Equivalent expressions with negative numbers | 7th grade | Khan Academy

Or ask which of the following expressions are equivalent to 2 minus 9.4 plus 0 plus 3.71, and we need to pick two answers. So pause this video and see if you can have a go at it before we do this together.

All right, now let's look through the choices. So this first choice we have 9.4. Here, we subtracted 9.4. Now, when you subtract a number, you can also view it as adding its opposite. So we could rewrite this top thing as 2 plus negative 9.4. Notice when we subtract 9.4, that's the same thing as adding negative 9.4.

So we could write it this way, and when you are adding numbers, and that's actually this property or this principle that we just used is really useful because when you're adding numbers like this, it doesn't matter what order you add them in. When you have some subtraction involved, now all of a sudden the order does matter. But when you're adding like this, it doesn't matter the order that you are doing it with.

So we can swap these numbers any way we want. We could write this, for example, as negative 9.4 plus 2 plus 3.71. We could ignore the zero or put that anywhere, obviously plus zero, something like that. But what we see over here, they don't have a negative 9.4; they have a positive 9.4. So I really don't know how I can reconcile this first thing and this right over here, so I'm going to rule that one out.

Now, this one over here, you have a two. We have a two there. We have a 3.71; we have a 3.71 there, and both of those things are just being added. And now we have a... and now from that, we are subtracting 9.4. And so here, they're subtracting 9.4, and it is indeed the case. Here you have to be very careful with subtraction. If you know we wouldn't want to make it 9.4 minus something else, but as long as we are only subtracting the 9.4, then the swapping of this order does work.

You can even verify that with the numbers over here. You could do 2 minus 9.4 plus 3.71, or you can do 2 plus 3.71 minus 9.4. So I like this choice.

Now let's look at this one. This has 3.71, 3.71. It has the two; it has the two, and now it's adding 9.4. Now, we already said we could rewrite subtracting 9.4 as adding negative 9.4, but we can't rewrite that as just adding 9.4. If this had a... if this was like this, if it had a negative there, then this would work, but it's not that. And so I'm going to... I'm going to rule that one out.

And I'm guessing this one's going to work, but let's see. So this one put parentheses around it, so it's really just telling us what we want to do first. So it says first do the 2 minus 9.4, which if you just went left to right, you would have done anyway, and then plus 0 plus 3.71. And 0 plus 3.71, obviously that's just going to be 3.71.

So of course, this does seem very reasonable to do. Then you could go 2 minus 9.4 plus 3.71, 2 minus 9.4. The 0 doesn't matter here, and then plus 3.71, so I like that choice.

Let's do another example. So once again, we want to come up with an equivalent expression. So let's look at this one. Actually, before I even look at the choices, let's just recognize that if you subtract a number, it's the same thing as adding its opposite. So for example, this could be rewritten as negative six-fifths plus one half plus the opposite of negative eight-fifths, which is positive eight-fifths. That's another way we could rewrite it that might be helpful.

Let's see over here. We have one half, which we see right over there. We have plus eight-fifths, and interestingly that looks like the one that we did in the second version right over here. And then we have minus six-fifths. So is minus six-fifths the same thing as this over here? Well, think about it. What we're really doing when we write this negative six-fifths out front, that's the same thing as we could do that as plus negative six-fifths, and that's going to be the same thing as subtracting the opposite of it, or another way of thinking about it as subtracting positive six-fifths.

Let me write it this way. Subtracting positive six-fifths. It's actually hard to tell because I put that... I have actually... let me write it this way. If we just took this version and we were to use a commutative property to swap the orders, we could write this as one-half plus eight-fifths and then plus negative six-fifths. All I did is swap the order, and now we know that this last part could be rewritten. So it's one-half plus eight-fifths. Instead of plus negative six-fifths, that can be rewritten as minus positive six-fifths.

Why can I do that? Because subtracting a number is the same thing as adding its opposite. And so all of these are equivalent, and this is exactly what they have written here, this last scenario. So I like that one.

Now let's see. This one has negative six-fifths, and it has a plus eight-fifths and a plus one-half. So all they did is they took this version, which means it is equivalent, and they swapped the order that we are doing. They actually swapped the order of the one-half and the eight-fifths. And when you're dealing with addition like this, you can move things around—the commutative property. You can do your addition in different orders, and that's all they're doing.

Actually, this is not just commutative; this is also associative. They're literally putting parentheses there and say, "Hey, let's just do those first two first." Let's swap these two, and then, of course, even if you're going left to right, you would do these first two even if the parentheses weren't there. So you can definitely... all of this is definitely equivalent to negative six-fifths swapping the one-half and the eight-fifths to eight-fifths plus one-half.

So I like that choice too. And so we had two choices, so these are probably not going to work. And let's see why they don't real fast. They both have the negative six-fifths. This has the plus one-half, but it has minus eight-fifths instead of minus negative eight-fifths. It could have plus eight-fifths here, but not minus eight-fifths, so we rule that one out. Negative six-fifths, and let's see.

Then it has... it's adding negative eight-fifths instead of subtracting negative eight-fifths. So I don't like that, and then it subtracts one-half instead of adding one-half, so it's no good on multiple dimensions.

Let's do one last example. Which of the expressions are equivalent? As always, pause and see if you can figure that out. So let's see. So actually before I look at the choices, I'm going to see if there's different ways to rewrite this. We could rewrite this as 1.7 minus 8.33. Why can I do that? Again, because subtracting a number is the same thing as adding its opposite.

And then I could say plus... I could write this as 8.33, and if I wanted to, I could even write this instead of minus 1.95. I could even write that as plus negative 1.95 if I choose because once again, subtracting a number is the same thing as adding its opposite. Now I'm tempted to try to compute this because we could just... we could just... when we're doing addition like this, and I could write it like this if I want everything to be addition: 1.7 plus negative 8.33 plus 8.33 plus negative 1.95.

When everything is expressed in addition, we know that the order doesn't matter. The commutative property tells us that, and even the associative property tells us how we put the parentheses doesn't matter. And so we could actually do that first, and then these two would cancel out.

Actually, maybe we will have to do that; that choice looks like that. But let's just go step by step. So this has 1.7, 1.7. Let me do this in a different color. One point... that's not a different color. 1.7, 1.7. They're subtracting 8.33; that's subtracting 8.33 here. So that's equivalent to this version right here. Then they're subtracting it again; no, this does not look good because even when we try to compute it, we know that these two terms will cancel out each other, so I don't like that one.

Let's see this one. If we take... if we take this one here, and we were to reorder it, because this is all addition of different integers here. But if we were to reorder it, we could reorder it as negative 8.33, negative 8.33 plus 8.33 plus 1.7 plus 1.7, and then we have plus negative 1.95, which we know we can write as minus 1.95 because subtracting a number is the same thing as adding its opposite.

And that's what they have here. They put these parentheses, but even if you just went right... if you just went left to right, I should say with this one, you're going to get the same thing. So I like that choice there. Now this choice, if you just cancel these two terms out, you're left with a 1.7 plus negative 1.95, which we already know is the same thing as minus 1.95. So I like this choice as well.

Now, why won't this one work? Let's see. 1.95, they're trying to change the order, but you can add a negative 1.95 and then put it out front. But this is a positive 1.95, so I'm already not liking this. And then they're subtracting 1.7. There's no reason why you should subtract 1.7. So this is definitely not looking good.