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- We have the equation 3/4x plus 2 is equal to 3/8x minus 4.
- Now, we could just, right from the get go, solve this the way
- we solved everything else, group the x terms, maybe on
- the left-hand side, group the constant terms on the
- right-hand side.
- But adding and subtracting fractions are messy.
- So what I'm going to do, right from the start of this video,
- is to multiply both sides of this equation by some number
- so I can get rid of the fractions.
- And the best number to do it by-- what number is the
- smallest number that if I multiply both of these
- fractions by it, they won't be fractions anymore, they'll be
- whole numbers?
- That smallest number is going to be 8.
- I'm going to multiply 8 times both sides of this equation.
- You say, hey, Sal, how did you get 8?
- And I got 8 because I said, well, what's the least common
- multiple of 4 and 8?
- Well, the smallest number that is divisible by 4 and 8 is 8.
- So when you multiply by 8, it's going to
- get rid of the fractions.
- And so let's see what happens.
- So 8 times 3/4, that's the same thing as 8
- times 3 over 4.
- Let me do it on the side over here.
- That's the same thing as 8 times 3 over 4, which is equal
- to 8 divided by 4 is just 2.
- So it's 2 times 3, which is 6.
- So the left-hand side becomes 8 times 3/4x is 6x.
- And then 8 times 2 is 16.
- You have to remember, when you multiply both sides, or a
- side, of an equation by a number, you multiply every
- term by that number.
- So you have to distribute the 8.
- So the left-hand side is 6x plus 16 is going to be equal
- to-- 8 times 3/8, that's pretty easy, the 8's cancel
- out and you're just left with 3x.
- And then 8 times negative 4 is negative 32.
- And now we've cleaned up the equation a good bit.
- Now the next thing, let's try to get all the x terms on the
- left-hand side, and all the constant terms on the right.
- So let's get rid of this 3x from the right.
- Let's subtract 3x from both sides to do it.
- That's the best way I can think of of getting rid of the
- 3x from the right.
- The left-hand side of this equation, 6x minus 3x is 3x.
- 6 minus 3 is 3.
- And then you have a plus 16 is equal to-- 3x minus 3x, that's
- the whole point of subtracting 3x, is so they cancel out.
- So those guys cancel out, and we're just left with a
- negative 32.
- Now, let's get rid of the 16 from the left-hand side.
- So to get rid of it, we're going to subtract 16 from both
- sides of this equation.
- Subtract 16 from both sides.
- The left-hand side of the equation just becomes-- you
- have this 3x here; these 16's cancel out, you don't have to
- write anything-- is equal to negative 32 minus 16 is
- negative 48.
- So we have 3x is equal to negative 48.
- To isolate the x, we can just divide both sides of this
- equation by 3.
- So let's divide both sides of that equation by 3.
- The left-hand side of the equation, 3x divided by
- 3 is just an x.
- That was the whole point behind dividing
- both sides by 3.
- And the right-hand side, negative 48 divided by 3 is
- negative 16.
- And we are done.
- x equals negative 16 is our solution.
- So let's make sure that this actually works by substituting
- to the original equation up here.
- And the original equation didn't have
- those 8's out front.
- So let's substitute in the original equation.
- We get 3/4-- 3 over 4-- times negative 16 plus 2 needs to be
- equal to 3/8 times negative 16 minus 4.
- So 3/4 of 16 is 12.
- And you can think of it this way.
- What's 16 divided by 4?
- It is 4.
- And then multiply that by 3, it's 12,
- just multiplying fractions.
- So this is going to be a negative 12.
- So we get negative 12 plus 2 on the left-hand side,
- negative 12 plus 2 is negative 10.
- So the left-hand side is a negative 10.
- Let's see what the right-hand side is.
- You have 3/8 times negative 16.
- If you divide negative 16 by 8, you get negative 2, times 3
- is a negative 6.
- So it's a negative 6 minus 4.
- Negative 6 minus 4 is negative 10.
- So when x is equal to negative 16, it does satisfy the
- original equation.
- Both sides of the equation become negative 10.
- And we are done.