Use number lines to find missing numbers in subtraction equations with integers. Arrows to the left equivalently mean adding a negative number and subtracting a positive number of the same magnitude. Arrows to the right equivalently mean adding a positive number and subtracting a negative number of the same magnitude. Created by Sal Khan.
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- [Instructor] So, if I were to ask you, or if I were to tell you that -3 minus blank is equal to -4, can you pause this video and figure out what this blank is? All right, now let's do this together, and I'm gonna do this by drawing out a number line because that's what my head tries to do with things like this. Let me draw a straighter line than that. All right, that's a pretty good number line. Let's see, I'm gonna be dealing with -3 and I'm gonna subtract something of it to get to -4, so let me focus on the negative end here. So, let's say this is zero, that could be +1. Then, I have -1, -2, -3, - 4, -5, let's go -6. That's the other end of it. So, this is -1, -2, -3, - 4, and -5. So, let's start at -3. So, -3 is that point on the number line and I want to end up at -4, that is that point on the number line. So, to go from -3 to -4, I have to go one step in the leftward direction. So, one step in the leftward direction, you could either view that as subtracting 1 or you could view that as adding a -1. Now, we have, we're already subtracting here, so the simplest thing to do would just say, "Okay, this is the same thing as subtracting 1." And we're done, -3 minus 1 is indeed equal to -4. Let's do another example. Let's say we had -1 is equal to -7, minus what? Pause a video and try to work through that. And try to do a number line, I always find that useful. All right, let's work through this together again, so let me draw a number line here. Let's see, I have a -1, I have a negative, I have a -7, so I'm gonna deal with the negate end of things. So, let me make this zero, I'll make that positive 1, - 1, -2, -3, - 4, -5, -6, -7. Let me write this, this is -7 right over here, - 8, -9, -10. I think that's probably enough. So, let's see. We're saying -1 is equal to -7 minus something, so essentially we're starting at -7 here, we're starting at -7, and we're subtracting something from that to end up over here at -1. This is -1 right over there. So, let's just think about what the arrow needs to go do to get from -7 to -1. Well, to do, that you're going to have to go, it looks like six units to the right. One, two, three, four, five, six, we go six units to the right. Now, there's two ways to describe going six units to the right. You could say that is just plus 6, or you could view that that's the same thing as minus -6. And we've seen that, subtracting a number is the same thing as adding its inverse. And since we already have a a negative sign here or minus sign here, we might as well say, "Well, this is the same thing as subtracting a -6," a -6, I'll put in parentheses to make it a little bit cleaner, and we're done. Let's do another example, another example here. So, let's say we make, give ourselves a clean slate. Let's say we wanted to figure out so I have blank minus -5 is equal to 13. How would you tackle that? Well, let me just draw my number line again. I have my whole real estate to use this time, so let me just go all the way there. And let's see, I'm dealing with something, it's gonna be roughly five away, it feels like from 13. So, let me just make my number line, I'm just gonna say this is zero, I have to get to 13, so actually let me give myself a little bit more space. So, this is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18. I think that is sufficient. So, we don't know where we're starting, we know we wanna end up at 13. So, 13, let me put that on our number line here, so it's 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. So, 13 is right over here. And now how did we get to 13? Well, we are subtracting a -5. Now, as I just mentioned, subtracting a number is the same thing as adding the inverse of the number, so this is the same thing as adding +5 is another way of thinking about that. So, or another way of depicting either of these, subtracting -5 or adding +5 would essentially be you're starting someplace and you're going 5 units to the right. So, you started someplace and you're going 5 units to the right to end up at 13. So, that means you started 5 units to the left, one, two, three, four, five. You started right over here, you go 5 units to the right and you ended up at 13. Well, this right over here, 5 units to the left of 13, this is going to be 8. And it is indeed the case that 8 minus -5 is equal to 13. Now, there's other ways that you might be able to think about this. If something minus -5 is equal to 13 then another way you could think about it, if I were to tell you that 3 minus 2 is equal to 1, you could also say that an equivalent statement is that 3 is equal to 1 plus 2. You could turn that subtraction equation into an addition equation. So, over here, if I say this minus this is equal to that then that means that 13 plus -5 must be equal to blank, our mystery number. And if I add 13 and -5, you might recognize that as being equal to 8. So, either way, there's a lot of ways that you could approach this, but they all get you to kind of the same conceptual place.