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# Examples of evaluating variable expressions

Created by Sal Khan and CK-12 Foundation.

Video transcript

Let's do some practice problems
dealing with variable expressions. So these first problems say
write the following in a more condensed form by leaving out
the multiplication symbol or leaving out a multiplication
symbol. So here we have 2 times 11x, so
if we have 11 x's and then we're going to have 2 times
those 11 x's, we're going to have 22 x's. So another way you could view
this, 2 times 11x, you could view this as being equal to 2
times 11, and all of that times x, and that's going
to be equal to 22 x's. You had 11 x's, you're going to
have 2 times as many x's, so you're going to
have 22 x's. Let's see, you have
1.35 times y. Now here we're just going to do
a straight simplifying how we write it. So 1.35 times y-- I'll do it
in a different color-- 1.35 times y-- that's a
little dot there. In algebra we can just get
rid of that dot symbol. If we have a variable following
a number, we know that means 1.35 times
that variable. So that, we could rewrite as
just being equal to 1.35y. We've condensed it by
getting rid of the multiplication sign. Let's see, here we
have 3 times 1/4. Well, this is just straight
up multiplying a fraction. So in problem 3-- this was
problem 1, this is problem 2, problem 3-- 3 times 1/4, that's
the same thing as 3 over 1 times 1/4. Multiply the numerators,
you get 3. Multiply the denominators,
1 times 4, you get 4. So number 3, I got 3/4. And then finally, you
have 1/4 times z. We could do the exact same
thing we did up here in problem number 2. This was the same
thing as 1.35y. That's the same thing
as 1.35 times y. So down here we could rewrite
this as either being equal to 1/4z, or we could view this as
being equal to 1 over 4 times z over 1, which is the same
thing as z times 1, over 4 times 1, or the same
thing as z over 4. So all of these are
equivalent. Now, what do they want
us to do down here? Evaluate the following
expressions for a is equal to 3, b is equal to 2, c is equal
to 5, and d is equal to minus 4-- or, actually, I should say
negative 4 is the correct terminology. Negative 4. So we just substitute. Every time we see an a, we're
going to put a minus 3 there, or a negative 3 there. Every time we see a b, we'll
put a positive 2 there. Every time we see a c,
we'll put a 5 there. And every time we see a d, we'll
put a minus 4 there. And I'll do a couple of these. I won't do all of them, just
for the sake of time. So let's say problem number 5. They gave us 2 times
a plus 3 times b. Well, this is the same thing as
2 times-- instead of an a, we know that a is going to
be equal to negative 3. So 2 times minus 3, plus
3 times b-- what's b? They're telling us that b is
equal to 2-- so 3 times 2. And what is this equal to? 2 times minus 3-- let me do it
in a different color-- 2 times negative 3 is negative
6, plus 3 times 2. 3 times 2 is 6. That's positive 6. So that is equal to 0. And notice the order
of operations. We did the multiplications, we
did the two multiplications before we added the
two numbers. Multiplication and division
takes precedence over addition and subtraction. Let's do problem 6. I'll do that right here. So you have 4 times c. 4 times-- now what's
c equal to? They tell us c is equal to 5. So 4 times 5, that's
our c, plus d. d is minus or negative 4. So we have 4 times 5 is 20, plus
negative 4-- that's the same thing as minus 4, so
that is equal to 16. Problem 6. Now, let's do one of the
harder ones down here. This problem 10 looks a little
bit more daunting. Problem 10 right there. So we have a minus 4b in the
numerator, if you can read it, it's kind of small. a is minus 3. So we have minus 3-- or negative
3-- minus 4 times b. b is 2. So 4 times 2. Remember, this right here is
a, that right there is b. How do I know? They're telling me up here. And then all of that over--
all of that is over 3c plus 2d. So 3 times-- what was c?
c is 5 plus 2 times d. What is d? d is negative 4. So let's figure this out. So we have to do order
of operations. Multiplication comes first
before addition and subtraction. So this is going to be equal
to minus 3 minus 4 times 2, minus 8, all of that over-- 3
times 5 is 15, plus 2 times negative 4 is negative 8,
or 15 plus negative 8 is 15 minus 8. And now, the numerator becomes
negative 3 minus 8, which is negative 11. And the denominator is 15
minus 8, which is 7. So problem 10, we simplified
it to negative 11 over 7. Right there. Let's do a couple of
these over here. OK, we see some exponents. I'll pick one of the
harder ones. Let's do this one over
here, problem 18. So 2x squared minus 3x squared,
plus 5x minus 4. OK, well, this wasn't
that hard. All of them are dealing
with x. But what we could do here--
let me write this down. 2x squared minus 3x squared,
plus 5x minus 4. And they tell us that x is
equal to negative 1. One thing we could do is
simplify this before we even substitute for negative 1. So what's 2 of something
minus 3 of something? Right? This is 2x squareds
minus 3x squareds. So 2 of something minus 3 of
something, that's going to be minus 1 of that something. So that right there-- or
negative 1 of that something-- that would be negative 1x
squared plus 5x minus 4. And they tell us x is
equal to negative 1. So this is negative 1 times x
squared, negative 1 squared, plus 5 times x, which is
negative 1, minus 4. So what is this? Negative 1 squared is just 1. That's just 1. So this whole expression
simplifies to negative 1 plus 5 times negative 1-- we do the multiplication first, of course. So that's minus 5, or
negative 5 minus 4. So negative 1 minus 5 is
negative 6, minus 4 is equal to negative 10. And I'll do these last two just
to get a sample of all of the types of problems in this
variable expression section. The weekly cost, c, of
manufacturing x remote controls-- so the cost is c, x
is the remote controls-- is given by this formula. The cost is equal to 2,000 plus
3 times the number of remote controls, where cost
is given in dollars. Question a, what is the
cost of producing 1,000 remote controls? Well, the number of remote
controls is x. So for part a-- I could write
it over here-- the cost is going to be equal to-- just use
this formula-- 2,000 plus 3 times the number of
remote controls. x is the number of
remote controls. So 3 times 1,000. So it's going to be equal to
2,000 plus 3 times 1,000 is 3,000, which is equal
to $5,000. So that's part a. $5,000. Now part b. What is the cost of producing
2,000 remote controls? Well, the cost-- just use the
same formula-- is equal to 2,000 plus 3 times the number
of remote controls. So 3 times 2,000. So that's equal to 2,000 plus
3 times 2,000 is 6,000. So that's equal to $8,000. Now we're at problem 22. The volume of a box without a
lid is given by the formula, volume is equal to 4x times 10
minus x squared, where x is a length in inches, and v is the
volume in cubic inches. What is the volume when
x is equal to 2? So part a, x is equal to 2. We just substitute 2 wherever
we see an x here. So the volume is going to be
equal to 4 times x, which is 2, times 10 minus 2 squared. And so this is going to be equal
to 4 times 2 is 8, times 10 minus 2 is 8 squared. So that's 8 times-- so this
is equal to 8 times-- 8 squared is 64. You could say this is going to
be 8 to the third power. And 64 times 8-- 4
times 8 is 32. 6 times 8 is 48, plus 3 is 51. So it's 512. Now, what is the volume
when x is equal to 3? I'll do it here in pink. b, when
is equal to 3, then the volume is equal to 4 times 3-- x
is equal to 3 now-- times 10 minus 3 squared. 4 times 3 is 12, times 10
minus 3 is 7 squared. So it's equal to 12 times 49. And just to get the exact
answer, let's multiply that. So 49 times 12-- 2 times
9 is 18, 2 times 4 is 8, plus 1 is 9. Put a 0. 1 times 9 is 9. 1 times 4 is 4. And then we add the two. We get 8, 9 plus 9 is 18, and
then 1 plus 4 is 588.