# 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`

## 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

`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.

## 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 / WorksheetEquations that require you to combine like terms before solving the equation. Equation CalculatorWill automatically combine like terms and solve the equation while showing all required work. (The equation calculator will not work with exponents.) Combining Like Terms CalcualtorSimplifies multiple signs and combines like terms in a given expression. Next Lesson: MultiplicationHow to multiply two or more terms. (Two monomials)

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