Equation with two rational expressions (old example 2)

solve the equation 5 over 2 X minus 4 over 3 X is equal to 7 over 18 and they tell us that X can't be equal to 0 because that would make these two expressions here undefined hopefully the answer here is not 0 and then this becomes just kind of extra unnecessary information so let's figure out how to solve this so a good place to start I don't like having X's in my denominators so let's multiply and in fact in general I don't like having fractions in my equations so let's see if we can multiply both sides of this equation by some things that will get rid of the fraction so let me just rewrite it so we have some space 5 over 2 X and then we have minus 4 over 3 X is equal to 7 over 7 over 18 now if we want to get rid of if we want to get rid of the 2 in the denominator here we could multiply everything by 2 if we want to get rid of this 3 in the denominator we could multiply everything by 3 if you want to get rid of this 18 in the denominator we can multiply everything by 18 and 18s also includes a 2 and a 3 you know the prime factorization of 18 is 2 times 9 which is 3 times 3 so when you're multiplying both sides of the equation by 18 you're actually multiplying it by a 2 and a 3 and another 3 so let's just multiply both sides of this equation by 18 so if you multiply it by 18 so I'll multiply this term right here by 18 and then this term right here by 18 that will get rid of all of these numbers in the denominator right 18 divided by 3 6 18 divided by 2 is 9 but we don't just have numbers in the denominator we also have these X's in the denominator so it's also multiply both sides of the equation by X so that we get rid of these so we're essentially going to multiply both sides of the equation by 18 X 18 X we're taking essentially the least common multiple of 2 X 3 X + 18 this is the smallest number that is divisible by all three of these characters now when we do that our denominators will disappear X divided by X is 1 18 divided by 2 is 9 so this term becomes 9 times 5 which is 45 and then this term right here X divided by X is 1 18 divided by 3 is 6 so you have 6 times 4 is 24 but you have a subtraction sign here so minus 24 is equal to is let me do that in that yellow is equal to and then you have this term 7 over 18 times 18 X well the 18s cancel out and you're just left with 7 times X is equal to 7x and now this becomes a much much simpler equation now what's 45 minus 24 it is see 45 minus 20 would be 25 45 minus 20 would be 25 then you subtract 4 more it's 21 so you get 21 is equal to 7x divide both sides by 7 and you get X is equal to 3 X is equal to 3 X is equal to 3 and let's verify that that works so we have 5 over 2 X so that's the same thing as 5 over 2 times 3 minus 4 over 3 times 3 so this is 5 6 5 6 minus 4 over 18 sorry for over 9 4 for 9 we want to find a common denominator 18 is the least common multiple of 6 & 9 so let's put it over 18 over 18 5 6 is the same thing as 15 over 18 multiply the numerator and denominator by 3 4/9 is the same thing as 8 over 18 multiply the numerator and denominator by 2 15 minus 8 so this becomes 15 minus 8 over 18 which is equal to 7 over 18 so it works out 5 over 2 X when X is equal to 3 minus 4 over 3 X when X is equal to 3 is indeed equal to 7 18s so we're done