Combining Like Terms Lessons
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
By adding 5 and 7, you can easily find that the expression is equivalent to
What Does Combining Like Terms Do?
Algebraic expressions can be simplified like the example above by Combining Like Terms. Consider the algebraic expression below:
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.
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.
2x, 45x, x, 0x, -26x, -xEach of the following are like terms because they are all constants.
15, -2, 27, 9043, 0.6Each of the following are like terms because they are all y^{2} with a coefficient.
3y^{2}, y^{2}, -y^{2}, 26y^{2}
For comparison, below are a few examples of unlike terms.
17x, 17zEach y variable in the terms below has a different exponent, therefore these are unlike terms.
15y, 19y^{2}, 31y^{5}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:
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 5x^{2}, -2x^{2}, and x^{2} 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
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.
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
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
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:
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.
Then we add the 8x and -6x to get 2x. The resulting equation is
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 Equation Calculator Combining Like Terms Calcualtor
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