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## Heart of algebra

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# Systems of linear equations word problems — Basic example

## Video transcript

- [Instructor] We are told that Hazel and Leo are
comparing the number of keys on their keychains. If Leo has four more keys on his keychain than Hazel does on hers and the two of them have 18 keys combined, how many keys does Hazel
have on her keychain? So pause this video and see
if you can figure that out. Okay now let's do it together. So there's two things
that we don't know here. We don't know how many keys
Hazel has on her keychain, and that's what they're asking us for, but we also don't know
how many keys Leo has on his keychain. And they give us two
pieces of information, Leo has four more keys on
his keychain than Hazel, and that the two of them
have 18 keys combined. So it's feeling like we
can set up two equations with two unknowns. If we let L be equal to
the number of keys Leo has, keys, I'll just say for Leo, and then H equaled the
number of keys for Hazel, and obviously if you were
doing this on a test, like the SAT, you wouldn't
have to write all this out, you would trying to be
getting to the solution, but I'm trying to explain it out for you, so I will do that. And so let's see how we can
set up these two constraints that they gave us as equations. So it says if Leo has four
more keys on his keychain than Hazel does. So how can we write that mathematically? So Leo has four more keys
on his keychain than Hazel, so we could write that Leo is equal to, the number of keys that Leo has
is four more than the number Hazel has, so it's Hazel plus four. So we could write it like that, we could also write it that the difference between the number of keys Leo has and the number of keys Hazel has is four, so another alternative, we
could have written it like this, Leo minus Hazel is equal to four, and these are algebraically equivalent, it doesn't take much
manipulation to go from one to the other. But that's ways that we
can mathematically write that first sentence. And now what about that second one? And the two of them have 18 keys combined. Well that just means that
L plus H is equal to 18. So we could just write it that
way L plus H is equal to 18. I could write it like that
again, L plus H is equal to 18, and obviously there's other ways that you could mathematically write this or that would be equivalent to this. But in either case now,
we have two equations with two unknowns, and there's two ways that we could go about approaching them. When I look at this version, these two, this system of equations right over here on the left, where I've already solved for L, to me this feels like substitution
might be really valuable. Because I have an L here, and we know that L is equal to H plus four, so if I substitute H plus four in for L, then I have one equation with one unknown, if L is equal to H plus four, I get H plus four plus
H plus H is equal to 18. And then if I add the
Hs I get two H plus four is equal to 18, and let me scroll down a little bit, and then if I subtract
four from both sides, and remember I'm doing
that to isolate the Hs on one side of the equation, well then I'm going to
get two H is equal to 14, and then dividing both sides by two gives me H is equal to seven. And we could then use that information to say okay, L is equal to H plus four, so it's seven plus four is equal to 11, but they're not even asking
us for the number of keys Leo has, they're just asking
us for the number of keys Hazel has on her keychain, and we just figured that
out, Hazel has seven keys. Now, we could go to this other system, and we could have solved this one as well, and here, it feels like elimination would be the more natural method. The reason why it jumps out at me that elimination would
be the natural method, we have a negative H here and
we have a positive H there. Now one issue to think about, and there's many ways to solve these, is if we just add the left hand sides of this equation and
add the right hand sides of this equation, then
we're gonna get an equation in terms of L, because
the Hs will cancel out. So then we would solve for L
and use one of these equations to figure out H. Another way you could do it is, you could multiply one of these equations by a negative one on both sides, and then when you do the elimination, it would eliminate the L immediately. And so let's try it out that way. So let's multiply this top equation by negative one on both sides. So negative one times L is negative L, negative one times a
negative H is positive H, and then this would
become a negative four. And now when we add the left hand sides, negative L plus L cancels
out, H plus H is two H, and that's going to be equal
to negative four plus 18, which is 14, and then
divide both sides by two, and we get once again,
H is equal to seven. You could have very easily
done it the other way, you could have said L
minus H is equal to four, and that L plus H is equal to 18, and then immediately just add
the two sides of the equation. On the left hand side
those would cancel out, you would get two L is equal to 22, divide both sides by two, you would have gotten L is equal to 11. But remember, they don't ask us about Leo, they're asking us about Hazel. So then we would have to
substitute that back in to one of these equations to figure out that H is equal to seven.