Solving Equations with Division

So far we've solved equations that have addition, subtraction, and multiplication. Now we're going to see how to solve equations that have division. Keep in mind that when we're in algebra, we're normally going to see division represented with a fraction line. So whenever you see an equation that has a fraction, just keep in mind that you can treat it as division.

Here we have x over two equals 15. And you'll notice that x over two is set up as a fraction. So that lets us know that we have division. Here we can read this as x divided by two is equal to 15. So we'll keep that in mind when we go to solve this problem, we're going to first draw a line down the middle so we can see both sides separately.

And now we work on isolating our variable x. So since x is being divided by two, we have to do the opposite, which is multiplication. So I have to multiply by two. Now notice I'm placing that two next to the numerator at the top.

If I were to place it at the bottom of the fraction, that would show that I'm dividing by that two. And I don't want to divide by two again, I want to do the opposite. So I need to show I'm multiplying by two. So I put it up top with the numerator. Of course, what I do to one side, I have to do to the other side. So multiply the right side by two.

Now we're going to simplify both sides. So on the left side we have the two that's placed with the numerator. We can treat that as part of the numerator and we have a two as the denominator. Well, two over two is equal to one, so that left side will simplify to one times x.

Now go to the middle and we bring down our equal sign and then we move to the right simplify 15 times two, which gives us 30. And remember, when we have a coefficient of one, it's saying one times x. Whenever we multiply something times one, it doesn't change the value of that thing. So it's really just equal to x.

We can write this over as x equals 30. And now that we've solved this equation, we figured out that x is equal to 30. But just to be sure, we're going to check our answer. So we start with our original equation. We're going to substitute 30 for x. So now we have 30 over two equals 15. Now we're going to simplify to see if we get a true equation.

So 30 divided by two will equal 15. So we have 15 on the left side, we already have a 15 on the right side. 15 equals 15. That's a true equation. So our answer is correct. And here we have negative A over five equals 14. Now we know that the fraction represents division, so it's saying a divided by five. But what's up with that negative sign in the front? What do we do with that?

So the interesting thing about negative fractions is that that negative sign can be placed one of three places. It can be in the front of the whole fraction, like we see here. Or we can put it in the numerator so we could put it with the A. But that might not be helpful because we want to get the A by itself.

So putting the negative sign with it doesn't help the A get by itself. If we put the negative sign in the denominator with the five, though, that would make this problem a little simpler for us. So I’m going to write this problem over just placing the negative sign in the denominator. And that doesn't change the value of the fraction at all, just makes it easier for us to solve.

There we go. Now we have the negative sign with the five. Now we can see how to isolate the variable. So let's go ahead and get started. We'll put our line down the middle to separate both sides of the equation. Now to isolate the variable, we're going to undo that negative five by doing the opposite of division. The opposite of division is multiplication. So I'm going to multiply by negative five.

Now notice I kept the negative sign with the five. I have to do that to make sure that I do isolate the variable A. That's the only way we're going to get rid of the negative and the five and we're going to do the same thing on the other side of the equation. Multiply the 14 by negative five. And now we simplify both sides.

Since we have negative five at the numerator and denominator of this fraction, that part simplifies to one. So that becomes one A brings down the equal sign in the middle 14 times. Negative five is negative 70. And as we've mentioned before, we don't really have to write that one as the coefficient of A because one A is just equal to A.

So we can write this as A equals negative 70. And we know our equation is solved because A isolated by itself on one side, equal sign is in the middle and we have our negative 70 on the other side. Let's check and see if this is the correct answer. We start with our original equation.

And notice I put the negative sign in front of the whole fraction because that's how it was in the beginning. But you could put that negative sign with the A or with the five, you can put in any of those three places, whichever makes it easier for you to work with. Now we substitute the negative 70 for A and now we simplify to see if we get a true equation.

So when we have a negative of a negative, we have a negative in front of the fraction and then we have a negative 70 in the numerator, a negative of a negative is a positive. Those negatives are going to cancel each other out. So let's go ahead and treat that as a positive. Now then, we just focus on the 70 divided by five, which will equal 14. We end up with a positive 14 on both sides. This is a true equation, which tells us that our answer is correct.