Solving for a Variable Using Addition, Subtraction, Multiplication, and Division with Fractions

Narrator: Ok we’re going to solve for the variable. [-⅔ = ⅔ U - ½] That means we need to isolate U or get U on one side of the equal sign by itself. And now if we knew what U was, if we knew what our variable was we could just plug it in right here and we would multiply by 2 thirds and subtract ½ and that would give us negative 2 thirds. [1. Multiply by ⅔ 2. Subtract ½] Ok so we can see those those steps here. It’s our order of operations but in this case we don’t know what our variable is, what it equals and so we need to work backwards, we need to do a reverse order of operations in order to isolate U or to solve for U.

And so we’re going to start at the bottom and work up and we’re also going to do the opposite of what it says so the last step is subtract ½ so we are going to begin by adding ½. Ok so we’re going to add ½ over here [⅔ U - ½ ] and what we do to one side we need to do to the other side so we’re going to add ½ over here. [½] So I can rewrite that as ½ minus 2 thirds equals 2 thirds times U and then we have minus ½ plus ½ which will just equal zero so we can leave that off.

And now I need to find a common denominator [½ ⅔] between these two factors in order to subtract and so we need to find the least common multiple between 2 and 3. And the least common multiple is going to be 6 so I know that I can multiply top and bottom by 3 over here. [½] And then I can multiply top and bottom by 2 over here. [⅔] And that is still all equal to 2 thirds times U. Ok and now I can multiply these 3 times 1 is 3 and 3 times 2 is 6. And then over there we have minus 2 times 2 is 4 and 3 times 2 is 6. So we got our least common multiple or our common denominator of 6 ok and that is all still equal to 2 thirds times U. Ok and now because I have the same denominator I can combine those. [3/6 4/6] So 3 minus 4 is a negative 1and that’s over 6. Ok and that’s equal to 2 thirds times U.

Ok so we have done step 2 and now we can look at step 3. The first step was to multiply by 2 thirds but we need to do the opposite so we’re going to multiply by the multiplicative inverse [3/2]  - excuse me, and what we do to one side we have to do to the other. Draw some parentheses here just to keep it easy. Ok and so here on the right side, the twos cancel and our threes cancel and we’re just left with U. And over here we have [left side] this was multiplied, right so we need to draw some parentheses there to make it clear. So we have 3 times a negative 1 and I’m going to come right over here so we can see clearly what we’re doing. So 3 times a negative 1 is a negative 3 and 2 times 6 is 12 [-3/12] and that is equal to U but we can take that a step further. We can see that we can pull out a 3 and up top we’ll have 3 times negative 1and on the bottom we’ll have 3 times 4 and the threes cancel. We can see that our answer is negative ¼. So U is equal to negative ¼.

Ok so now I’m going to plug this back into our original equation as U and check our work to make sure that we did things correctly. Ok so our original problem was negative, let’s write that over here, negative 2 thirds which was equal to 2 thirds times U which we found to be negative ¼ minus ½. So by plugging in what we found for U, we should be able to solve this and it should equal negative 2 thirds and so let’s try that. We have 2 times a negative 1 is negative 2 and 3 times 4 is 12 ok minus ½  and then let's take this further. We can reduce this. We can pull out 2 times 2 and up top we have 2 times a negative 1 and on the bottom we have 2 times 6 and then we still have minus our ½ out here and all this should still be equal to negative 2 thirds. I should probably rewrite that everytime.                                             

Ok and let’s take this a little bit further so these twos cancel. We’re left with negative 1 over 6 minus 1 over 2 and if I find a common denominator here. I know that 6 can be a common denominator so I can multiply ½ by 3 on top and on bottom. Ok and that’s going to equal, let’s rewrite it down here. Negative 1 over 6 minus 3 over 6. Ok and that’s going to give me, now that I have a common denominator, I can combine my numerators. So negative 1 minus 3 is negative 4 over 6. Ok and that can be written, I can take out a 2 top and bottom. Ok and that can be 2 times 2 and then the denominator 2 times 3, oh excuse me there’s a negative up here. Ok and our twos cancel and we are left with negative 2 thirds which is what we needed to find over here so yep. Our answer of U equals negative 1 fourth is correct.

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