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