WEBVTT 00:00:00.600 --> 00:00:02.341 When we're dealing with basic arithmetic, 00:00:02.341 --> 00:00:04.592 we see the concrete numbers there. 00:00:04.592 --> 00:00:07.406 We'll see 23 + 5. 00:00:07.406 --> 00:00:08.715 We know what these numbers are right over here 00:00:08.715 --> 00:00:10.005 and we can calculate them. 00:00:10.005 --> 00:00:11.661 It's going to be 28. 00:00:11.661 --> 00:00:13.898 We can say 2 x 7. 00:00:13.898 --> 00:00:17.476 We could say 3 divided by 4 (3 / 4). 00:00:17.476 --> 00:00:19.059 In all of these cases, we know exactly 00:00:19.059 --> 00:00:20.872 what numbers we're dealing with. 00:00:20.872 --> 00:00:23.776 As we start entering into the algebratic world – 00:00:23.776 --> 00:00:25.873 (And you probably have seen this a little bit already.) 00:00:25.873 --> 00:00:30.051 – we start dealing with the idea of variables. 00:00:30.051 --> 00:00:31.533 And variables, there are a bunch of ways 00:00:31.533 --> 00:00:32.283 you can think about them. 00:00:32.283 --> 00:00:34.502 but they're really just values and expressions 00:00:34.502 --> 00:00:36.252 where they can change. 00:00:36.252 --> 00:00:38.145 The values in those expressions can change. 00:00:38.145 --> 00:00:42.201 So for example, if I write 00:00:42.201 --> 00:00:44.781 'x + 5.' 00:00:44.781 --> 00:00:46.647 this is an expression right over here. 00:00:46.647 --> 00:00:48.305 This can take on some value, 00:00:48.305 --> 00:00:51.466 depending on what the value of x is. 00:00:51.466 --> 00:00:56.656 If x is equal to 1, 00:00:56.656 --> 00:01:01.723 then x + 5 – our expression right over here – 00:01:01.723 --> 00:01:06.049 Is going to be equal to 1 – 00:01:06.049 --> 00:01:07.070 because now x is 1. 00:01:07.070 --> 00:01:08.321 It'll be 1 + 5. 00:01:08.321 --> 00:01:11.101 So x + 5 will be equal to 6. (x + 5 = 6) 00:01:11.101 --> 00:01:16.821 If x is equal to, I don't know, -7, (x = -7) 00:01:16.821 --> 00:01:22.183 then x + 5, is going to be equal to – 00:01:22.183 --> 00:01:24.120 Well now x is -7. 00:01:24.120 --> 00:01:28.842 It's going to be -7 + 5, which is -2. 00:01:28.842 --> 00:01:29.441 So notice. 00:01:29.441 --> 00:01:34.019 x here is a variable, x here is the variable, 00:01:34.019 --> 00:01:37.705 and its value can change depending on the context. 00:01:37.705 --> 00:01:39.946 And this is in the context of an expression. 00:01:39.946 --> 00:01:42.174 You'll also see that in the context of an equation. 00:01:42.174 --> 00:01:44.299 It's actually important to realize the distinction 00:01:44.299 --> 00:01:46.897 between an expression and an equation. 00:01:46.897 --> 00:01:49.827 An expression is really just a statement of value – 00:01:49.827 --> 00:01:51.734 a statement of some type of quantity. 00:01:51.734 --> 00:01:54.327 So this is an expression. 00:01:54.327 --> 00:01:56.639 An expression would be something like. 00:01:56.639 --> 00:01:57.976 well, what we saw over here: 00:01:57.976 --> 00:01:59.260 x + 5 00:01:59.260 --> 00:02:01.052 The value of this expression will change 00:02:01.052 --> 00:02:05.745 depending on what the value of this variable is. 00:02:05.745 --> 00:02:09.058 And you could just evaluate it for different values of x 00:02:09.058 --> 00:02:11.270 Another expression could be something like ... 00:02:11.270 --> 00:02:13.150 I don't know ... y + z. 00:02:13.150 --> 00:02:14.340 Now everything is a variable. 00:02:14.340 --> 00:02:16.554 If y is 1 and z is 2, 00:02:16.554 --> 00:02:18.560 it's going to be 1 + 2. 00:02:18.560 --> 00:02:21.392 If y is 0 and z is -1, 00:02:21.392 --> 00:02:24.068 it's going to be 0 + (-1). NOTE Paragraph 00:02:24.068 --> 00:02:25.897 These can all be evaluated 00:02:25.897 --> 00:02:27.416 and they'll essentially give you a value 00:02:27.416 --> 00:02:30.811 depending on the values of each of these variables 00:02:30.811 --> 00:02:32.327 that make up the expression. 00:02:32.327 --> 00:02:34.285 In an equation, you're essentially setting expressions 00:02:34.285 --> 00:02:35.472 to be equal to each other. 00:02:35.472 --> 00:02:38.100 That's why they're called 'equations.' 00:02:38.100 --> 00:02:40.122 You're equating two things. 00:02:40.122 --> 00:02:42.919 In an equation, you'll see one expression 00:02:42.919 --> 00:02:44.643 being equal to another expression. 00:02:44.643 --> 00:02:47.869 So, for example, you could say something like 00:02:47.869 --> 00:02:52.062 x + 3 = 1. 00:02:52.062 --> 00:02:54.459 And in this situation where you have one equation, 00:02:54.459 --> 00:02:57.883 with only one unknown, 00:02:57.883 --> 00:02:59.273 you could actually figure out 00:02:59.273 --> 00:03:01.622 what x needs to be in this scenario. 00:03:01.622 --> 00:03:03.210 And you could possibly even do it in your head. 00:03:03.210 --> 00:03:05.327 'What' + 3 is equal to 1? ( __ + 3 = 1?) 00:03:05.327 --> 00:03:06.432 Well, you can do that in your head. 00:03:06.432 --> 00:03:08.871 Ff I have -2, -2 + 3 is equal to 1. (-2 +3 = 1) 00:03:08.871 --> 00:03:12.033 So in this context, an equation is starting to constrain 00:03:12.033 --> 00:03:15.134 the value that this variable can take on. 00:03:15.134 --> 00:03:17.411 But it doesn't have necessarily constrain as much. 00:03:17.411 --> 00:03:18.932 You could have something like: 00:03:18.932 --> 00:03:25.734 x + y + z = 5. 00:03:25.734 --> 00:03:27.784 Now – this expression is 00:03:27.784 --> 00:03:29.368 equal to this other expression. 00:03:29.368 --> 00:03:31.645 5 is really just an expression right over here. 00:03:31.645 --> 00:03:32.901 And there are some constraints. 00:03:32.901 --> 00:03:35.004 If someone tells you what y and z is, 00:03:35.004 --> 00:03:36.314 then that constrains what x is. 00:03:36.314 --> 00:03:38.226 If someone tells you what x and y are, 00:03:38.226 --> 00:03:39.925 then that constrains what z is. 00:03:39.925 --> 00:03:42.381 But it depends on what the different things are. 00:03:42.381 --> 00:03:44.060 So for example, 00:03:44.060 --> 00:03:51.637 if we said y = 3, and z = 2, 00:03:51.637 --> 00:03:53.393 then what would x be in that situation? 00:03:53.393 --> 00:03:58.102 So if y = 3, and z =2, 00:03:58.102 --> 00:03:58.608 then you're going to have – 00:03:58.608 --> 00:04:00.487 the left hand expression is going to be 00:04:00.487 --> 00:04:02.148 x + 3 + 2 – 00:04:02.148 --> 00:04:04.998 which is going to be x + 5 – NOTE Paragraph 00:04:04.998 --> 00:04:06.813 This part right over here is going to be 5. 00:04:06.813 --> 00:04:08.975 x + 5 = 5 00:04:08.975 --> 00:04:11.198 And so what + 5 = 5? 00:04:11.198 --> 00:04:12.632 Well now, we're constraining x to be – 00:04:12.632 --> 00:04:14.378 x would have to be – 00:04:14.378 --> 00:04:16.938 x would have to be equal to 0. (x = 0) 00:04:16.938 --> 00:04:18.235 But the important point here – 00:04:18.235 --> 00:04:19.789 1) hopefully, you realize the difference 00:04:19.789 --> 00:04:20.803 between an expression and an equation. 00:04:20.803 --> 00:04:21.850 In an equation, essentially, 00:04:21.850 --> 00:04:23.669 you're equating two expressions. 00:04:23.669 --> 00:04:25.370 The important thing to take away from here, 00:04:25.370 --> 00:04:27.994 is that a variable can take on different values, 00:04:27.994 --> 00:04:31.365 depending on the context of the problem. 00:04:31.365 --> 00:04:32.778 And to hit the point home, 00:04:32.778 --> 00:04:35.218 let‘s just evaluate a bunch of expressions, 00:04:35.218 --> 00:04:38.056 when the variables have different values. 00:04:38.056 --> 00:04:41.595 So for example, if we had the expression 00:04:41.595 --> 00:04:43.309 if we had the expression. 00:04:43.309 --> 00:04:47.799 x to the y power, 00:04:47.799 --> 00:04:51.955 if x is equal to 5, 00:04:51.955 --> 00:04:54.311 and y is equal to 2 00:04:54.311 --> 00:04:55.791 y is equal to 2. 00:04:55.791 --> 00:04:58.908 then our expression here is going to evaluate to – 00:04:58.908 --> 00:05:01.506 Well x is now going to be 5. 00:05:01.506 --> 00:05:02.888 x is going to be 5. 00:05:02.888 --> 00:05:04.363 y is going to be 2. 00:05:04.363 --> 00:05:06.612 it's going to be 5 to the second power. 00:05:06.612 --> 00:05:08.154 or it's going to evaluate to 00:05:08.154 --> 00:05:09.785 25. 00:05:09.785 --> 00:05:11.633 If we change the values – 00:05:11.633 --> 00:05:14.360 If we said x – 00:05:14.360 --> 00:05:16.292 (Let me do it in that same color.) 00:05:16.292 --> 00:05:20.965 If we said x is equal to -2, 00:05:20.965 --> 00:05:24.772 and y is equal to 3, 00:05:24.772 --> 00:05:27.839 then this expression would evaluate to, 00:05:27.839 --> 00:05:30.469 (Let me do in that color.) 00:05:30.469 --> 00:05:32.386 – so it would evaluate to -2. 00:05:32.386 --> 00:05:35.376 (That's what we're going to substitute for x now, NOTE Paragraph 00:05:35.376 --> 00:05:36.705 in this context.) 00:05:36.705 --> 00:05:38.172 – and y is now 3 – 00:05:38.172 --> 00:05:42.080 -2 to the third power – 00:05:42.080 --> 00:05:44.577 which is -2 x -2 x -2, 00:05:44.577 --> 00:05:46.895 which is -8. 00:05:46.895 --> 00:05:48.567 -2 × -2 = +4. 00:05:48.567 --> 00:05:52.154 × -2 again is equal to -8. 00:05:52.154 --> 00:05:53.367 is equal to -8 00:05:53.367 --> 00:05:55.713 So you see, depending on what the values of these are – 00:05:55.713 --> 00:05:58.280 (And we could even do more complex things.) 00:05:58.280 --> 00:05:59.681 We could have an expression like 00:05:59.681 --> 00:06:06.609 "the square root of x + y and then minus x" ... like that. 00:06:06.609 --> 00:06:11.878 If x is equal to – Let's say that x is equal to 1, 00:06:11.878 --> 00:06:16.013 and y is equal to 8, 00:06:16.013 --> 00:06:18.571 then this expression would evaluate to – 00:06:18.571 --> 00:06:21.422 (Well every time we see an x, we want to put a 1 there.) 00:06:21.422 --> 00:06:23.008 – so we would have a 1 there. 00:06:23.008 --> 00:06:24.812 And you would have a 1 over there. 00:06:24.812 --> 00:06:26.746 And every time you would see a y, 00:06:26.746 --> 00:06:28.413 you would put an 8 in its place – 00:06:28.413 --> 00:06:30.819 – in this context. We're setting these variables to specific numbers. 00:06:30.819 --> 00:06:32.087 So you would see an 8. 00:06:32.087 --> 00:06:34.611 So under the radical sign, you would have a 1+8 – 00:06:34.611 --> 00:06:37.821 so you would have the principal root of 9 – which is 3. 00:06:37.821 --> 00:06:40.974 So this whole thing would simplify in this context. 00:06:40.974 --> 00:06:43.119 When we set these variables to be these things, 00:06:43.119 --> 00:06:45.586 this whole thing would simplify to be 3. 00:06:45.586 --> 00:06:46.503 1 + 8 is 9. 00:06:46.503 --> 00:06:48.685 Principal root of that is 3. 00:06:48.685 --> 00:06:50.769 And then you'd have 3 - 1. 00:06:50.769 --> 00:06:54.769 Which is equal to 2.