< Return to Video

Variables Expressions and Equations

  • 0:01 - 0:02
    When we're dealing with basic arithmetic,
  • 0:02 - 0:05
    we see the concrete numbers there.
  • 0:05 - 0:07
    We'll see 23 + 5.
  • 0:07 - 0:09
    We know what these numbers are right over here
  • 0:09 - 0:10
    and we can calculate them.
  • 0:10 - 0:12
    It's going to be 28.
  • 0:12 - 0:14
    We can say 2 x 7.
  • 0:14 - 0:17
    We could say 3 divided by 4 (3 / 4).
  • 0:17 - 0:19
    In all of these cases, we know exactly
  • 0:19 - 0:21
    what numbers we're dealing with.
  • 0:21 - 0:24
    As we start entering into the algebratic world –
  • 0:24 - 0:26
    (And you probably have seen this a little bit already.)
  • 0:26 - 0:30
    – we start dealing with the idea of variables.
  • 0:30 - 0:32
    And variables, there are a bunch of ways
  • 0:32 - 0:32
    you can think about them.
  • 0:32 - 0:35
    but they're really just values and expressions
  • 0:35 - 0:36
    where they can change.
  • 0:36 - 0:38
    The values in those expressions can change.
  • 0:38 - 0:42
    So for example, if I write
  • 0:42 - 0:45
    'x + 5.'
  • 0:45 - 0:47
    this is an expression right over here.
  • 0:47 - 0:48
    This can take on some value,
  • 0:48 - 0:51
    depending on what the value of x is.
  • 0:51 - 0:57
    If x is equal to 1,
  • 0:57 - 1:02
    then x + 5 – our expression right over here –
  • 1:02 - 1:06
    Is going to be equal to 1 –
  • 1:06 - 1:07
    because now x is 1.
  • 1:07 - 1:08
    It'll be 1 + 5.
  • 1:08 - 1:11
    So x + 5 will be equal to 6. (x + 5 = 6)
  • 1:11 - 1:17
    If x is equal to, I don't know, -7, (x = -7)
  • 1:17 - 1:22
    then x + 5, is going to be equal to –
  • 1:22 - 1:24
    Well now x is -7.
  • 1:24 - 1:29
    It's going to be -7 + 5, which is -2.
  • 1:29 - 1:29
    So notice.
  • 1:29 - 1:34
    x here is a variable, x here is the variable,
  • 1:34 - 1:38
    and its value can change depending on the context.
  • 1:38 - 1:40
    And this is in the context of an expression.
  • 1:40 - 1:42
    You'll also see that in the context of an equation.
  • 1:42 - 1:44
    It's actually important to realize the distinction
  • 1:44 - 1:47
    between an expression and an equation.
  • 1:47 - 1:50
    An expression is really just a statement of value –
  • 1:50 - 1:52
    a statement of some type of quantity.
  • 1:52 - 1:54
    So this is an expression.
  • 1:54 - 1:57
    An expression would be something like.
  • 1:57 - 1:58
    well, what we saw over here:
  • 1:58 - 1:59
    x + 5
  • 1:59 - 2:01
    The value of this expression will change
  • 2:01 - 2:06
    depending on what the value of this variable is.
  • 2:06 - 2:09
    And you could just evaluate it for different values of x
  • 2:09 - 2:11
    Another expression could be something like ...
  • 2:11 - 2:13
    I don't know ... y + z.
  • 2:13 - 2:14
    Now everything is a variable.
  • 2:14 - 2:17
    If y is 1 and z is 2,
  • 2:17 - 2:19
    it's going to be 1 + 2.
  • 2:19 - 2:21
    If y is 0 and z is -1,
  • 2:21 - 2:24
    it's going to be 0 + (-1).
  • 2:24 - 2:26
    These can all be evaluated
  • 2:26 - 2:27
    and they'll essentially give you a value
  • 2:27 - 2:31
    depending on the values of each of these variables
  • 2:31 - 2:32
    that make up the expression.
  • 2:32 - 2:34
    In an equation, you're essentially setting expressions
  • 2:34 - 2:35
    to be equal to each other.
  • 2:35 - 2:38
    That's why they're called 'equations.'
  • 2:38 - 2:40
    You're equating two things.
  • 2:40 - 2:43
    In an equation, you'll see one expression
  • 2:43 - 2:45
    being equal to another expression.
  • 2:45 - 2:48
    So, for example, you could say something like
  • 2:48 - 2:52
    x + 3 = 1.
  • 2:52 - 2:54
    And in this situation where you have one equation,
  • 2:54 - 2:58
    with only one unknown,
  • 2:58 - 2:59
    you could actually figure out
  • 2:59 - 3:02
    what x needs to be in this scenario.
  • 3:02 - 3:03
    And you could possibly even do it in your head.
  • 3:03 - 3:05
    'What' + 3 is equal to 1? ( __ + 3 = 1?)
  • 3:05 - 3:06
    Well, you can do that in your head.
  • 3:06 - 3:09
    Ff I have -2, -2 + 3 is equal to 1. (-2 +3 = 1)
  • 3:09 - 3:12
    So in this context, an equation is starting to constrain
  • 3:12 - 3:15
    the value that this variable can take on.
  • 3:15 - 3:17
    But it doesn't have necessarily constrain as much.
  • 3:17 - 3:19
    You could have something like:
  • 3:19 - 3:26
    x + y + z = 5.
  • 3:26 - 3:28
    Now – this expression is
  • 3:28 - 3:29
    equal to this other expression.
  • 3:29 - 3:32
    5 is really just an expression right over here.
  • 3:32 - 3:33
    And there are some constraints.
  • 3:33 - 3:35
    If someone tells you what y and z is,
  • 3:35 - 3:36
    then that constrains what x is.
  • 3:36 - 3:38
    If someone tells you what x and y are,
  • 3:38 - 3:40
    then that constrains what z is.
  • 3:40 - 3:42
    But it depends on what the different things are.
  • 3:42 - 3:44
    So for example,
  • 3:44 - 3:52
    if we said y = 3, and z = 2,
  • 3:52 - 3:53
    then what would x be in that situation?
  • 3:53 - 3:58
    So if y = 3, and z =2,
  • 3:58 - 3:59
    then you're going to have –
  • 3:59 - 4:00
    the left hand expression is going to be
  • 4:00 - 4:02
    x + 3 + 2 –
  • 4:02 - 4:05
    which is going to be x + 5 –
  • 4:05 - 4:07
    This part right over here is going to be 5.
  • 4:07 - 4:09
    x + 5 = 5
  • 4:09 - 4:11
    And so what + 5 = 5?
  • 4:11 - 4:13
    Well now, we're constraining x to be –
  • 4:13 - 4:14
    x would have to be –
  • 4:14 - 4:17
    x would have to be equal to 0. (x = 0)
  • 4:17 - 4:18
    But the important point here –
  • 4:18 - 4:20
    1) hopefully, you realize the difference
  • 4:20 - 4:21
    between an expression and an equation.
  • 4:21 - 4:22
    In an equation, essentially,
  • 4:22 - 4:24
    you're equating two expressions.
  • 4:24 - 4:25
    The important thing to take away from here,
  • 4:25 - 4:28
    is that a variable can take on different values,
  • 4:28 - 4:31
    depending on the context of the problem.
  • 4:31 - 4:33
    And to hit the point home,
  • 4:33 - 4:35
    let‘s just evaluate a bunch of expressions,
  • 4:35 - 4:38
    when the variables have different values.
  • 4:38 - 4:42
    So for example, if we had the expression
  • 4:42 - 4:43
    if we had the expression.
  • 4:43 - 4:48
    x to the y power,
  • 4:48 - 4:52
    if x is equal to 5,
  • 4:52 - 4:54
    and y is equal to 2
  • 4:54 - 4:56
    y is equal to 2.
  • 4:56 - 4:59
    then our expression here is going to evaluate to –
  • 4:59 - 5:02
    Well x is now going to be 5.
  • 5:02 - 5:03
    x is going to be 5.
  • 5:03 - 5:04
    y is going to be 2.
  • 5:04 - 5:07
    it's going to be 5 to the second power.
  • 5:07 - 5:08
    or it's going to evaluate to
  • 5:08 - 5:10
    25.
  • 5:10 - 5:12
    If we change the values –
  • 5:12 - 5:14
    If we said x –
  • 5:14 - 5:16
    (Let me do it in that same color.)
  • 5:16 - 5:21
    If we said x is equal to -2,
  • 5:21 - 5:25
    and y is equal to 3,
  • 5:25 - 5:28
    then this expression would evaluate to,
  • 5:28 - 5:30
    (Let me do in that color.)
  • 5:30 - 5:32
    – so it would evaluate to -2.
  • 5:32 - 5:35
    (That's what we're going to substitute for x now,
  • 5:35 - 5:37
    in this context.)
  • 5:37 - 5:38
    – and y is now 3 –
  • 5:38 - 5:42
    -2 to the third power –
  • 5:42 - 5:45
    which is -2 x -2 x -2,
  • 5:45 - 5:47
    which is -8.
  • 5:47 - 5:49
    -2 × -2 = +4.
  • 5:49 - 5:52
    × -2 again is equal to -8.
  • 5:52 - 5:53
    is equal to -8
  • 5:53 - 5:56
    So you see, depending on what the values of these are –
  • 5:56 - 5:58
    (And we could even do more complex things.)
  • 5:58 - 6:00
    We could have an expression like
  • 6:00 - 6:07
    "the square root of x + y and then minus x" ... like that.
  • 6:07 - 6:12
    If x is equal to –
    Let's say that x is equal to 1,
  • 6:12 - 6:16
    and y is equal to 8,
  • 6:16 - 6:19
    then this expression would evaluate to –
  • 6:19 - 6:21
    (Well every time we see an x, we want to put a 1 there.)
  • 6:21 - 6:23
    – so we would have a 1 there.
  • 6:23 - 6:25
    And you would have a 1 over there.
  • 6:25 - 6:27
    And every time you would see a y,
  • 6:27 - 6:28
    you would put an 8 in its place –
  • 6:28 - 6:31
    – in this context.
    We're setting these variables to specific numbers.
  • 6:31 - 6:32
    So you would see an 8.
  • 6:32 - 6:35
    So under the radical sign, you would have a 1+8 –
  • 6:35 - 6:38
    so you would have the principal root of 9 – which is 3.
  • 6:38 - 6:41
    So this whole thing would simplify in this context.
  • 6:41 - 6:43
    When we set these variables to be these things,
  • 6:43 - 6:46
    this whole thing would simplify to be 3.
  • 6:46 - 6:47
    1 + 8 is 9.
  • 6:47 - 6:49
    Principal root of that is 3.
  • 6:49 - 6:51
    And then you'd have 3 - 1.
  • 6:51 - 6:55
    Which is equal to 2.
Title:
Variables Expressions and Equations
Description:

Introduction and examples of variables, expressions and equations

more » « less
Video Language:
English
Duration:
06:55

English subtitles

Revisions