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When we're dealing with basic arithmetic,

2
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we see the concrete numbers there.

3
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We'll see 23 + 5.

4
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We know what these numbers are right over here

5
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and we can calculate them.

6
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It's going to be 28.

7
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We can say 2 x 7.

8
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We could say 3 divided by 4 (3 / 4).

9
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In all of these cases, we know exactly

10
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what numbers we're dealing with.

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As we start entering into the algebratic world –

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(And you probably have seen this a little bit already.)

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– we start dealing with the idea of variables.

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And variables, there are a bunch of ways

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you can think about them.

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but they're really just values and expressions

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where they can change.

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The values in those expressions can change.

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So for example, if I write

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'x + 5.'

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this is an expression right over here.

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This can take on some value,

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depending on what the value of x is.

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If x is equal to 1,

25
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then x + 5 – our expression right over here –

26
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Is going to be equal to 1 –

27
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because now x is 1.

28
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It'll be 1 + 5.

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So x + 5 will be equal to 6. (x + 5 = 6)

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If x is equal to, I don't know, -7, (x = -7)

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then x + 5, is going to be equal to –

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Well now x is -7.

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It's going to be -7 + 5, which is -2.

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So notice.

35
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x here is a variable, x here is the variable,

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and its value can change depending on the context.

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And this is in the context of an expression.

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You'll also see that in the context of an equation.

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It's actually important to realize the distinction

40
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between an expression and an equation.

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An expression is really just a statement of value –

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a statement of some type of quantity.

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So this is an expression.

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An expression would be something like.

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well, what we saw over here:

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x + 5

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The value of this expression will change

48
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depending on what the value of this variable is.

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And you could just evaluate it for different values of x

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Another expression could be something like ...

51
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I don't know ... y + z.

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Now everything is a variable.

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If y is 1 and z is 2,

54
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it's going to be 1 + 2.

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If y is 0 and z is -1,

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it's going to be 0 + (-1).

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These can all be evaluated

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and they'll essentially give you a value

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depending on the values of each of these variables

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that make up the expression.

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In an equation, you're essentially setting expressions

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to be equal to each other.

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That's why they're called 'equations.'

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You're equating two things.

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In an equation, you'll see one expression

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being equal to another expression.

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So, for example, you could say something like

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x + 3 = 1.

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And in this situation where you have one equation,

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with only one unknown,

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you could actually figure out

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what x needs to be in this scenario.

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And you could possibly even do it in your head.

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'What' + 3 is equal to 1? ( __ + 3 = 1?)

75
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Well, you can do that in your head.

76
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Ff I have -2, -2 + 3 is equal to 1. (-2 +3 = 1)

77
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So in this context, an equation is starting to constrain

78
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the value that this variable can take on.

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But it doesn't have necessarily constrain as much.

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You could have something like:

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x + y + z = 5.

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Now – this expression is

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equal to this other expression.

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5 is really just an expression right over here.

85
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And there are some constraints.

86
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If someone tells you what y and z is,

87
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then that constrains what x is.

88
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If someone tells you what x and y are,

89
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then that constrains what z is.

90
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But it depends on what the different things are.

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So for example,

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if we said y = 3, and z = 2,

93
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then what would x be in that situation?

94
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So if y = 3, and z =2,

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

96
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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
which is going to be x + 5 –

99
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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 x to be –

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. (x = 0)

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

106
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1) hopefully, 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
In an equation, essentially,

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

110
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The important thing to take away from here,

111
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is that a variable can take on different values,

112
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depending on the context of the problem.

113
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And to hit the point home,

114
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let‘s just evaluate a bunch of expressions,

115
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when the variables have different values.

116
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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 y power,

119
00:04:47,799 --> 00:04:51,955
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
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If we said x –

131
00:05:14,360 --> 00:05:16,292
(Let me do it in that same color.)

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

133
00:05:20,965 --> 00:05:24,772
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
(Let me do in that color.)

136
00:05:30,469 --> 00:05:32,386
– so it would evaluate to -2.

137
00:05:32,386 --> 00:05:35,376
(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 –

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 × -2 = +4.

144
00:05:48,567 --> 00:05:52,154
× -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
(And 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 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 want to 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
– in this context.
We're setting these variables to specific numbers.

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 + 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'd have 3 - 1.

168
00:06:50,769 --> 00:06:54,769
Which is equal to 2.