Evaluating expressions with unknown variables
Evaluating expressions where individual variable values are unknown Working through a few examples where we need to evaluate an expression, but we don't know the individual variable values
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- I'm now going to do a few classic problems
- that show up a lot at math competetions
- and sometimes on standardized tests,
- and they seem like very daunting
- and intimidating problems, but hopefully
- over the course of this video,
- you'll realize that
- if you kind of see what they're asking for,
- it's really not that bad.
- So let's try this first one right over here.
- They tell us
- a
- a +
- a + b
- a + b +
- a + b + c
- a + b + c =
- a + b + c = 7
- and then they ask us, what is
- 5a
- 5a +
- 5a + 5
- 5a + 5b
- 5a + 5b +
- 5a + 5b + 5
- 5a + 5b + 5c equal to?
- And the first reaction is like,
- "Well, you just gave me one equation right over here,
- one equation with three unknowns,
- how do I solve for a, b, or c?
- Don't I have to be able to solve for
- a, b, or c in order to figure out
- what 5a + 5b + 5c is?"
- So I'll let you think about that for a second.
- So, the big idea here
- is to realize that 5a + 5b + 5c
- that's just the same thing as
- 5
- 5(
- 5(a
- 5(a +
- 5(a + b
- 5(a + b +
- 5(a + b + c)
- If you distribute the 5 here,
- 5 times a,
- 5 times b,
- 5 times c,
- you get 5a + 5b + 5c,
- another way you can think of it is
- we are factoring a 5 out.
- If you factor a 5 out,
- you are left with 5(a + b + c)
- Now, how do we evaluate this?
- Well, that first equation gave us all
- the information that we needed.
- They told us that a + b + c
- They told us that a + b + c = 7
- a + b + c
- a + b + c = 7
- so 5(a + b + c) is the same exact thing,
- and I think you see
- where this is going now,
- is 5(
- is 5(7)
- is 5(7), and now this becomes
- pretty straightforward,
- and that's going to be equal to 35.
- Let's do one more of these,
- and this is going to be
- a little bit more involved,
- but hopefully you see the same idea.
- So we're told,
- a
- a +
- a + b
- a + b +
- a + b + c
- a + b + c is equal to -1
- And we have two more variables here,
- we're told that x
- we're told that x + y
- we're told that x + y is equal to 7
- and then they ask us this big hairy thing,
- what is this equal to?
- And I'll give you a few seconds
- to give it a little bit of thought.
- Well, you can imagine,
- you're given three variables
- with one equation up here,
- another two variables with another equation,
- we have five unknowns with two equations,
- there's no way we're going
- to be able to individually figure out what
- a
- a or
- a or b
- a or b or
- a or b or c
- a or b or c or
- a or b or c or x
- a or b or c or x or
- a or b or c or x or y
- a or b or c or x or y is,
- but maybe we can use
- what we saw in the last example
- to solve this.
- And so what we might want to do
- is rearrange it so that we have the
- xs and ys kind of grouped together,
- and the as, bs, and cs
- kinda grouped together.
- So let's try that out.
- Let's focus first on the
- as,
- as, bs,
- as, bs, and
- as, bs, and cs
- So we have -9a, -9b --
- I'll go in alphabetical order --
- -9b and -9c, and I think you see
- what's emerging.
- And then let's work on the xs and the ys.
- And then we have -7x, and
- then we have a -7y
- So, all I have done is rearranged
- this expression here,
- but this makes it, in a way,
- a little bit clearer of what's
- going on here.
- These first three terms,
- I can factor our a -9,
- and I get -9
- and I get -9(a
- and I get -9(a +
- and I get -9(a + b
- and I get -9(a + b +
- and I get -9(a + b + c)
- and these second two terms,
- I can factor out a 7.
- -7
- -7(
- -7(x
- -7(x +
- -7(x + y)
- and just to verify,
- if you wanted to go the other way,
- multiply this -7 times (x + y)
- and you'll get this
- Multiply -9 times (a + b + c),
- (distribute it)
- you'll get this right over here.
- And so this makes it a little bit clearer.
- What is (a + b + c) equal to?
- Well they tell us right over here,
- a + b + c is equal to -1
- this whole expression is -1,
- at least in the parentheses,
- and what is (x + y) equal to?
- (Let me do this in a new color)
- What is (x + y) equal to?
- Well, they tell us right over here,
- x + y is equal to 7
- So this whole thing simplifies to
- -
- -9
- -9(
- -9(-
- -9(-1)
- -9(-1) -
- -9(-1) - 7
- -9(-1) - 7(
- -9(-1) - 7(7)
- and so this is equal to--
- we're in the home stretch--
- -9(-1), well that's positive 9,
- and then -7(7) is negative 49
- so it's 9 - 49,
- which is equal to
- which is equal too
- which is equal tooo
- which is equal toooo
- which is equal toooo!
- negative 40.
- And we are done.
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