just have to pay $5. And when you look a little bit more carefully, you see that there actually was a sign there that says $5 toll to cross the bridge. Now, unfortunately for you, you do not have any money in your pocket. And the troll says, well, you can't cross. But you say I need to really, really get to that castle. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | And when you look a little bit more carefully, you see that there actually was a sign there that says $5 toll to cross the bridge. Now, unfortunately for you, you do not have any money in your pocket. And the troll says, well, you can't cross. But you say I need to really, really get to that castle. And so the troll says, well, I'll take some pity on you. Instead of paying the $5, I will give you a riddle. And the riddle is this. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | But you say I need to really, really get to that castle. And so the troll says, well, I'll take some pity on you. Instead of paying the $5, I will give you a riddle. And the riddle is this. And now I'm speaking as the troll. I am a rich troll because I get to charge $5 from everyone who crosses the bridge. And actually, I only accept $5 or $10 bills. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | And the riddle is this. And now I'm speaking as the troll. I am a rich troll because I get to charge $5 from everyone who crosses the bridge. And actually, I only accept $5 or $10 bills. It's a bit of a riddle why they accept American currency in this fantasy land. But let's just take that as a given for now. So I only take $5 or $10 bills. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | And actually, I only accept $5 or $10 bills. It's a bit of a riddle why they accept American currency in this fantasy land. But let's just take that as a given for now. So I only take $5 or $10 bills. I'm being the troll. Obviously, if you give me a 10, I'll give you 5 back. And I know, because I count my money on a daily basis, I like to save my money as the troll, I know that I have a total of 900 bills. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | So I only take $5 or $10 bills. I'm being the troll. Obviously, if you give me a 10, I'll give you 5 back. And I know, because I count my money on a daily basis, I like to save my money as the troll, I know that I have a total of 900 bills. Let me write that down. I have a total of 900 bills. Total of 900 $5 and $10 bills. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | And I know, because I count my money on a daily basis, I like to save my money as the troll, I know that I have a total of 900 bills. Let me write that down. I have a total of 900 bills. Total of 900 $5 and $10 bills. And he says, I will give you, because I'm very sympathetic, I'll give you another piece of information. He says, if you were to add up the value of all of my money, which is all in $5 and $10 bills, value of all the $5 and $10 bills that I have, I, speaking as a troll, bought dollar bills, is $5,500. This is a rich troll. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | Total of 900 $5 and $10 bills. And he says, I will give you, because I'm very sympathetic, I'll give you another piece of information. He says, if you were to add up the value of all of my money, which is all in $5 and $10 bills, value of all the $5 and $10 bills that I have, I, speaking as a troll, bought dollar bills, is $5,500. This is a rich troll. Is $5,500. And so the riddle is, exactly, and if you give the wrong answer, and if you're not able to solve it in 10 minutes, he's just going to push you into the river and do something horrible to you. He says, exactly, how many $5s and $10s do I, the troll, have? |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | This is a rich troll. Is $5,500. And so the riddle is, exactly, and if you give the wrong answer, and if you're not able to solve it in 10 minutes, he's just going to push you into the river and do something horrible to you. He says, exactly, how many $5s and $10s do I, the troll, have? So the first thing I'm going to have you think about is, is this even a solvable problem? Because it's not a solvable problem. You should probably run as fast as you can in the other direction. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | He says, exactly, how many $5s and $10s do I, the troll, have? So the first thing I'm going to have you think about is, is this even a solvable problem? Because it's not a solvable problem. You should probably run as fast as you can in the other direction. So now I will tell you, yes, it is a solvable problem. And let's start thinking about it a little bit algebraically. And to do that, let's just set some variables. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | You should probably run as fast as you can in the other direction. So now I will tell you, yes, it is a solvable problem. And let's start thinking about it a little bit algebraically. And to do that, let's just set some variables. And I will set the variables to be what we're really trying to solve for. We're trying to solve for the number of $5 bills we have and the number of $10 bills that we have. So let's just define some variables. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | And to do that, let's just set some variables. And I will set the variables to be what we're really trying to solve for. We're trying to solve for the number of $5 bills we have and the number of $10 bills that we have. So let's just define some variables. I'll say f for 5. Let's let f equal the number of $5 bills that we have. And I'll use the same idea. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | So let's just define some variables. I'll say f for 5. Let's let f equal the number of $5 bills that we have. And I'll use the same idea. Let's let t is equal to the number of $10 bills that we have. Now given this information, and now I'm not sure if I'm speaking as a troll. Let's say I'm still speaking as a troll. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | And I'll use the same idea. Let's let t is equal to the number of $10 bills that we have. Now given this information, and now I'm not sure if I'm speaking as a troll. Let's say I'm still speaking as a troll. I'm a very sympathetic troll. And I'm going to give you hints. Given this information, setting these variables in this way, can I represent the clues in the riddle mathematically? |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | Let's say I'm still speaking as a troll. I'm a very sympathetic troll. And I'm going to give you hints. Given this information, setting these variables in this way, can I represent the clues in the riddle mathematically? So let's focus on the first clue. Can I represent this clue, that the total of 900 $5 and $10 bills, or can I represent that mathematically, that I have a total of 900 $5 and $10 bills? Well, what's going to be our total of bills? |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | Given this information, setting these variables in this way, can I represent the clues in the riddle mathematically? So let's focus on the first clue. Can I represent this clue, that the total of 900 $5 and $10 bills, or can I represent that mathematically, that I have a total of 900 $5 and $10 bills? Well, what's going to be our total of bills? It's going to be the number of 5's that we have, which is f. The number of 5's that we have is f. And then the number of 10's that we have is t. The total number of 5's plus the total number of 10's, that's our total number of bills. So that's going to be equal to 900. So this statement, this first clue in our riddle, can be written mathematically like this if we defined the variables like that. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | Well, what's going to be our total of bills? It's going to be the number of 5's that we have, which is f. The number of 5's that we have is f. And then the number of 10's that we have is t. The total number of 5's plus the total number of 10's, that's our total number of bills. So that's going to be equal to 900. So this statement, this first clue in our riddle, can be written mathematically like this if we defined the variables like that. And I just said f for 5, because f for 5 and t for 10. Now let's look at the second clue. Can we represent this one mathematically, given these variable definitions that we created? |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | So this statement, this first clue in our riddle, can be written mathematically like this if we defined the variables like that. And I just said f for 5, because f for 5 and t for 10. Now let's look at the second clue. Can we represent this one mathematically, given these variable definitions that we created? Well, let's think separately about the value of the $5 bills and the value of the $10 bills. What's the value of all of the $5 bills? Well, each $5 bill is worth $5. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | Can we represent this one mathematically, given these variable definitions that we created? Well, let's think separately about the value of the $5 bills and the value of the $10 bills. What's the value of all of the $5 bills? Well, each $5 bill is worth $5. So it's going to be 5 times the number of $5 bills that we have. So if I have one $5 bill, it will be $5. If I have 100 $5 bills, that's going to be $500. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | Well, each $5 bill is worth $5. So it's going to be 5 times the number of $5 bills that we have. So if I have one $5 bill, it will be $5. If I have 100 $5 bills, that's going to be $500. However many $5 bills, I just multiply it by 5. That's the value of the $5 bills. Let me write that down. |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | If I have 100 $5 bills, that's going to be $500. However many $5 bills, I just multiply it by 5. That's the value of the $5 bills. Let me write that down. Value of the $5 bills. Now, same logic. What's the value of the $10 bills? |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | Let me write that down. Value of the $5 bills. Now, same logic. What's the value of the $10 bills? Well, the value of the $10 bills is just going to be 10 times however many bills I have. Value of the $10 bills. So what's going to be the total value of my bills? |
Trolls, tolls, and systems of equations Algebra II Khan Academy.mp3 | What's the value of the $10 bills? Well, the value of the $10 bills is just going to be 10 times however many bills I have. Value of the $10 bills. So what's going to be the total value of my bills? What's going to be the value of the $5 bills plus the value of the $10 bills? And he tells me what that total value is. It's $5,500. |
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