1
00:00:00,816 --> 00:00:02,341
When we're dealing with basic arithmetic

2
00:00:02,341 --> 00:00:04,592
we see the concrete numbers there.

3
00:00:04,592 --> 00:00:07,514
We'll see 23 + 5.

4
00:00:07,514 --> 00:00:08,715
We know what these numbers are right over here

5
00:00:08,715 --> 00:00:10,005
and we can calculate them.

6
00:00:10,005 --> 00:00:11,661
It's going to be 28.

7
00:00:11,661 --> 00:00:13,898
We can say 2 x 7.

8
00:00:13,898 --> 00:00:17,476
We could say 3 / 4.

9
00:00:17,476 --> 00:00:19,059
And all of these cases we know exactly

10
00:00:19,059 --> 00:00:20,872
what numbers we're dealing with.

11
00:00:20,872 --> 00:00:23,776
As we start entering into the Algebratic world

12
00:00:23,776 --> 00:00:25,873
and you probably have seen this a little bit already.

13
00:00:25,873 --> 00:00:30,051
We start dealing with the ideas of variables.

14
00:00:30,051 --> 00:00:31,533
And variables, there's a bunch of ways

15
00:00:31,533 --> 00:00:32,283
you can think about them

16
00:00:32,283 --> 00:00:34,502
but they're really just values and expressions

17
00:00:34,502 --> 00:00:36,252
where they can change.

18
00:00:36,252 --> 00:00:38,145
The values in those expressions can change.

19
00:00:38,145 --> 00:00:42,201
So for example, if I write, if I write,

20
00:00:42,201 --> 00:00:44,781
x + 5

21
00:00:44,781 --> 00:00:46,647
this is an expression right over here.

22
00:00:46,647 --> 00:00:48,305
This can take on some value

23
00:00:48,305 --> 00:00:51,466
depending on what the value of x is.

24
00:00:51,466 --> 00:00:56,656
If x is equal to, so if x is equal to 1,

25
00:00:56,656 --> 00:01:01,723
then x + 5, our expression right over here.

26
00:01:01,723 --> 00:01:06,049
Is going to be equal to 1.

27
00:01:06,049 --> 00:01:07,070
Because now x is 1.

28
00:01:07,070 --> 00:01:08,321
It'll be 1 + 5.

29
00:01:08,321 --> 00:01:11,101
So x + 5 will be equal to 6.

30
00:01:11,101 --> 00:01:16,821
If x is equal to, I don't know, -7

31
00:01:16,821 --> 00:01:22,183
then x plus 5, is going to be equal to,

32
00:01:22,183 --> 00:01:24,120
well now x is -7.

33
00:01:24,120 --> 00:01:28,842
It's going to be -7 + 5 which is negative 2.

34
00:01:28,842 --> 00:01:29,441
So notice.

35
00:01:29,441 --> 00:01:34,019
x here is a variable, x here is the variable,

36
00:01:34,019 --> 00:01:37,705
and its value can change depending on the context.

37
00:01:37,705 --> 00:01:39,946
And this is in the context of an expression.

38
00:01:39,946 --> 00:01:42,174
You'll also see that in the context of an equation.

39
00:01:42,174 --> 00:01:44,299
It's actually important to realize this distinction

40
00:01:44,299 --> 00:01:46,897
between the expression and an equation.

41
00:01:46,897 --> 00:01:49,827
An expression is really just a statement of values

42
00:01:49,827 --> 00:01:51,734
a statement of some type of quantity.

43
00:01:51,734 --> 00:01:54,327
So this is an expression.

44
00:01:54,327 --> 00:01:56,639
An expression would be something like.

45
00:01:56,639 --> 00:01:57,976
Well, what we saw over here.

46
00:01:57,976 --> 00:01:59,260
x + 5

47
00:01:59,260 --> 00:02:01,052
the value of this expression will change

48
00:02:01,052 --> 00:02:05,745
depending on what the value of this variable is.

49
00:02:05,745 --> 00:02:09,058
And you could just evaluate for different values of x

50
00:02:09,058 --> 00:02:11,270
Another expression could be something like

51
00:02:11,270 --> 00:02:13,150
I don't know y + z.

52
00:02:13,150 --> 00:02:14,340
Now everything is a variable.

53
00:02:14,340 --> 00:02:16,554
If y is 1 and z is 2,

54
00:02:16,554 --> 00:02:18,560
it's going to be 1 + 2.

55
00:02:18,560 --> 00:02:21,392
If y is 0 and z is -1

56
00:02:21,392 --> 00:02:24,068
it's going to be 0 + (-1).

57
00:02:24,068 --> 00:02:25,897
These can all be evaluated

58
00:02:25,897 --> 00:02:27,416
and they'll essentially give you a value

59
00:02:27,416 --> 00:02:30,811
depending on the values of each of these variables

60
00:02:30,811 --> 00:02:32,327
that make up the expression.

61
00:02:32,327 --> 00:02:34,285
An equation, you're essentially setting expressions

62
00:02:34,285 --> 00:02:35,472
to be equal to each other.

63
00:02:35,472 --> 00:02:38,100
that's why they're called 'equations'

64
00:02:38,100 --> 00:02:40,122
you're equating two things.

65
00:02:40,122 --> 00:02:42,919
an equation you'll see one expression

66
00:02:42,919 --> 00:02:44,643
being equal to another expression.

67
00:02:44,643 --> 00:02:47,869
So for example, you could say something like...

68
00:02:47,869 --> 00:02:52,062
x + 3 = 1

69
00:02:52,062 --> 00:02:54,459
and in this situation where you have an equation

70
00:02:54,459 --> 00:02:57,883
where you have one equation with only one unknown.

71
00:02:57,883 --> 00:02:59,273
You could actually figure out

72
00:02:59,273 --> 00:03:01,622
what x needs to be in this scenario.

73
00:03:01,622 --> 00:03:03,210
and you could might even do it in your heads.

74
00:03:03,210 --> 00:03:05,327
What plus 3 is equal to 1?

75
00:03:05,327 --> 00:03:06,432
well you can do that in your head.

76
00:03:06,432 --> 00:03:08,871
if I have -2 + 3 is equal to 1.

77
00:03:08,871 --> 00:03:12,033
so in this context an equation is starting to constrain

78
00:03:12,033 --> 00:03:15,134
what value that this variable can take on.

79
00:03:15,134 --> 00:03:17,411
but it doesn't have necessarily constrain as much.

80
00:03:17,411 --> 00:03:18,932
You could have something like,

81
00:03:18,932 --> 00:03:25,734
x + y + z = 5

82
00:03:25,734 --> 00:03:27,784
now you have this expression that is

83
00:03:27,784 --> 00:03:29,368
equal to this other expression.

84
00:03:29,368 --> 00:03:31,645
5 is really just an expression right over here.

85
00:03:31,645 --> 00:03:32,901
and there are some constraints.

86
00:03:32,901 --> 00:03:35,004
If someone tells you what y and z is

87
00:03:35,004 --> 00:03:36,314
and you're going to get an x.

88
00:03:36,314 --> 00:03:38,226
If someone tells you what x and y is

89
00:03:38,226 --> 00:03:39,925
and that constrains what z is.

90
00:03:39,925 --> 00:03:42,381
But it depends on what the different things are.

91
00:03:42,381 --> 00:03:44,060
So for example

92
00:03:44,060 --> 00:03:51,637
if we said y = 3, and z = 2

93
00:03:51,637 --> 00:03:53,393
then what would be x in that situation?

94
00:03:53,393 --> 00:03:58,102
so if y = 3, and z =2

95
00:03:58,102 --> 00:03:58,608
then you're going to have

96
00:03:58,608 --> 00:04:00,487
the left hand expression is going to be

97
00:04:00,487 --> 00:04:02,148
x + 3 + 2

98
00:04:02,148 --> 00:04:04,998
is going to be x + 5

99
00:04:04,998 --> 00:04:06,813
this part right over here is going to be 5

100
00:04:06,813 --> 00:04:08,975
x + 5 = 5

101
00:04:08,975 --> 00:04:11,198
and so what + 5 = 5?

102
00:04:11,198 --> 00:04:12,632
well now we're constraining that

103
00:04:12,632 --> 00:04:14,378
x would have to be...

104
00:04:14,378 --> 00:04:16,938
x would have to be equal to 0

105
00:04:16,938 --> 00:04:18,235
But the important point here, one

106
00:04:18,235 --> 00:04:19,789
you hope you realize the difference

107
00:04:19,789 --> 00:04:20,803
between an expression and an equation.

108
00:04:20,803 --> 00:04:21,850
an Equation is essentially

109
00:04:21,850 --> 00:04:23,669
you're equating two expressions.

110
00:04:23,669 --> 00:04:25,370
the important thing to take away from here,

111
00:04:25,370 --> 00:04:27,994
is that a variable can take on different values

112
00:04:27,994 --> 00:04:31,365
depending on the context of the problem.

113
00:04:31,365 --> 00:04:32,778
And to hit the point home,

114
00:04:32,778 --> 00:04:35,218
let‘s just evaluate a bunch of expressions,

115
00:04:35,218 --> 00:04:38,056
when the variables have different values.

116
00:04:38,056 --> 00:04:41,595
So for example, if we had the expression

117
00:04:41,595 --> 00:04:43,309
if we had the expression,

118
00:04:43,309 --> 00:04:47,799
x to the... x to the y power

119
00:04:47,799 --> 00:04:51,955
if x is equal to... if x is equal to 5

120
00:04:51,955 --> 00:04:54,311
and y is equal to 2

121
00:04:54,311 --> 00:04:55,791
y is equal to 2.

122
00:04:55,791 --> 00:04:58,908
then our expression here is going to evaluate to...

123
00:04:58,908 --> 00:05:01,506
well x is now going to be 5.

124
00:05:01,506 --> 00:05:02,888
x is going to be 5.

125
00:05:02,888 --> 00:05:04,363
y is going to be 2.

126
00:05:04,363 --> 00:05:06,612
it's going to be 5 to the second power.

127
00:05:06,612 --> 00:05:08,154
or it's going to evaluate to

128
00:05:08,154 --> 00:05:09,785
25.

129
00:05:09,785 --> 00:05:11,633
If we change the values,

130
00:05:11,633 --> 00:05:14,360
if we said, x... if we said,

131
00:05:14,360 --> 00:05:16,292
let me do that in the same colour.

132
00:05:16,292 --> 00:05:20,965
If we said x is equal to... x is equal to -2

133
00:05:20,965 --> 00:05:24,772
and y... and y is equal to 3

134
00:05:24,772 --> 00:05:27,839
then this expression would evaluate to,

135
00:05:27,839 --> 00:05:30,469
it would evaluate to, let me do in that (colour)

136
00:05:30,469 --> 00:05:32,386
so it would evaluate to

137
00:05:32,386 --> 00:05:35,376
-2, that's what we're going to substitute for x now

138
00:05:35,376 --> 00:05:36,705
in this context.

139
00:05:36,705 --> 00:05:38,172
and y is now 3

140
00:05:38,172 --> 00:05:42,080
-2 to the third power... -2 to the third power

141
00:05:42,080 --> 00:05:44,577
which is -2 x -2 x -2

142
00:05:44,577 --> 00:05:46,895
which is -8

143
00:05:46,895 --> 00:05:48,567
-2 x -2 = +4

144
00:05:48,567 --> 00:05:52,154
x -2 again is equal to -8

145
00:05:52,154 --> 00:05:53,367
is equal to -8

146
00:05:53,367 --> 00:05:55,713
so you see depending on what the values of these are

147
00:05:55,713 --> 00:05:58,280
you know we could even do more complex things.

148
00:05:58,280 --> 00:05:59,681
We could have an expression like,

149
00:05:59,681 --> 00:06:06,609
the square root of x + y and then minus x, like that.

150
00:06:06,609 --> 00:06:11,878
If x is equal to, let's say that x is equal to 1

151
00:06:11,878 --> 00:06:16,013
and y... y is equal to 8

152
00:06:16,013 --> 00:06:18,571
then this expression would evaluate to

153
00:06:18,571 --> 00:06:21,422
well every time we see an x we wanna put a 1 there.

154
00:06:21,422 --> 00:06:23,008
so we would have a 1 there.

155
00:06:23,008 --> 00:06:24,812
and you would have a 1 over there.

156
00:06:24,812 --> 00:06:26,746
And every time you would see a y,

157
00:06:26,746 --> 00:06:28,413
you would put an 8 in its place.

158
00:06:28,413 --> 00:06:30,819
And in this context, we're setting these variables

159
00:06:30,819 --> 00:06:32,087
so you would see an 8.

160
00:06:32,087 --> 00:06:34,611
So under the radical sign you would have a 1+8.

161
00:06:34,611 --> 00:06:37,821
So you would have the principal root of 9, which is 3

162
00:06:37,821 --> 00:06:40,974
So this whole thing would simplify in this context.

163
00:06:40,974 --> 00:06:43,119
When we set these variables to be these things,

164
00:06:43,119 --> 00:06:45,586
this whole thing would simplify to be 3.

165
00:06:45,586 --> 00:06:46,503
1 plus 8 is 9

166
00:06:46,503 --> 00:06:48,685
principal root of that is 3

167
00:06:48,685 --> 00:06:50,769
and then you would have 3 - 1

168
00:06:50,769 --> 99:59:59,999
which is equal to... which is equal to 2.