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6th-8th Grade Math: Practice & Review55 chapters | 468 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will know how to set up and solve addition problems where you have two or more variables. You will learn what steps you need to take to solve for each variable.

In this lesson, we'll talk about what we need to do to solve addition equations with two or more variables. **Addition equations with two or more variables** are math problems where we use the addition operator and that have two or more variables to solve for. Right now, we may be used to solving a simple addition equation, such as *x* + 3 = 8. We know that we need to isolate the variable to solve our equation. We know that we need to perform inverse operations to get our variable by itself. We will use these same skills to help us solve addition equations with more than one variable.

One big difference we will see is that we will have more than one equation to use. For each variable that we need to solve for, we will need one equation. Each equation may have one variable or more in it. So, if we have two variables, then we will need two equations. Each equation may have just one variable in it, or it may have up to all of the variables in it. For example, for a problem with the variables *x* and *y*, we will need two equations. Our problem may look like this.

*x* + *y* = 9

*x* + 2*y* = 14

We have two variables, so we likewise have two equations. Each equation just happens to have both variables in it.

Now, let's see how we can get started on solving these equations. We begin by solving for one of the variables. We can pick any one of the variables. There is no rule as to which variable to pick. We just have to pick one. Usually, if you are looking at the equations, there will be one variable that will look like it is easier to solve for. In our case, either variable will work. Both are easy to solve for.

Let's pick the variable *x* to solve for first. When we do this first round of solving, we are not necessarily looking for a number answer, we just want to isolate our chosen variable. We choose one equation to use. Let's choose the first one. Solving *x* + *y* = 9 for *x*, we subtract the *y* from both sides to get *x* + *y* - *y* = 9 - *y*. This gives us *x* = 9 - *y*. We have isolated our chosen variable, and we will leave it at that. We will mark this new equation that we have created. This new equation actually holds the key to the whole problem. Without this step, we wouldn't be able to find our answer.

Since we used the first equation already, we will now use the second equation. We will plug our newly created equation into our second equation. Can you guess how we will do that? We will plug in 9 - *y* for the *x* in the second equation. Let's see what happens when we do that. 9 - *y* + 2*y* = 14. Hey, look at that! We now have an equation with just one variable.

Simplifying and combining like terms, we get 9 + *y* = 14. This looks an awful lot like those addition equations that we already know how to solve. Well, let's just go ahead and solve this for *y* then. Our *y* is being added to a 9, so to get the *y* by itself, we need to subtract the 9 from both sides of the equation. We get 9 + *y* - 9 = 14 - 9. This simplifies to *y* = 5. Hey, look at that! We have solved for *y*!

Now we need to solve for *x*. How do we do that? We will use the equation that we created in the beginning, the *x* = 9 - *y*. We will plug in what *y* equals to find out what *x* equals. We get *x* = 9 - 5. This simplifies to *x* = 4. And there we have it! We found our answers! Our *x* = 4 and our *y* = 5.

We can plug these values back into our original equations to check whether these answers are correct or not. Plugging them into *x* + *y* = 9, we get 4 + 5 = 9. We get 9 = 9. That works. Plugging them into *x* + 2*y* = 14, we get 4 + 2*5 = 14. This turns into 14 = 14. That works, too. This tells me that my answers are correct!

The method that we have just used is called the substitution method. Other methods of solving are discussed in other lessons.

Let's look at another problem.

*x* + *z* = 4

*y* + *z* = 5

*x* + *y* + 3*z* = 12

This problem has three variables, so it has three equations. Looking at these equations, we see that if we solved the first equation for *x* and then the second equation for *y*, we can then plug these into the third equation to solve for *z*. Our goal in creating new equations is so that we can plug them into another equation so that this equation is left with only one variable. Doing this, we can solve for this one variable and then we can use this new information to help us solve for the other variables. So, solving the first equation for *x*, we get this:

*x* + *z* = 4

*x* + *z* - *z* = 4 - *z*

*x* = 4 - *z*

Solving the second for *y*, we get this.

*y* + *z* = 5

*y* + *z* - *z* = 5 - *z*

*y* = 5 - *z*

Now we can plug these two new equations into the third for *x* and *y*, respectively. We then solve this for the variable *z*.

*x* + *y* + 3*z* = 12

4 - *z* + 5 - *z* + 3*z* = 12

9 + *z* = 12

9 + *z* - 9 = 12 - 9

*z* = 3

Now, we can use our answer for the *z* variable and plug it into the other equations that we have created for *x* and *y* to help us find those answers.

Plugging in *z* = 3 in the equation *y* = 5 - *z* gives us *y* = 5 - 3 = 2. So, *y* = 2.

Plugging in *z* = 3 in the equation *x* = 4 - *z* gives us *x* = 4 - 3 = 1. So, *x* = 1.

Our complete answer is *x* = 1, *y* = 2, and *z* = 3. And we are done!

Let's review what we've learned. **Addition equations with two or more variables** are math problems where we use the addition operator and that have two or more variables to solve for. We will have one equation for every variable. So if we have two variables, our problem will then have two equations. If we have four variables, then our problem will have four equations.

To solve these problems, we solve for each variable. We create new equations when we solve for our variables. Our goal in creating these new equations is so that we can plug them into another equation so that we are left with just one variable. We can then solve for this one variable. Then we will use this answer to help us find the other variables by using the equations that we have created.

Once you are finished, you should be able to use substitution to solve an addition problem that has two or more variables.

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6th-8th Grade Math: Practice & Review55 chapters | 468 lessons

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