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Let's do a couple of word problems. This tells us that

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Nadia's at home and Peter is at school, which is 6 miles

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away from home.

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So let me draw a little diagram here.

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So Nadia is there.

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Peter is over there, and they're 6 miles apart.

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So this is the distance between home and school.

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That is 6 miles.

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They start traveling to each other at the same time.

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So they both leave at the same time, traveling

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towards each other.

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Nadia is walking at 3 and 1/2 miles per hour.

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So she's travelling at 3 and 1/2 miles per

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hour in that direction.

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And Peter is skateboarding at 6 miles per

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hour in that direction.

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So Peter is going in that direction at 6 miles per hour.

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When will they meet and how far from home is

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their meeting place?

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So let's say that their meeting place is right-- I

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don't know.

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It's going to be closer to home, because this guy is

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going faster.

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Peter is going faster.

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So let's say, let x be equal to the distance

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from home they meet.

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So this distance right here is going to be x.

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And then what's this distance going to be?

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Well, if that's x, and this whole thing is 6 miles, then

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this distance right there is going to be 6 minus x.

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So Nadia will travel x miles and Peter will

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travel 6 minus x miles.

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And they're going to travel the same amount of time.

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Remember, they both leave at time 0, and after time t, they

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will both meet right there.

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So they're both going to take the same amount of time.

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So Nadia's equation, if we just remember that distance is

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equal to rate times time, we could use that for both of

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them, in Nadia's case, the distance she travels is x.

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x is going to be equal to her rate, which is 3.5 miles per

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hour, times t, which is the time that she travels.

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And then Peter's equation is going to be his distance.

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Well, he travels 6 minus x miles, so it's 6 minus x is

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going to be equal to his rate, which is 6 miles per hour

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times the time he travels.

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Well, he also travels t hours, so times t.

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So we have two equations and two unknowns, so we

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can solve for them.

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We've already solved for x here.

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So let's substitute this for x right there.

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So we can rewrite this equation as 6 minus, but

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instead of an x, we can put 3.5t, 6 minus 3.5t, because we

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know that x is equal to 3.5t, is equal to 6t.

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And then let's see, if we add 3.5t to both sides of this

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equation, we get 6 is equal to 3.5t plus 6t is 9.5t, or if we

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divide both sides by 9.5, you get t is equal to 6 over 9.5

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hours, which is a bit of a bizarre number.

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But 9.5, we can rewrite that.

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That is the same thing as 6 over-- see, 9.5, that's the

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same thing as 19/2, right?

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19/2, you divide that and you would get 9.5.

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So this is equal to 6 times 2 over 19.

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So this is equal to 12/19 of an hour.

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That's how long it'll take them to meet.

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And then if you ask the question how far from home

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will they meet?

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If you want to figure out the x value, x is going to be

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equal to 3.5-- actually, instead of writing 3.5, I can

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write 3.5 as 7/2; that's the same thing as 3.5-- times our

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time, times 12/19.

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So you divide the numerator and the denominator by 2, and

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we get 7 times 6 is 42 over 19 miles.

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And this right here, that is in hours.

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So in 12/19 of an hour-- it's a weird fraction-- they will

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meet exactly 42/19 miles from home.

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So it's a little over 2 miles from home.

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Next problem.

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Peter bought several notebooks at Staples for $2.25.

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And he bought a few more notebooks at

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Rite Aid for $2 each.

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He spent the same amount of money in both places and he

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bought 17 notebooks in total.

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How many notebooks did he buy in each store?

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So let's define some variables.

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Let's let S equal number bought at Staples.

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And then we could say that R is equal to the number bought

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at Rite Aid.

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So what do we know?

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So he bought a total of 17 notebooks.

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Let me do that in a different color.

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He bought a total of 17 notebooks.

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So that tells us that S plus R is equal to 17.

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And we also know that he spent the same amount of money in

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both places.

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So how much did he spend at Rite Aid?

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Let me do this in orange.

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Spent the same amount of money in both places.

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So at Staples, how much did he spend?

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He bought S notebooks for $2.25 each, so 2.25S, that's

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how much he spent at Staples.

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That's going to be equal to the amount he

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spent at Rite Aid.

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He spent $2 per notebook at Rite Aid.

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So $2 times the number of notebooks from Rite Aid, so

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this is spent at Rite Aid.

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So once again, two equations with two unknowns.

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We can do a little bit of substitution maybe.

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So what's the best way to substitute here?

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Well, let's divide both sides of this

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equation by 2 right here.

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So you have-- I'm going to rewrite 2.2-- well, I'll just

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do it like this.

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So you have R.

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If you divide both sides of this by 2, you have 2.25 over

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2S is equal to R, right?

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I just divided both sides of this by 2.

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So if you take this value and you substitute it in for R

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right there, this equation becomes S plus-- instead of an

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R we have 2.25 over 2S is equal to 17.

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And let's just simplify.

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So this is 1-- we could view this as 2 over 2S, right?

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The coefficient there is just 1.

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So we can view this as we have a common denominator.

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This is the same thing is 4.25 over 2S is equal to 17.

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I'm avoiding doing any hard math just yet.

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Let me multiply both sides by the inverse.

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2/4.25.

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It's a little bizarre to have an expression with both a

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fraction and a decimal, but it's not illegal.

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Let's multiply both times 2/4.25.

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These cancel out, and so you get S is equal to 17 times 2

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is 34 over 4.25.

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And actually, I can eyeball that.

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That looks like that should be equal to 8, right?

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4.25 is 4 and 1/4, which is the same thing as 17/4 over 4.

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So this is the same thing as 34 over 17/4, which is the

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same thing as 34 times 4/17.

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Put a 1 there, divide by 17, you get a 2, divide by 17, you

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get a 1, 2 times 4 is 8.

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So he bought 8 notebooks at Staples, this is 8, and then

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at Rite Aid, he bought-- well, 8 plus R is equal to 17.

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Subtract 8 from both sides.

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He must have bought 9 notebooks at Rite Aid.

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Let's do one more.

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This one is especially fun-looking.

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Peter is outside, looking at the pigs and

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chickens in the yard.

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Nadia is indoors and cannot see the animals.

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Peter gives her a puzzle.

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He tells her that he counts 13 heads and 36 feet and asks her

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how many pigs and how many chickens are in the yard.

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So let's once again define our variables.

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P is equal to the number of pigs.

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And let C is equal to the number of chickens.

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So the number of heads will essentially be the number of

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pigs and chickens, assuming they each have one head.

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So the number of pigs plus the number of chickens will be

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equal to the number of heads, so that is right here.

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That is 13 heads.

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And then 4 times the number of pigs, right?

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Each pig has 4 legs, so 4 times the number of pigs plus

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2 times the number of chickens, assuming we're

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dealing with two-legged chickens, is going to be equal

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to the number of feet, is equal to 36.

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So once again, we have two equations with two unknowns.

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Instead of solving it with substitution, I'm going to do

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it by adding and subtracting the two equations.

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So what we can do, this equation, we can

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multiply it times 2.

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And when I say multiply it times 2, we have to multiply

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the entire equation times 2.

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And so it's still true.

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So we multiply the entire equation times 2.

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This becomes 2P plus 2C is equal to 26.

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And now what I'm going to do is I'm going to subtract this

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equation from that equation, which you can do.

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Because that is equal to that, that is equal to that, and so

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we're not doing anything that violates the laws of algebra.

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So if you subtract the bottom equation from the top

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equation, or I could just put a negative sign everywhere

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here, just so you know we're subtracting,

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4P minus 2P is 2P.

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2C minus 2C is 0.

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That's why I did this, so that these would cancel out.

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And then 36 minus 26 is equal to 10.

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So 2P is equal to 10.

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Divide both sides by 2.

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We have 5 pigs.

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And then 5 plus the number of chickens is equal to 13, so we

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must have-- subtract 5 from both sides-- 8 chickens.

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Now if this confuses you, you might want to try it with

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substitution, but I have many videos on solving systems of

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equations where I go a little bit slower and explain a

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little bit more of the logic of it.

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So whatever floats your boat.

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But either way, Nadia should hopefully guess or hopefully

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solve for 5 pigs and 8 chickens.

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And it should work out, right?

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You have 13 animals.

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You have 13 heads.

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And if you multiply 5 times 4, that's 20 pig feet plus 16

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chicken feet.

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20 plus 16 is 36 total feet.

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