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# Systems of linear equations word problems — Basic example

## Video transcript

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 then 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 much 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 key chain 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 equal to 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 key chain than hazel does so how could we write that in mathematically so Leo has four more keys on his key chain 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 then the number of keys hazel has is four so another alternative we could have written it like this Leo - 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 and the two of them have 18 keys combined well that just means that L + H is equal to 80 so we could just write it that way l plus h is equal to 18 and 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 not 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 4 so if I substitute H plus 4 in for L then I have one equation with one unknown if L is equal to H plus 4 I get h plus 4 plus h plus h is equal to 18 and then if I add the HS I get 2h plus 4 is equal to 18 and let me scroll down a little bit and then if I subtract 4 from both sides I and remember I'm doing that to isolate the HS on one side of the equation well then I'm going to get 2 H is equal to 14 and then dividing both sides by 2 gives me H is equal to 7 and we could then use that information to say okay L is equal to H plus 4 so it's 7 plus 4 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 our key chain and we just figured that out hazel has 7 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 going to get an equation in terms of L because the H's will cancel out so then we would solve for L and then use one of these equations to figure out H another way that you could do it is you could multiply one of these equations by a [ __ ] one on both sides and then when you do the elimination it will eliminate the ELL 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 1 times L is negative L negative 1 times a negative H is positive H and then this would become a negative 4 and now when we add the left-hand sides negative al plus L cancels out h plus h is 2h and that's going to be equal to negative 4 plus 18 which is 14 and then divide both sides by 2 and we get once again H is equal to 7 you could have very easily done it the other way you could have said L minus H is equal to 4 and that L + 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 2 L is equal to 22 divide both sides by 2 you would have gotten L is equal to 11 but remember we're not they don't ask us about Leo they're asking us about hazel so then we would have to substitute that back into one of these equations to figure out that H is equal to 7