One-step multiplication and division equations
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Let's say we have the equation 7 times x is equal to 14. Now before even trying to solve this equation, what I want to do is think a little bit about what this actually means. 7x equals 14, this is the exact same thing as saying 7 times x -- let me write it this way -- 7 times x -- we'll do the x in orange again -- 7 times x is equal to 14. Now you might be able to do this in your head. You could literally go through the 7 times table. You say well 7 times 1 is equal to 7, so that won't work. 7 times 2 is equal to 14, so 2 works here. So you would immediately be able to solve it. You would immediately, just by trying different numbers out, say hey, that's going to be a 2. But what we're going to do in this video is to think about how to solve this systematically. Because what we're going to find is as these equations get more and more complicated, you're not going to be able to just think about it and do it in your head. So it's really important that one, you understand how to manipulate these equations, but even more important to understand what they actually represent. This literally just says 7 times x is equal to 14. In algebra we don't write the times there. When you write two numbers next to each other or a number next to a variable like this, it just means that you are multiplying. It's just a shorthand, a shorthand notation. And in general we don't use the multiplication sign because it's confusing, because x is the most common variable used in algebra. And if I were to write 7 times x is equal to 14, if I write my times sign or my x a little bit strange, it might look like xx or times times. So in general when you're dealing with equations, especially when one of the variables is an x, you wouldn't use the traditional multiplication sign. You might use something like this -- you might use dot to represent multiplication. So you might have 7 times is equal to 14. But this is still a little unusual. If you have something multiplying by a variable you'll just write 7x. That literally means 7 times x. Now, to understand how you can manipulate this equation to solve it, let's visualize this. So 7 times x, what is that? That's the same thing -- so I'm just going to re-write this equation, but I'm going to re-write it in visual form. So 7 times x. So that literally means x added to itself 7 times. That's the definition of multiplication. So it's literally x plus x plus x plus x plus x -- let's see, that's 5 x's -- plus x plus x. So that right there is literally 7 x's. This is 7x right there. Let me re-write it down. This right here is 7x. Now this equation tells us that 7x is equal to 14. So just saying that this is equal to 14. Let me draw 14 objects here. So let's say I have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. So literally we're saying 7x is equal to 14 things. These are equivalent statements. Now the reason why I drew it out this way is so that you really understand what we're going to do when we divide both sides by 7. So let me erase this right here. So the standard step whenever -- I didn't want to do that, let me do this, let me draw that last circle. So in general, whenever you simplify an equation down to a -- a coefficient is just the number multiplying the variable. So some number multiplying the variable or we could call that the coefficient times a variable equal to something else. What you want to do is just divide both sides by 7 in this case, or divide both sides by the coefficient. So if you divide both sides by 7, what do you get? 7 times something divided by 7 is just going to be that original something. 7's cancel out and 14 divided by 7 is 2. So your solution is going to be x is equal to 2. But just to make it very tangible in your head, what's going on here is when we're dividing both sides of the equation by 7, we're literally dividing both sides by 7. This is an equation. It's saying that this is equal to that. Anything I do to the left hand side I have to do to the right. If they start off being equal, I can't just do an operation to one side and have it still be equal. They were the same thing. So if I divide the left hand side by 7, so let me divide it into seven groups. So there are seven x's here, so that's one, two, three, four, five, six, seven. So it's one, two, three, four, five, six, seven groups. Now if I divide that into seven groups, I'll also want to divide the right hand side into seven groups. One, two, three, four, five, six, seven. So if this whole thing is equal to this whole thing, then each of these little chunks that we broke into, these seven chunks, are going to be equivalent. So this chunk you could say is equal to that chunk. This chunk is equal to this chunk -- they're all equivalent chunks. There are seven chunks here, seven chunks here. So each x must be equal to two of these objects. So we get x is equal to, in this case -- in this case we had the objects drawn out where there's two of them. x is equal to 2. Now, let's just do a couple more examples here just so it really gets in your mind that we're dealing with an equation, and any operation that you do on one side of the equation you should do to the other. So let me scroll down a little bit. So let's say I have I say I have 3x is equal to 15. Now once again, you might be able to do is in your head. You're saying this is saying 3 times some number is equal to 15. You could go through your 3 times tables and figure it out. But if you just wanted to do this systematically, and it is good to understand it systematically, say OK, this thing on the left is equal to this thing on the right. What do I have to do to this thing on the left to have just an x there? Well to have just an x there, I want to divide it by 3. And my whole motivation for doing that is that 3 times something divided by 3, the 3's will cancel out and I'm just going to be left with an x. Now, 3x was equal to 15. If I'm dividing the left side by 3, in order for the equality to still hold, I also have to divide the right side by 3. Now what does that give us? Well the left hand side, we're just going to be left with an x, so it's just going to be an x. And then the right hand side, what is 15 divided by 3? Well it is just 5. Now you could also done this equation in a slightly different way, although they are really equivalent. If I start with 3x is equal to 15, you might say hey, Sal, instead of dividing by 3, I could also get rid of this 3, I could just be left with an x if I multiply both sides of this equation by 1/3. So if I multiply both sides of this equation by 1/3 that should also work. You say look, 1/3 of 3 is 1. When you just multiply this part right here, 1/3 times 3, that is just 1, 1x. 1x is equal to 15 times 1/3 third is equal to 5. And 1 times x is the same thing as just x, so this is the same thing as x is equal to 5. And these are actually equivalent ways of doing it. If you divide both sides by 3, that is equivalent to multiplying both sides of the equation by 1/3. Now let's do one more and I'm going to make it a little bit more complicated. And I'm going to change the variable a little bit. So let's say I have 2y plus 4y is equal to 18. Now all of a sudden it's a little harder to do it in your head. We're saying 2 times something plus 4 times that same something is going to be equal to 18. So it's harder to think about what number that is. You could try them. Say if y was 1, it'd be 2 times 1 plus 4 times 1, well that doesn't work. But let's think about how to do it systematically. You could keep guessing and you might eventually get the answer, but how do you do this systematically. Let's visualize it. So if I have two y's, what does that mean? It literally means I have two y's added to each other. So it's literally y plus y. And then to that I'm adding four y's. To that I'm heading four y's, which are literally four y's added to each other. So it's y plus y plus y plus y. And that has got to be equal to 18. So that is equal to 18. Now, how many y's do I have here on the left hand side? How many y's do I have? I have one, two, three, four, five, six y's. So you could simplify this as 6y is equal to 18. And if you think about it it makes complete sense. So this thing right here, the 2y plus the 4y is 6y. So 2y plus 4y is 6y, which makes sense. If I have 2 apples plus 4 apples, I'm going to have 6 apples. If I have 2 y's plus 4 y's I'm going to have 6 y's. Now that's going to be equal to 18. And now, hopefully, we understand how to do this. If I have 6 times something is equal to 18, if I divide both sides of this equation by 6, I'll solve for the something. So divide the left hand side by 6, and divide the right hand side by 6. And we are left with y is equal to 3. And you could try it out. That's what's cool about an equation. You can always check to see if you got the right answer. Let's see if that works. 2 times 3 plus 4 times 3 is equal to what? 2 times 3, this right here is 6. And then 4 times 3 is 12. 6 plus 12 is, indeed, equal to 18. So it works out.