[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.60,0:00:02.34,Default,,0000,0000,0000,,When we're dealing with basic arithmetic,
Dialogue: 0,0:00:02.34,0:00:04.59,Default,,0000,0000,0000,,we see the concrete numbers there.
Dialogue: 0,0:00:04.59,0:00:07.41,Default,,0000,0000,0000,,We'll see 23 + 5.
Dialogue: 0,0:00:07.41,0:00:08.72,Default,,0000,0000,0000,,We know what these numbers are right over here
Dialogue: 0,0:00:08.72,0:00:10.00,Default,,0000,0000,0000,,and we can calculate them.
Dialogue: 0,0:00:10.00,0:00:11.66,Default,,0000,0000,0000,,It's going to be 28.
Dialogue: 0,0:00:11.66,0:00:13.90,Default,,0000,0000,0000,,We can say 2 x 7.
Dialogue: 0,0:00:13.90,0:00:17.48,Default,,0000,0000,0000,,We could say 3 divided by 4 (3 / 4).
Dialogue: 0,0:00:17.48,0:00:19.06,Default,,0000,0000,0000,,In all of these cases, we know exactly
Dialogue: 0,0:00:19.06,0:00:20.87,Default,,0000,0000,0000,,what numbers we're dealing with.
Dialogue: 0,0:00:20.87,0:00:23.78,Default,,0000,0000,0000,,As we start entering into the algebratic world –
Dialogue: 0,0:00:23.78,0:00:25.87,Default,,0000,0000,0000,,(And you probably have seen this a little bit already.)
Dialogue: 0,0:00:25.87,0:00:30.05,Default,,0000,0000,0000,,– we start dealing with the idea of variables.
Dialogue: 0,0:00:30.05,0:00:31.53,Default,,0000,0000,0000,,And variables, there are a bunch of ways
Dialogue: 0,0:00:31.53,0:00:32.28,Default,,0000,0000,0000,,you can think about them.
Dialogue: 0,0:00:32.28,0:00:34.50,Default,,0000,0000,0000,,but they're really just values and expressions
Dialogue: 0,0:00:34.50,0:00:36.25,Default,,0000,0000,0000,,where they can change.
Dialogue: 0,0:00:36.25,0:00:38.14,Default,,0000,0000,0000,,The values in those expressions can change.
Dialogue: 0,0:00:38.14,0:00:42.20,Default,,0000,0000,0000,,So for example, if I write
Dialogue: 0,0:00:42.20,0:00:44.78,Default,,0000,0000,0000,,'x + 5.'
Dialogue: 0,0:00:44.78,0:00:46.65,Default,,0000,0000,0000,,this is an expression right over here.
Dialogue: 0,0:00:46.65,0:00:48.30,Default,,0000,0000,0000,,This can take on some value,
Dialogue: 0,0:00:48.30,0:00:51.47,Default,,0000,0000,0000,,depending on what the value of x is.
Dialogue: 0,0:00:51.47,0:00:56.66,Default,,0000,0000,0000,,If x is equal to 1,
Dialogue: 0,0:00:56.66,0:01:01.72,Default,,0000,0000,0000,,then x + 5 – our expression right over here –
Dialogue: 0,0:01:01.72,0:01:06.05,Default,,0000,0000,0000,,Is going to be equal to 1 –
Dialogue: 0,0:01:06.05,0:01:07.07,Default,,0000,0000,0000,,because now x is 1.
Dialogue: 0,0:01:07.07,0:01:08.32,Default,,0000,0000,0000,,It'll be 1 + 5.
Dialogue: 0,0:01:08.32,0:01:11.10,Default,,0000,0000,0000,,So x + 5 will be equal to 6. (x + 5 = 6)
Dialogue: 0,0:01:11.10,0:01:16.82,Default,,0000,0000,0000,,If x is equal to, I don't know, -7, (x = -7)
Dialogue: 0,0:01:16.82,0:01:22.18,Default,,0000,0000,0000,,then x + 5, is going to be equal to –
Dialogue: 0,0:01:22.18,0:01:24.12,Default,,0000,0000,0000,,Well now x is -7.
Dialogue: 0,0:01:24.12,0:01:28.84,Default,,0000,0000,0000,,It's going to be -7 + 5, which is -2.
Dialogue: 0,0:01:28.84,0:01:29.44,Default,,0000,0000,0000,,So notice.
Dialogue: 0,0:01:29.44,0:01:34.02,Default,,0000,0000,0000,,x here is a variable, x here is the variable,
Dialogue: 0,0:01:34.02,0:01:37.70,Default,,0000,0000,0000,,and its value can change depending on the context.
Dialogue: 0,0:01:37.70,0:01:39.95,Default,,0000,0000,0000,,And this is in the context of an expression.
Dialogue: 0,0:01:39.95,0:01:42.17,Default,,0000,0000,0000,,You'll also see that in the context of an equation.
Dialogue: 0,0:01:42.17,0:01:44.30,Default,,0000,0000,0000,,It's actually important to realize the distinction
Dialogue: 0,0:01:44.30,0:01:46.90,Default,,0000,0000,0000,,between an expression and an equation.
Dialogue: 0,0:01:46.90,0:01:49.83,Default,,0000,0000,0000,,An expression is really just a statement of value –
Dialogue: 0,0:01:49.83,0:01:51.73,Default,,0000,0000,0000,,a statement of some type of quantity.
Dialogue: 0,0:01:51.73,0:01:54.33,Default,,0000,0000,0000,,So this is an expression.
Dialogue: 0,0:01:54.33,0:01:56.64,Default,,0000,0000,0000,,An expression would be something like.
Dialogue: 0,0:01:56.64,0:01:57.98,Default,,0000,0000,0000,,well, what we saw over here:
Dialogue: 0,0:01:57.98,0:01:59.26,Default,,0000,0000,0000,,x + 5
Dialogue: 0,0:01:59.26,0:02:01.05,Default,,0000,0000,0000,,The value of this expression will change
Dialogue: 0,0:02:01.05,0:02:05.74,Default,,0000,0000,0000,,depending on what the value of this variable is.
Dialogue: 0,0:02:05.74,0:02:09.06,Default,,0000,0000,0000,,And you could just evaluate it for different values of x
Dialogue: 0,0:02:09.06,0:02:11.27,Default,,0000,0000,0000,,Another expression could be something like ...
Dialogue: 0,0:02:11.27,0:02:13.15,Default,,0000,0000,0000,,I don't know ... y + z.
Dialogue: 0,0:02:13.15,0:02:14.34,Default,,0000,0000,0000,,Now everything is a variable.
Dialogue: 0,0:02:14.34,0:02:16.55,Default,,0000,0000,0000,,If y is 1 and z is 2,
Dialogue: 0,0:02:16.55,0:02:18.56,Default,,0000,0000,0000,,it's going to be 1 + 2.
Dialogue: 0,0:02:18.56,0:02:21.39,Default,,0000,0000,0000,,If y is 0 and z is -1,
Dialogue: 0,0:02:21.39,0:02:24.07,Default,,0000,0000,0000,,it's going to be 0 + (-1).
Dialogue: 0,0:02:24.07,0:02:25.90,Default,,0000,0000,0000,,These can all be evaluated
Dialogue: 0,0:02:25.90,0:02:27.42,Default,,0000,0000,0000,,and they'll essentially give you a value
Dialogue: 0,0:02:27.42,0:02:30.81,Default,,0000,0000,0000,,depending on the values of each of these variables
Dialogue: 0,0:02:30.81,0:02:32.33,Default,,0000,0000,0000,,that make up the expression.
Dialogue: 0,0:02:32.33,0:02:34.28,Default,,0000,0000,0000,,In an equation, you're essentially setting expressions
Dialogue: 0,0:02:34.28,0:02:35.47,Default,,0000,0000,0000,,to be equal to each other.
Dialogue: 0,0:02:35.47,0:02:38.10,Default,,0000,0000,0000,,That's why they're called 'equations.'
Dialogue: 0,0:02:38.10,0:02:40.12,Default,,0000,0000,0000,,You're equating two things.
Dialogue: 0,0:02:40.12,0:02:42.92,Default,,0000,0000,0000,,In an equation, you'll see one expression
Dialogue: 0,0:02:42.92,0:02:44.64,Default,,0000,0000,0000,,being equal to another expression.
Dialogue: 0,0:02:44.64,0:02:47.87,Default,,0000,0000,0000,,So, for example, you could say something like
Dialogue: 0,0:02:47.87,0:02:52.06,Default,,0000,0000,0000,,x + 3 = 1.
Dialogue: 0,0:02:52.06,0:02:54.46,Default,,0000,0000,0000,,And in this situation where you have one equation,
Dialogue: 0,0:02:54.46,0:02:57.88,Default,,0000,0000,0000,,with only one unknown,
Dialogue: 0,0:02:57.88,0:02:59.27,Default,,0000,0000,0000,,you could actually figure out
Dialogue: 0,0:02:59.27,0:03:01.62,Default,,0000,0000,0000,,what x needs to be in this scenario.
Dialogue: 0,0:03:01.62,0:03:03.21,Default,,0000,0000,0000,,And you could possibly even do it in your head.
Dialogue: 0,0:03:03.21,0:03:05.33,Default,,0000,0000,0000,,'What' + 3 is equal to 1? ( __ + 3 = 1?)
Dialogue: 0,0:03:05.33,0:03:06.43,Default,,0000,0000,0000,,Well, you can do that in your head.
Dialogue: 0,0:03:06.43,0:03:08.87,Default,,0000,0000,0000,,Ff I have -2, -2 + 3 is equal to 1. (-2 +3 = 1)
Dialogue: 0,0:03:08.87,0:03:12.03,Default,,0000,0000,0000,,So in this context, an equation is starting to constrain
Dialogue: 0,0:03:12.03,0:03:15.13,Default,,0000,0000,0000,,the value that this variable can take on.
Dialogue: 0,0:03:15.13,0:03:17.41,Default,,0000,0000,0000,,But it doesn't have necessarily constrain as much.
Dialogue: 0,0:03:17.41,0:03:18.93,Default,,0000,0000,0000,,You could have something like:
Dialogue: 0,0:03:18.93,0:03:25.73,Default,,0000,0000,0000,,x + y + z = 5.
Dialogue: 0,0:03:25.73,0:03:27.78,Default,,0000,0000,0000,,Now – this expression is
Dialogue: 0,0:03:27.78,0:03:29.37,Default,,0000,0000,0000,,equal to this other expression.
Dialogue: 0,0:03:29.37,0:03:31.64,Default,,0000,0000,0000,,5 is really just an expression right over here.
Dialogue: 0,0:03:31.64,0:03:32.90,Default,,0000,0000,0000,,And there are some constraints.
Dialogue: 0,0:03:32.90,0:03:35.00,Default,,0000,0000,0000,,If someone tells you what y and z is,
Dialogue: 0,0:03:35.00,0:03:36.31,Default,,0000,0000,0000,,then that constrains what x is.
Dialogue: 0,0:03:36.31,0:03:38.23,Default,,0000,0000,0000,,If someone tells you what x and y are,
Dialogue: 0,0:03:38.23,0:03:39.92,Default,,0000,0000,0000,,then that constrains what z is.
Dialogue: 0,0:03:39.92,0:03:42.38,Default,,0000,0000,0000,,But it depends on what the different things are.
Dialogue: 0,0:03:42.38,0:03:44.06,Default,,0000,0000,0000,,So for example,
Dialogue: 0,0:03:44.06,0:03:51.64,Default,,0000,0000,0000,,if we said y = 3, and z = 2,
Dialogue: 0,0:03:51.64,0:03:53.39,Default,,0000,0000,0000,,then what would x be in that situation?
Dialogue: 0,0:03:53.39,0:03:58.10,Default,,0000,0000,0000,,So if y = 3, and z =2,
Dialogue: 0,0:03:58.10,0:03:58.61,Default,,0000,0000,0000,,then you're going to have –
Dialogue: 0,0:03:58.61,0:04:00.49,Default,,0000,0000,0000,,the left hand expression is going to be
Dialogue: 0,0:04:00.49,0:04:02.15,Default,,0000,0000,0000,,x + 3 + 2 –
Dialogue: 0,0:04:02.15,0:04:04.100,Default,,0000,0000,0000,,which is going to be x + 5 –
Dialogue: 0,0:04:04.100,0:04:06.81,Default,,0000,0000,0000,,This part right over here is going to be 5.
Dialogue: 0,0:04:06.81,0:04:08.98,Default,,0000,0000,0000,,x + 5 = 5
Dialogue: 0,0:04:08.98,0:04:11.20,Default,,0000,0000,0000,,And so what + 5 = 5?
Dialogue: 0,0:04:11.20,0:04:12.63,Default,,0000,0000,0000,,Well now, we're constraining x to be –
Dialogue: 0,0:04:12.63,0:04:14.38,Default,,0000,0000,0000,,x would have to be –
Dialogue: 0,0:04:14.38,0:04:16.94,Default,,0000,0000,0000,,x would have to be equal to 0. (x = 0)
Dialogue: 0,0:04:16.94,0:04:18.24,Default,,0000,0000,0000,,But the important point here –
Dialogue: 0,0:04:18.24,0:04:19.79,Default,,0000,0000,0000,,1) hopefully, you realize the difference
Dialogue: 0,0:04:19.79,0:04:20.80,Default,,0000,0000,0000,,between an expression and an equation.
Dialogue: 0,0:04:20.80,0:04:21.85,Default,,0000,0000,0000,,In an equation, essentially,
Dialogue: 0,0:04:21.85,0:04:23.67,Default,,0000,0000,0000,,you're equating two expressions.
Dialogue: 0,0:04:23.67,0:04:25.37,Default,,0000,0000,0000,,The important thing to take away from here,
Dialogue: 0,0:04:25.37,0:04:27.99,Default,,0000,0000,0000,,is that a variable can take on different values,
Dialogue: 0,0:04:27.99,0:04:31.36,Default,,0000,0000,0000,,depending on the context of the problem.
Dialogue: 0,0:04:31.36,0:04:32.78,Default,,0000,0000,0000,,And to hit the point home,
Dialogue: 0,0:04:32.78,0:04:35.22,Default,,0000,0000,0000,,let‘s just evaluate a bunch of expressions,
Dialogue: 0,0:04:35.22,0:04:38.06,Default,,0000,0000,0000,,when the variables have different values.
Dialogue: 0,0:04:38.06,0:04:41.60,Default,,0000,0000,0000,,So for example, if we had the expression
Dialogue: 0,0:04:41.60,0:04:43.31,Default,,0000,0000,0000,,if we had the expression.
Dialogue: 0,0:04:43.31,0:04:47.80,Default,,0000,0000,0000,,x to the y power,
Dialogue: 0,0:04:47.80,0:04:51.96,Default,,0000,0000,0000,,if x is equal to 5,
Dialogue: 0,0:04:51.96,0:04:54.31,Default,,0000,0000,0000,,and y is equal to 2
Dialogue: 0,0:04:54.31,0:04:55.79,Default,,0000,0000,0000,,y is equal to 2.
Dialogue: 0,0:04:55.79,0:04:58.91,Default,,0000,0000,0000,,then our expression here is going to evaluate to –
Dialogue: 0,0:04:58.91,0:05:01.51,Default,,0000,0000,0000,,Well x is now going to be 5.
Dialogue: 0,0:05:01.51,0:05:02.89,Default,,0000,0000,0000,,x is going to be 5.
Dialogue: 0,0:05:02.89,0:05:04.36,Default,,0000,0000,0000,,y is going to be 2.
Dialogue: 0,0:05:04.36,0:05:06.61,Default,,0000,0000,0000,,it's going to be 5 to the second power.
Dialogue: 0,0:05:06.61,0:05:08.15,Default,,0000,0000,0000,,or it's going to evaluate to
Dialogue: 0,0:05:08.15,0:05:09.78,Default,,0000,0000,0000,,25.
Dialogue: 0,0:05:09.78,0:05:11.63,Default,,0000,0000,0000,,If we change the values –
Dialogue: 0,0:05:11.63,0:05:14.36,Default,,0000,0000,0000,,If we said x –
Dialogue: 0,0:05:14.36,0:05:16.29,Default,,0000,0000,0000,,(Let me do it in that same color.)
Dialogue: 0,0:05:16.29,0:05:20.96,Default,,0000,0000,0000,,If we said x is equal to -2,
Dialogue: 0,0:05:20.96,0:05:24.77,Default,,0000,0000,0000,,and y is equal to 3,
Dialogue: 0,0:05:24.77,0:05:27.84,Default,,0000,0000,0000,,then this expression would evaluate to,
Dialogue: 0,0:05:27.84,0:05:30.47,Default,,0000,0000,0000,,(Let me do in that color.)
Dialogue: 0,0:05:30.47,0:05:32.39,Default,,0000,0000,0000,,– so it would evaluate to -2.
Dialogue: 0,0:05:32.39,0:05:35.38,Default,,0000,0000,0000,,(That's what we're going to substitute for x now,
Dialogue: 0,0:05:35.38,0:05:36.70,Default,,0000,0000,0000,,in this context.)
Dialogue: 0,0:05:36.70,0:05:38.17,Default,,0000,0000,0000,,– and y is now 3 –
Dialogue: 0,0:05:38.17,0:05:42.08,Default,,0000,0000,0000,,-2 to the third power –
Dialogue: 0,0:05:42.08,0:05:44.58,Default,,0000,0000,0000,,which is -2 x -2 x -2,
Dialogue: 0,0:05:44.58,0:05:46.90,Default,,0000,0000,0000,,which is -8.
Dialogue: 0,0:05:46.90,0:05:48.57,Default,,0000,0000,0000,,-2 × -2 = +4.
Dialogue: 0,0:05:48.57,0:05:52.15,Default,,0000,0000,0000,,× -2 again is equal to -8.
Dialogue: 0,0:05:52.15,0:05:53.37,Default,,0000,0000,0000,,is equal to -8
Dialogue: 0,0:05:53.37,0:05:55.71,Default,,0000,0000,0000,,So you see, depending on what the values of these are –
Dialogue: 0,0:05:55.71,0:05:58.28,Default,,0000,0000,0000,,(And we could even do more complex things.)
Dialogue: 0,0:05:58.28,0:05:59.68,Default,,0000,0000,0000,,We could have an expression like
Dialogue: 0,0:05:59.68,0:06:06.61,Default,,0000,0000,0000,,"the square root of x + y and then minus x" ... like that.
Dialogue: 0,0:06:06.61,0:06:11.88,Default,,0000,0000,0000,,If x is equal to –\NLet's say that x is equal to 1,
Dialogue: 0,0:06:11.88,0:06:16.01,Default,,0000,0000,0000,,and y is equal to 8,
Dialogue: 0,0:06:16.01,0:06:18.57,Default,,0000,0000,0000,,then this expression would evaluate to –
Dialogue: 0,0:06:18.57,0:06:21.42,Default,,0000,0000,0000,,(Well every time we see an x, we want to put a 1 there.)
Dialogue: 0,0:06:21.42,0:06:23.01,Default,,0000,0000,0000,,– so we would have a 1 there.
Dialogue: 0,0:06:23.01,0:06:24.81,Default,,0000,0000,0000,,And you would have a 1 over there.
Dialogue: 0,0:06:24.81,0:06:26.75,Default,,0000,0000,0000,,And every time you would see a y,
Dialogue: 0,0:06:26.75,0:06:28.41,Default,,0000,0000,0000,,you would put an 8 in its place –
Dialogue: 0,0:06:28.41,0:06:30.82,Default,,0000,0000,0000,,– in this context.\NWe're setting these variables to specific numbers.
Dialogue: 0,0:06:30.82,0:06:32.09,Default,,0000,0000,0000,,So you would see an 8.
Dialogue: 0,0:06:32.09,0:06:34.61,Default,,0000,0000,0000,,So under the radical sign, you would have a 1+8 –
Dialogue: 0,0:06:34.61,0:06:37.82,Default,,0000,0000,0000,,so you would have the principal root of 9 – which is 3.
Dialogue: 0,0:06:37.82,0:06:40.97,Default,,0000,0000,0000,,So this whole thing would simplify in this context.
Dialogue: 0,0:06:40.97,0:06:43.12,Default,,0000,0000,0000,,When we set these variables to be these things,
Dialogue: 0,0:06:43.12,0:06:45.59,Default,,0000,0000,0000,,this whole thing would simplify to be 3.
Dialogue: 0,0:06:45.59,0:06:46.50,Default,,0000,0000,0000,,1 + 8 is 9.
Dialogue: 0,0:06:46.50,0:06:48.68,Default,,0000,0000,0000,,Principal root of that is 3.
Dialogue: 0,0:06:48.68,0:06:50.77,Default,,0000,0000,0000,,And then you'd have 3 - 1.
Dialogue: 0,0:06:50.77,0:06:54.77,Default,,0000,0000,0000,,Which is equal to 2.