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Variables Expressions and Equations
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0:01 - 0:02When we're dealing with basic arithmetic,
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0:02 - 0:05we see the concrete numbers there.
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0:05 - 0:07We'll see 23 + 5.
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0:07 - 0:09We know what these numbers are right over here
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0:09 - 0:10and we can calculate them.
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0:10 - 0:12It's going to be 28.
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0:12 - 0:14We can say 2 x 7.
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0:14 - 0:17We could say 3 divided by 4 (3 / 4).
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0:17 - 0:19In all of these cases, we know exactly
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0:19 - 0:21what numbers we're dealing with.
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0:21 - 0:24As we start entering into the algebratic world –
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0:24 - 0:26(And you probably have seen this a little bit already.)
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0:26 - 0:30– we start dealing with the idea of variables.
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0:30 - 0:32And variables, there are a bunch of ways
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0:32 - 0:32you can think about them.
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0:32 - 0:35but they're really just values and expressions
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0:35 - 0:36where they can change.
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0:36 - 0:38The values in those expressions can change.
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0:38 - 0:42So for example, if I write
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0:42 - 0:45'x + 5.'
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0:45 - 0:47this is an expression right over here.
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0:47 - 0:48This can take on some value,
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0:48 - 0:51depending on what the value of x is.
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0:51 - 0:57If x is equal to 1,
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0:57 - 1:02then x + 5 – our expression right over here –
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1:02 - 1:06Is going to be equal to 1 –
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1:06 - 1:07because now x is 1.
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1:07 - 1:08It'll be 1 + 5.
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1:08 - 1:11So x + 5 will be equal to 6. (x + 5 = 6)
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1:11 - 1:17If x is equal to, I don't know, -7, (x = -7)
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1:17 - 1:22then x + 5, is going to be equal to –
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1:22 - 1:24Well now x is -7.
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1:24 - 1:29It's going to be -7 + 5, which is -2.
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1:29 - 1:29So notice.
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1:29 - 1:34x here is a variable, x here is the variable,
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1:34 - 1:38and its value can change depending on the context.
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1:38 - 1:40And this is in the context of an expression.
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1:40 - 1:42You'll also see that in the context of an equation.
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1:42 - 1:44It's actually important to realize the distinction
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1:44 - 1:47between an expression and an equation.
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1:47 - 1:50An expression is really just a statement of value –
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1:50 - 1:52a statement of some type of quantity.
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1:52 - 1:54So this is an expression.
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1:54 - 1:57An expression would be something like.
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1:57 - 1:58well, what we saw over here:
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1:58 - 1:59x + 5
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1:59 - 2:01The value of this expression will change
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2:01 - 2:06depending on what the value of this variable is.
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2:06 - 2:09And you could just evaluate it for different values of x
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2:09 - 2:11Another expression could be something like ...
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2:11 - 2:13I don't know ... y + z.
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2:13 - 2:14Now everything is a variable.
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2:14 - 2:17If y is 1 and z is 2,
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2:17 - 2:19it's going to be 1 + 2.
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2:19 - 2:21If y is 0 and z is -1,
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2:21 - 2:24it's going to be 0 + (-1).
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2:24 - 2:26These can all be evaluated
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2:26 - 2:27and they'll essentially give you a value
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2:27 - 2:31depending on the values of each of these variables
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2:31 - 2:32that make up the expression.
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2:32 - 2:34In an equation, you're essentially setting expressions
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2:34 - 2:35to be equal to each other.
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2:35 - 2:38That's why they're called 'equations.'
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2:38 - 2:40You're equating two things.
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2:40 - 2:43In an equation, you'll see one expression
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2:43 - 2:45being equal to another expression.
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2:45 - 2:48So, for example, you could say something like
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2:48 - 2:52x + 3 = 1.
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2:52 - 2:54And in this situation where you have one equation,
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2:54 - 2:58with only one unknown,
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2:58 - 2:59you could actually figure out
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2:59 - 3:02what x needs to be in this scenario.
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3:02 - 3:03And you could possibly even do it in your head.
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3:03 - 3:05'What' + 3 is equal to 1? ( __ + 3 = 1?)
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3:05 - 3:06Well, you can do that in your head.
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3:06 - 3:09Ff I have -2, -2 + 3 is equal to 1. (-2 +3 = 1)
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3:09 - 3:12So in this context, an equation is starting to constrain
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3:12 - 3:15the value that this variable can take on.
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3:15 - 3:17But it doesn't have necessarily constrain as much.
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3:17 - 3:19You could have something like:
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3:19 - 3:26x + y + z = 5.
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3:26 - 3:28Now – this expression is
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3:28 - 3:29equal to this other expression.
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3:29 - 3:325 is really just an expression right over here.
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3:32 - 3:33And there are some constraints.
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3:33 - 3:35If someone tells you what y and z is,
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3:35 - 3:36then that constrains what x is.
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3:36 - 3:38If someone tells you what x and y are,
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3:38 - 3:40then that constrains what z is.
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3:40 - 3:42But it depends on what the different things are.
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3:42 - 3:44So for example,
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3:44 - 3:52if we said y = 3, and z = 2,
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3:52 - 3:53then what would x be in that situation?
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3:53 - 3:58So if y = 3, and z =2,
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3:58 - 3:59then you're going to have –
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3:59 - 4:00the left hand expression is going to be
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4:00 - 4:02x + 3 + 2 –
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4:02 - 4:05which is going to be x + 5 –
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4:05 - 4:07This part right over here is going to be 5.
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4:07 - 4:09x + 5 = 5
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4:09 - 4:11And so what + 5 = 5?
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4:11 - 4:13Well now, we're constraining x to be –
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4:13 - 4:14x would have to be –
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4:14 - 4:17x would have to be equal to 0. (x = 0)
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4:17 - 4:18But the important point here –
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4:18 - 4:201) hopefully, you realize the difference
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4:20 - 4:21between an expression and an equation.
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4:21 - 4:22In an equation, essentially,
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4:22 - 4:24you're equating two expressions.
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4:24 - 4:25The important thing to take away from here,
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4:25 - 4:28is that a variable can take on different values,
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4:28 - 4:31depending on the context of the problem.
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4:31 - 4:33And to hit the point home,
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4:33 - 4:35let‘s just evaluate a bunch of expressions,
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4:35 - 4:38when the variables have different values.
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4:38 - 4:42So for example, if we had the expression
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4:42 - 4:43if we had the expression.
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4:43 - 4:48x to the y power,
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4:48 - 4:52if x is equal to 5,
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4:52 - 4:54and y is equal to 2
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4:54 - 4:56y is equal to 2.
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4:56 - 4:59then our expression here is going to evaluate to –
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4:59 - 5:02Well x is now going to be 5.
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5:02 - 5:03x is going to be 5.
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5:03 - 5:04y is going to be 2.
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5:04 - 5:07it's going to be 5 to the second power.
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5:07 - 5:08or it's going to evaluate to
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5:08 - 5:1025.
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5:10 - 5:12If we change the values –
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5:12 - 5:14If we said x –
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5:14 - 5:16(Let me do it in that same color.)
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5:16 - 5:21If we said x is equal to -2,
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5:21 - 5:25and y is equal to 3,
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5:25 - 5:28then this expression would evaluate to,
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5:28 - 5:30(Let me do in that color.)
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5:30 - 5:32– so it would evaluate to -2.
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5:32 - 5:35(That's what we're going to substitute for x now,
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5:35 - 5:37in this context.)
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5:37 - 5:38– and y is now 3 –
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5:38 - 5:42-2 to the third power –
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5:42 - 5:45which is -2 x -2 x -2,
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5:45 - 5:47which is -8.
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5:47 - 5:49-2 × -2 = +4.
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5:49 - 5:52× -2 again is equal to -8.
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5:52 - 5:53is equal to -8
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5:53 - 5:56So you see, depending on what the values of these are –
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5:56 - 5:58(And we could even do more complex things.)
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5:58 - 6:00We could have an expression like
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6:00 - 6:07"the square root of x + y and then minus x" ... like that.
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6:07 - 6:12If x is equal to –
Let's say that x is equal to 1, -
6:12 - 6:16and y is equal to 8,
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6:16 - 6:19then this expression would evaluate to –
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6:19 - 6:21(Well every time we see an x, we want to put a 1 there.)
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6:21 - 6:23– so we would have a 1 there.
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6:23 - 6:25And you would have a 1 over there.
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6:25 - 6:27And every time you would see a y,
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6:27 - 6:28you would put an 8 in its place –
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6:28 - 6:31– in this context.
We're setting these variables to specific numbers. -
6:31 - 6:32So you would see an 8.
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6:32 - 6:35So under the radical sign, you would have a 1+8 –
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6:35 - 6:38so you would have the principal root of 9 – which is 3.
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6:38 - 6:41So this whole thing would simplify in this context.
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6:41 - 6:43When we set these variables to be these things,
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6:43 - 6:46this whole thing would simplify to be 3.
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6:46 - 6:471 + 8 is 9.
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6:47 - 6:49Principal root of that is 3.
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6:49 - 6:51And then you'd have 3 - 1.
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6:51 - 6:55Which is equal to 2.
Mike Ridgway edited English subtitles for Variables Expressions and Equations | ||
Alex Mou edited English subtitles for Variables Expressions and Equations | ||
Alex Mou edited English subtitles for Variables Expressions and Equations | ||
Alex Mou edited English subtitles for Variables Expressions and Equations | ||
Alex Mou edited English subtitles for Variables Expressions and Equations | ||
wangxf edited English subtitles for Variables Expressions and Equations | ||
wangxf edited English subtitles for Variables Expressions and Equations | ||
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