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# Ex 2: Distributive property to simplify

## Video transcript

Solve for x. And we have x minus 8 is equal to x/3 plus 1/6. Now the first thing I want to do here-- and there's multiple ways to do this problem-- but what I want to do is just to simplify the fraction. I'm going to multiply everything times the least common multiple of all of these guys' denominators. This is essentially x/1. This is 8/1, x/3, 1/6. The least common multiple of 1, 3, and 6 is 6. So if I multiply everything times 6, what that's going to do is going to clear out these fractions. So these weren't fractions to begin with, so we're just multiplying them by 6. So it becomes 6x minus 6 times negative 8, or 6 times 8 is 48. And we're subtracting it right over there. And then we have x/3 times 6. Let me just write it out here. So that's going to be 6 times x/3 plus 6 times 1/6. Or we get 6x minus 48 is equal to 6 times something divided by 3. That's the same thing as 6 divided by 3 times that something. That's just going to be equal to 2x plus 6 times 1/6 or 6 divided by 6 is just going to be 1. So that first step cleared out all of the fractions and now this is just a straightforward problem with all integer coefficients or integers on either side of the equation. And what we want to do is we want to isolate all of the x's on one side or the other. And we might as well isolate them all on the left hand side. So let's subtract 2x from both sides. We want to get rid of this 2x here. That's why I'm subtracting the 2x. So let's subtract 2x from both sides. And on the right hand side, I have 2x plus 1 minus 2x. Those cancel out. That was the whole point. So I'm left with just this 1 over here. On the left hand side I have 6x minus 2x. Well, that's just going to be 4x. If I have 6 of something minus 2 of that something, I have 4 of that something. Minus 48. And now I can-- let's see, I want to get rid of this 48 on the left hand side because I want only x's here. So let me add 48 to both sides of the equation. I'll do this in a new color. So let me add 48 to both sides of this equation. And on the left hand side 4x minus 48 plus 48, those cancel out. I'm left with just a 4x. And on the right hand side, 1 plus 48 is going to be 49. And now I've isolated the x but it's still multiplied by a 4. So to make that a 1 coefficient, let's multiply both sides by 1/4. Or you could also say, let's divide both sides by 4. Anything you do to one side you have to do to the other. And so you have-- what do we have over here? 4x/4 is just x. x is equal to 49 over 4. And that's about as far as we can simplify it because these don't have any common factors, 49 and 4. Let's check to see whether 49/4 is indeed the answer. So let's put it into the original equation. Remember, the original equation is what we have in green here before we multiplied it by six. But in theory, we should be able to put it into any of these steps and the x should satisfy. But let's do it in our original equation. So we have x minus 8. So we have 49/4 minus 8 should be equal to 49/4 over 3 plus 1/6. So let's see what we can do here. So we can multiply. Well, like we did before. We can multiply both sides of this equation by 6. That'll help simplify a lot of the fractions here. So if we multiply both sides of this equation by 6-- so we're going to multiply everything by 6-- what do we get on the left hand side? 6/4 is the same thing as 3/2, right? So this is going to be 3 times 49 over 2. 3 times 49 over 2 minus 48 will be equal to 6 divided by 3 is 2. So it's going to be 2 times 49 over 4, which is the same thing as 49/2. And then 6 times 1/6 is plus 1. And let's see. Well, I'm just going to actually just evaluate them out. I could just essentially have to subtract 49/2 from both sides and that'll simplify things. But let me just figure out what these evaluate to. So 3 times 49, 49 times 3. You could think about it. It's going to be 3 less than 50 times 3. So it's 147. But let's just multiply it out. 9 times 3 is 27 4 times 3 is 12, plus 2 is 14. So 147. So this is 147/2. And then let's put this over a denominator of 2. So 48 is equal to 96/2, right? 96 divided by 2 is 48 I just multiplied this by 2. So this is minus 96/2 needs to be equal to 49/2 plus-- and instead of having this 1 let's write that as 2/2. And what's 47 minus 96? So 147 minus 100 would be 47, but we're going to subtract 4 more than that. So we're going to have-- we're going to subtract 4 less than 100. So it's going to be 147 minus 96 is going to be 51. 51/2 is equal to-- 49 plus 2 is 51/2. So it all checks out.