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# Equivalent expressions with negative numbers

## Thuyết minh video

what I hope to do with this video is several examples from the Khan Academy exercise on negative number addition and subtraction equivalent expressions and this is just to really make sure that we we fully digested how we can compute with negative and positive numbers and also the properties that we the the properties of arithmetic that we can apply when we compute with them so let's get started so it says Raul try to evaluate an expression step by step alright so they start with the expression they give us the steps and then they say find Raul's mistake and we have step 1 step 2 step 3 or maybe Raul did not make a mistake and like always I encourage you to pause the video and try to solve it on your own before I work through it alright so let's look at this first step it says 3 minus negative 5 minus 7 and it looks like he correctly recognized that subtracting a negative 5 is the same thing as adding a positive 5 so he replaced the minus negative 5 with the plus 5 right over there so that seems right all right so step one that seems alright then what does he do next so he has two options he could say 3 plus 5 is equal to 8 and then subtract 7 from that and he would get 1 or he could even say 5 minus 7 is negative 2 and then add that to 3 and get 1 and it looks like we're I'll try to go that second way but he made a slight mistake because he said 5 minus 7 is equal to 2 not negative 2 5 minus 7 should be negative 2 but he wrote a positive 2 right over here so his mistake is in step 2 and then obviously step 3 becomes wrong because of step 2 but this is this is where he first this is where he first makes the mistake let's do several more of these which of the following expressions are equivalent to 6 plus negative 4 minus 5 all right so let's look at all the options here so this first one let's see it has a negative 5 well the negative 5 this minus 5 if you put it out front you could you could represent it as a negative 5 then it has a Nega before here and here we're adding a negative for so or I guess you could say where we have negative 5 we have minus 4 here which is the same thing as adding a negative 4 so that seems reasonable and then you have plus 6 and you just had a positive 6 right over here so this one actually seems equivalent it's just the order they just switch the order instead of instead of adding the six first they add this positive six last instead of having the minus five at the end they put a negative five in the beginning and instead of saying adding negative four they just said minus four but these are all equivalent so this one is going to be an equivalent expression now let's see this one has so it has the six and the minus four there's change the order but now it has positive five and that's not going to work because we have a we're subtracting five out here so if we want to put the five out front it needs to be a negative five so this isn't going to be right I'm not going to select none of the above because I found a choice that is equivalent let's keep going it's a lot of fun which of the following expressions are equivalent to negative 4 plus 4 plus 5 in parentheses so if I were to compute this I would say negative 4 plus and I would add the 4 and the 5 first and I would get 9 so it'd be negative 4 plus 9 which is equal to 5 so let's see 0 plus 5 well that's going to be equal to 5 so that's equivalent I mean I can just do it again negative 4 plus 9 negative 4 plus 9 is 5 it's the same thing as 9 minus 4 and this right here is 5 now this is negative 4 minus 4 which would be negative 8 and then plus 5 would be negative 3 so that's not equivalent in order to be equivalent this should evaluate out to be 5 so I'm not going to select that and I'm not gonna select none of the above because I already found a choice that I like let's do a few more of these so once again which are equivalent negative 5 plus 7 minus negative 3 so here you have negative 5 plus 7 plus 3 well that makes sense if you subtract a negative 3 that's the same thing as adding a 3 and they didn't change the negative 5 plus 7 they didn't change the negative 5 plus 7 so I'll definitely select this one definitely select that now let's look at this one so negative five here they're subtracting a five so that seems reasonable adding seven they're adding seven here they're subtracting a negative three which is the same thing as adding a positive three here they just have a they just they just have a negative three here if this was a positive three then we'd be in business but this is a negative three so this isn't going to work so I will not select that and I won't select none of the above because I found a choice now let's do let's do one more of these which of the following expressions are equivalent to X plus y and since they're parentheses around it I would do that first and then I would subtract Z select all that apply all right so here here they they have Z plus X plus y so they have the X plus y part but then they have a positive there or they have a Z here while over here they have minus Z so if they are at a negative Z here this and this would have been equivalent they don't have a negative Z here so I would not I would not select that one now this one you have X plus y minus Z well this they just changed the parentheses instead of doing the X plus y first they're doing the Y minus Z first and this is you know this this goes back to addition and subtraction you have the associative property you could associate these you could start with these two adding these two first and then you do the subtraction or you could do the subtraction first and then you add this one and if you don't believe me you could try it out with some numbers and encourage you to try it out with some negative numbers but it will indeed I'll work out all they did is they changed where the parentheses are they did the Y minus Z first and then they add they do the Y minus Z and then they add it to the X as opposed to doing the X plus y first and then subtracting the Z so I would go I would go with that one