Solving Equations with Addition and Subtraction

In our last video lesson, we learned how to recognize equations. We also learned how to tell if an equation is solved or not. Well, now we're really going to start getting into the process of how we solve equations.

So the first thing that we're going to work on is solving equations with addition and subtraction in them. So we'll look at a couple of sample problems and we're also going to practice checking our answers just to make sure that we are correct.

Here's our first example. We have n plus nine equals 15. Now, before we start solving this problem, I like to draw a line in the middle, right down through that equal sign to help us see the left side of the equation clearly from the right side of the equation. This is going to really help you to keep your work nice and organized.

Now remember, we want to try to get the variable by itself. We know that our equation will be solved when the variable n is by itself on that left side and we have just the number on the other side. So we call that isolating the variable, getting it by itself. Well, we have that plus nine. That's kind of in the way there.

So how do we get rid of that plus nine? Well, to undo that number nine, we use the opposite operation. Since nine is being added, we have plus there. We have to do the opposite, which is subtract, so we're going to subtract nine.

Now, another thing to remember when solving equations, which is very important, is that we have to do the same thing on both sides of the equation. If we take nine away from one side, we also have to take nine away from the other side. Think about an equation as being a scale that has to stay balanced on both sides.

So whatever we do to one side, we have to do to the other side to keep it balanced. Now we need to simplify both sides of the equation and we're going to do that by combining like terms. So I'm just going to draw a horizontal line there so that I can show that what I write under that line is the result of simplifying what's above the line.

So on the left side, I have n plus nine, and then I have a minus nine as well. Well there's only one term that has n, there's no like terms to combine with the n, so that's going to stay the same. I'll just bring that down. But we have the plus nine and minus nine. Those are like terms, they're just numbers, they're constants, so we can combine them together.

So when I add a plus nine and a minus nine, they just become zero. So I can just write that as plus zero. So that left side just becomes n plus zero. Now I go to the middle where my equal sign is that's just going to stay the same. So you always just bring that equal sign down in the middle and then we move to the right side and combine like terms.

Over there we have 15 minus nine, which gives us six. Okay, so now we have n plus zero equals six. Well, do we really need that plus zero though? Whenever we're adding zero to something, it's not changing it at all. So we really don't need to write it.

That left side is really just equal to n. So we can just simplify this even more and write it as n equals six. And look at what we have here. We have our solved equation. We have a variable on one side, an equal sign in the middle, and a number on the other side. Remember, that's how we recognize that our equation is solved when we have those three parts.

So now we know that n is equal to six. But before we move on to the next problem, let's just make sure that our answer is correct. So we're just going to check our answer on the side here.

Now when we check our answer, we start off with our original equation, the n plus nine equals 15. And then we substitute our number for our variable. And since we said that n is six, I substitute six four n in the equation. So now it becomes six plus nine equals 15.

And now I simplify both sides and see if I get a true equation on the left side. Six plus nine will give us 15. And of course on the right side we just have 15. So that's going to stay as it is. So I end up with 15 equals 15 and that is a true equation. When we get a true equation, that tells us that our answer was correct. So now we know for sure that n is equal to six.

Let's look at another example. Here we have twelve equals x minus eight. Let's draw a line through the equal sign so we can see the two sides separately. Now if you notice, this one's set up a little bit differently this time. Our variable is on the right side of the equation.

We have x on the right side, but that's okay, it's totally fine. Sometimes the variable will be on the left, sometimes on the right, sometimes it will even be on both sides. But we can still work out the problem with the same steps. So remember, we start by isolating the variable. So we want to get x by itself on that right side.

So that means we need to get rid of or undo the minus eight. And we do that by using the opposite operation. So the opposite of minus eight is plus eight. And of course, whatever we do to one side, we have to do the same thing to the other side. We have to add eight to the left side as well. And then we're going to simplify both sides. On the left side, we combine like terms. Twelve plus eight gives us 20.

Move over to the middle, we bring down the equal sign, and then we combine like terms to simplify the right side. X will come down. There's nothing else to combine that with. We have minus eight plus eight that's just going to end up as zero. Now we have 20 equals x plus zero.

But like we said in the previous problem, we don't even need to write the plus zero because it doesn't change the value of that right side at all. That right side is just going to become x. So we can just write it as 20 equals x and it is solved.

Now we have the number on one side by itself. We have the equal sign in the middle and the variable by itself on the other side. Even though our variable x is on the right side, it's still okay because it is by itself.

Now let's check our answer to make sure that 20 really is the answer to this equation. We write down our original equation, substitute 20 for x, and then we simplify both sides on that right side. When we simplify 20 minus eight, that will give us twelve.

So now we have twelve on the left and then the right side. Simplifies to 12. 12 equals twelve is a true equation. So we know that our answer is correct and 20 is the value of x. O

kay, so now you can see as long as you keep your work neat and organized and take your time, follow each step one at a time, you can solve your equations. And of course, don't forget to check your answer to make sure it's correct.