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Combining Like Terms Lessons

A term is a constant or a variable in an expression. In the equation 12+3x+2x2=5x-1, the terms on the left are 12, 3x and 2x2, while the terms on the right are 5x, and -1.

Combining Like Terms is a process used to simplify an expression or an equation using addition and subtraction of the coefficients of terms. Consider the expression below

5 + 7

By adding 5 and 7, you can easily find that the expression is equivalent to

12

What Does Combining Like Terms Do?

Algebraic expressions can be simplified like the example above by Combining Like Terms. Consider the algebraic expression below:

12x + 7 + 5x

As you will soon learn, 12x and 5x are like terms. Therefore the coefficients, 12 and 5, can be added. This is a simple example of Combining Like Terms.

17x + 7

What are Like Terms?

The key to using and understanding the method of Combining Like Terms is to understand like terms and be able to identify when a pair of terms is a pair of like terms. Some examples of like terms are presented below.

The following are like terms because each term consists of a single variable, x, and a numeric coefficient.
2x, 45x, x, 0x, -26x, -x
Each of the following are like terms because they are all constants.
15, -2, 27, 9043, 0.6
Each of the following are like terms because they are all y2 with a coefficient.
3y2, y2, -y2, 26y2

For comparison, below are a few examples of unlike terms.

The following two terms both have a single variable with an exponent of 1, but the terms are not alike since different variables are used.
17x, 17z
Each y variable in the terms below has a different exponent, therefore these are unlike terms.
15y, 19y2, 31y5
Although both terms below have an x variable, only one term has the y variable, thus these are not like terms either.
19x, 14xy

Combining Like Terms

In an Expression

Consider the expression below:

5x2 + 7x + 2 - 2x2 + 7 + x2

We will demonstrate how to simplify this expression by combining like terms. First, we identify sets of like terms. Both 2 and 7 are like terms because they are both constants. The terms 5x2, -2x2, and x2 are like terms because they each consist of a constant times x squared.

Now the coefficients of each set of like terms are added. The coefficients of the first set are the constants themselves, 2 and 7. When added the result is 9. The coefficients of the second set of like terms are 5, -2, and 1. Therefore, when added the result is 4.

With the like terms combined, the expression becomes

9 + 7x + 4x2

The Combining Like Terms process is also used to make equations easier to solve.

While Solving an Equation

The equation which we will be simplifying and solving is below.

x + 3x + 7 = 42 + x - 12

When combining like terms it is important to preserve the equality of the equation by only combining like terms on one side at a time. We will simplify the left hand side first. The first step is to find pairs of like terms, the second step is to add. The x and 3x are like terms, so they are added resulting in 4x. (HINT: when a variable such as x has no coefficient, its coefficient is 1 so x is the same as 1x.) The 7 does not have a like term, so it is not changed. The equation now reads

4x + 7 = 42 + x - 12

The next step is to simplify the right hand side of the equation. This time there is no term which can be added with x, but there are two constants which are like terms. The 42 and the -12 are added, resulting in 30. The equation now reads.

4x + 7 = x + 30

The equation is now similar to those presented in the Equation Basics lesson, therefore the solution can be completed using the methods learned there.

Combining Like Terms

A Second Equation Example

The next example equation is shown below. Solving this equation will require both Simplifying Multiple Signs and Combining Like Terms.

-9 + 12x - 10 - 4x = 8x - 6x + 46 - - 1

The first step to simplifying this equation is to simplify the double negative sign in front of the 1. The second negative sign cancels out the first one, so there are no signs left, meaning that the 1 is positive. Review the Simplifying Multiple Signs lesson if this concept is unfamiliar to you. When this step is completed, the equation becomes

-9 + 12x - 10 - 4x = 8x - 6x + 46 + 1

We will start combining like terms on the left side with -9, a constant. The only other constant on the left side is -10, so we can add the two together as shown below. The sum of -9 and -10 is -19, thus the equation becomes

-19 + 12x - 4x = 8x - 6x + 46 + 1

Next we will add together 12x and -4x because they are like terms (x to the first power is the only variable in each). The resulting equation is shown below:

-19 + 8x = 8x - 6x + 46 + 1

Now that all like terms on the left side have been combined, we start working on the right side by adding the constants 46 and 1 to get 47.

-19 + 8x = 8x - 6x + 46 + 1

Then we add the 8x and -6x to get 2x. The resulting equation is

8x - 19 = 2x + 47

Now, the equation can be solved using addition, subtraction, and division, as presented in the Equation Basics lesson.

Combining Like Terms Resources

Equation Practice Problems / Worksheet
Equations that require you to combine like terms before solving the equation.

Equation Calculator
Will automatically combine like terms and solve the equation while showing all required work. (The equation calculator will not work with exponents.)

Combining Like Terms Calcualtor
Simplifies multiple signs and combines like terms in a given expression.

Next Lesson: Multiplication
How to multiply two or more terms. (Two monomials)

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