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## Two-step equation word problems

## Video transcript

Now that we've set
up our equations, let's see if we can answer some
more interesting questions. Just as a warm-up,
we're going to do a little bit of what we
did in the last video. But I'll do it with
the equations now. The equations really are just
a mathematical representation of the same information
that we had up here. So as a starting
point, let's think about how much,
under the basic plan, it'll cost for us to attend
3 sessions in a given month. Well, we've got our
equation right over here. Our total cost in
that month is going to be equal to 20 plus 8
times the number of sessions. The number of sessions is S.
S is 3 in this circumstances, so we'll replace
the S with the 3. And so you get our cost
in this circumstance is going to be 20 plus 8 times
3, or 20 plus 24, or $44. Let me erase this
right over here. So this is going to cost us $44. Now let's do the same
thing under the trial plan. How much is it going to cost
us to attend 3 sessions? Well, this is more
straightforward, or slightly more
straightforward. Our total cost is going to be
equal to 12 times the number of sessions. The number of sessions in this
circumstance are 3 sessions, so it's going to be 12 times 3. Our total cost is going
to be equal to $36. Let me write that down. It's going to be equal to $36. But we could also answer
more interesting questions with these equations. For example, how many sessions
can I attend in a month for, oh, I don't know, let's
throw out something, $54. Let's say that's just my budget. When I look at my salary
and how much money I have budgeted to
health and fitness, this is what I can afford. And so based on that, I want
to ask, how many sessions can I attend in each of
these situations? First, let's think about
it for the trial plan. How many sessions
can I attend for $54? Now let's just assume that this
is some type of a special yoga studio where you can even
attend half a session. So if you attend half a
session, they'll charge you $6. If you attend a fourth of a
session, they'll charge you $3. So they'll actually keep
track of exactly what fraction of a session you're attending. You don't have to just attend
whole number of sessions. You could attend 2 and 1/2, and
they'll bill you accordingly. With that said, how many
sessions under the trial plan can I attend for $54? Well, let's think about
this a little bit. My cost now is going to be $54. $54 is C. It's my monthly cost. So 54 is going to be equal
to $12 per session times the number of sessions. Now I set up this equation. In order to answer
this question, I just need to figure out
what's the number of sessions that satisfy it. Now this S is an
unknown variable. What number of sessions
times 12 is equal to 54? Well, you might say,
well, if I could get just a S on the
right-hand side here, then I'll have an answer. S will be equal to some value. And if I want just a S
on the right-hand side, I'd ideally just like
to divide this by 12. 12S divided by 12 is
just going to give me a S. It's just going to give
me the number of sessions. But if I have an equal
sign, I can't just willy-nilly divide
one side by 12 without doing the exact same
thing to the other side. If that is equal to
that, in order for them to both still be equal, I
have to do the same thing to both sides. So on the right-hand side,
I'm left with just a S. And on the left-hand side,
I have 54 divided by 12. Let me think about that. I'll just work it out. 12 goes into 54 4 times. 4 times 12 is 48. You're left with
a remainder of 6. So you could say
12 goes into 54, or you could say 54/12 is equal
to 4 and 6/12 or 4 and 1/2. So the number of sessions that
I could attend are 4 and 1/2. If I have $54 to spend, I
could attend 4 and 1/2 sessions under the trial plan. If this was a yoga
studio that said, oh, you can't attend
half sessions, then you could
say, well, at most I could attend four sessions,
because five sessions would get me above my budget. But I said that you can
attend fractional sessions, so we're going to say 4 and 1/2. Now let's answer the same
question for the basic plan. Under the basic plan, how many
sessions can I attend for $54? Well, once again,
this is going to be our total cost for that month. So we say $54 is equal to
20 plus 8 times the number of sessions. And once again, I'd like
to just have sessions maybe on the right-hand
side of this equation. So I could say, oh, sessions are
going to be equal to something. This is the number of
sessions that I could attend. But I have this 20
here and this 8 here, and so I have to think about
how I can get rid of them. The first thing I've
got to think about is, how can I get rid of this
20 from the right-hand side of this equation? What am I going to have to
do to the entire equation? Well, the simplest thing
is I could subtract 20 from the right-hand side. But if I subtract 20
from the right-hand side, I also have to subtract
20 from the left-hand side in order to maintain
the equality. So I'm left with, on the
left-hand side, 54 minus 20 gets me 34. And that's going to
be equal to-- well, the whole point
of subtracting 20 is so that these
negate each other. And then I have just an 8S. So 8 times the number of
sessions has to be equal to 34 in order for 8 times the
number of sessions plus 20 to be equal to 54. And then you could add 20 to
both sides of this equation to get our original. We're almost done. We just want to isolate
the S right here. And you can imagine
the easiest way would be to divide the
right-hand side by 8. But if we're dividing
the right-hand side by 8, we have to do the
same to the left. And so what do we get for
the number of sessions that we would have to attend? Well, these cancel out. And so you get the
number of sessions is equal to 34 divied by 8. 34 divided by 8 is
equal to, let's see, 8 goes into 34 four times with
a remainder of 2, so 4 and 2/8, or it's 4 and 1/4. So I could attend 4 and 1/4
sessions under the basic plan. Assuming I can do fractional
sessions, for $54, which one can I attend
more sessions in, the basic plan or
the trial plan? Well, for $54 I could attend
a slightly longer amount of sessions under
the trial plan.