Let's do a couple of word problems. This tells us that
Nadia's at home and Peter is at school, which is 6 miles
away from home.
So let me draw a little diagram here.
So Nadia is there.
Peter is over there, and they're 6 miles apart.
So this is the distance between home and school.
That is 6 miles.
They start traveling to each other at the same time.
So they both leave at the same time, traveling
towards each other.
Nadia is walking at 3 and 1/2 miles per hour.
So she's travelling at 3 and 1/2 miles per
hour in that direction.
And Peter is skateboarding at 6 miles per
hour in that direction.
So Peter is going in that direction at 6 miles per hour.
When will they meet and how far from home is
their meeting place?
So let's say that their meeting place is right-- I
don't know.
It's going to be closer to home, because this guy is
going faster.
Peter is going faster.
So let's say, let x be equal to the distance
from home they meet.
So this distance right here is going to be x.
And then what's this distance going to be?
Well, if that's x, and this whole thing is 6 miles, then
this distance right there is going to be 6 minus x.
So Nadia will travel x miles and Peter will
travel 6 minus x miles.
And they're going to travel the same amount of time.
Remember, they both leave at time 0, and after time t, they
will both meet right there.
So they're both going to take the same amount of time.
So Nadia's equation, if we just remember that distance is
equal to rate times time, we could use that for both of
them, in Nadia's case, the distance she travels is x.
x is going to be equal to her rate, which is 3.5 miles per
hour, times t, which is the time that she travels.
And then Peter's equation is going to be his distance.
Well, he travels 6 minus x miles, so it's 6 minus x is
going to be equal to his rate, which is 6 miles per hour
times the time he travels.
Well, he also travels t hours, so times t.
So we have two equations and two unknowns, so we
can solve for them.
We've already solved for x here.
So let's substitute this for x right there.
So we can rewrite this equation as 6 minus, but
instead of an x, we can put 3.5t, 6 minus 3.5t, because we
know that x is equal to 3.5t, is equal to 6t.
And then let's see, if we add 3.5t to both sides of this
equation, we get 6 is equal to 3.5t plus 6t is 9.5t, or if we
divide both sides by 9.5, you get t is equal to 6 over 9.5
hours, which is a bit of a bizarre number.
But 9.5, we can rewrite that.
That is the same thing as 6 over-- see, 9.5, that's the
same thing as 19/2, right?
19/2, you divide that and you would get 9.5.
So this is equal to 6 times 2 over 19.
So this is equal to 12/19 of an hour.
That's how long it'll take them to meet.
And then if you ask the question how far from home
will they meet?
If you want to figure out the x value, x is going to be
equal to 3.5-- actually, instead of writing 3.5, I can
write 3.5 as 7/2; that's the same thing as 3.5-- times our
time, times 12/19.
So you divide the numerator and the denominator by 2, and
we get 7 times 6 is 42 over 19 miles.
And this right here, that is in hours.
So in 12/19 of an hour-- it's a weird fraction-- they will
meet exactly 42/19 miles from home.
So it's a little over 2 miles from home.
Next problem.
Peter bought several notebooks at Staples for $2.25.
And he bought a few more notebooks at
Rite Aid for $2 each.
He spent the same amount of money in both places and he
bought 17 notebooks in total.
How many notebooks did he buy in each store?
So let's define some variables.
Let's let S equal number bought at Staples.
And then we could say that R is equal to the number bought
at Rite Aid.
So what do we know?
So he bought a total of 17 notebooks.
Let me do that in a different color.
He bought a total of 17 notebooks.
So that tells us that S plus R is equal to 17.
And we also know that he spent the same amount of money in
both places.
So how much did he spend at Rite Aid?
Let me do this in orange.
Spent the same amount of money in both places.
So at Staples, how much did he spend?
He bought S notebooks for $2.25 each, so 2.25S, that's
how much he spent at Staples.
That's going to be equal to the amount he
spent at Rite Aid.
He spent $2 per notebook at Rite Aid.
So $2 times the number of notebooks from Rite Aid, so
this is spent at Rite Aid.
So once again, two equations with two unknowns.
We can do a little bit of substitution maybe.
So what's the best way to substitute here?
Well, let's divide both sides of this
equation by 2 right here.
So you have-- I'm going to rewrite 2.2-- well, I'll just
do it like this.
So you have R.
If you divide both sides of this by 2, you have 2.25 over
2S is equal to R, right?
I just divided both sides of this by 2.
So if you take this value and you substitute it in for R
right there, this equation becomes S plus-- instead of an
R we have 2.25 over 2S is equal to 17.
And let's just simplify.
So this is 1-- we could view this as 2 over 2S, right?
The coefficient there is just 1.
So we can view this as we have a common denominator.
This is the same thing is 4.25 over 2S is equal to 17.
I'm avoiding doing any hard math just yet.
Let me multiply both sides by the inverse.
2/4.25.
It's a little bizarre to have an expression with both a
fraction and a decimal, but it's not illegal.
Let's multiply both times 2/4.25.
These cancel out, and so you get S is equal to 17 times 2
is 34 over 4.25.
And actually, I can eyeball that.
That looks like that should be equal to 8, right?
4.25 is 4 and 1/4, which is the same thing as 17/4 over 4.
So this is the same thing as 34 over 17/4, which is the
same thing as 34 times 4/17.
Put a 1 there, divide by 17, you get a 2, divide by 17, you
get a 1, 2 times 4 is 8.
So he bought 8 notebooks at Staples, this is 8, and then
at Rite Aid, he bought-- well, 8 plus R is equal to 17.
Subtract 8 from both sides.
He must have bought 9 notebooks at Rite Aid.
Let's do one more.
This one is especially fun-looking.
Peter is outside, looking at the pigs and
chickens in the yard.
Nadia is indoors and cannot see the animals.
Peter gives her a puzzle.
He tells her that he counts 13 heads and 36 feet and asks her
how many pigs and how many chickens are in the yard.
So let's once again define our variables.
P is equal to the number of pigs.
And let C is equal to the number of chickens.
So the number of heads will essentially be the number of
pigs and chickens, assuming they each have one head.
So the number of pigs plus the number of chickens will be
equal to the number of heads, so that is right here.
That is 13 heads.
And then 4 times the number of pigs, right?
Each pig has 4 legs, so 4 times the number of pigs plus
2 times the number of chickens, assuming we're
dealing with two-legged chickens, is going to be equal
to the number of feet, is equal to 36.
So once again, we have two equations with two unknowns.
Instead of solving it with substitution, I'm going to do
it by adding and subtracting the two equations.
So what we can do, this equation, we can
multiply it times 2.
And when I say multiply it times 2, we have to multiply
the entire equation times 2.
And so it's still true.
So we multiply the entire equation times 2.
This becomes 2P plus 2C is equal to 26.
And now what I'm going to do is I'm going to subtract this
equation from that equation, which you can do.
Because that is equal to that, that is equal to that, and so
we're not doing anything that violates the laws of algebra.
So if you subtract the bottom equation from the top
equation, or I could just put a negative sign everywhere
here, just so you know we're subtracting,
4P minus 2P is 2P.
2C minus 2C is 0.
That's why I did this, so that these would cancel out.
And then 36 minus 26 is equal to 10.
So 2P is equal to 10.
Divide both sides by 2.
We have 5 pigs.
And then 5 plus the number of chickens is equal to 13, so we
must have-- subtract 5 from both sides-- 8 chickens.
Now if this confuses you, you might want to try it with
substitution, but I have many videos on solving systems of
equations where I go a little bit slower and explain a
little bit more of the logic of it.
So whatever floats your boat.
But either way, Nadia should hopefully guess or hopefully
solve for 5 pigs and 8 chickens.
And it should work out, right?
You have 13 animals.
You have 13 heads.
And if you multiply 5 times 4, that's 20 pig feet plus 16
chicken feet.
20 plus 16 is 36 total feet.