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- Let's say we have the equation seven times x is equal to fourteen.
- Now before even trying to solve this equation,
- what I want to do is think a little bit about what this actually means.
- Seven x equals fourteen,
- this is the exact same thing as saying seven 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 x 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.
- ocourse.org

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