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Equation with variables on both sides: fractions

CCSS.Math:

Video transcript

we have the equation 3/4 X plus 2 is equal to 3/8 X minus 4 now we could just write from the get-go solve this 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 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 there'll be whole numbers and that smallest number is going to be 8 that smallest number is going to be 8 I'm gonna multiply 8 times both sides of this equation and 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 though the smallest number that is divisible by what 4 and 8 is 8 so when you multiply by 8 it's going to get rid of the fractions and see 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 3/4 X 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 eights 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 getting the 3x get getting rid of the 3x 4 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 so they cancel out so those guys cancel out and we're just left with a negative 32 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 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 in the original equation didn't have those eights out front so let's substitute in the original equation we get 3/4 3 over 4 times negative 16 times negative 16 plus 2 plus 2 needs to be equal to 3/8 times negative 16 times negative 16 minus 4 minus 4 so 3/4 of 16 is 12 and you could think of it this way what's 16 divided by 4 it is 4 and then multiply that by 3 that's 12 this mostess multiplying fraction 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 do at the right hand let's see what the right hand side is with three-eighths 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