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2 Tutorials that teach Combining Like Terms
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Combining Like Terms

Combining Like Terms

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In this lesson, students will learn how to combine like terms to simplify an expression.

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Tutorial
This tutorial covers combining like terms, through the definition and discussion of:
  1. Terms and Like Terms
  2. Combining Like Terms


1. Terms and Like Terms

Solving an equation for a single variable is not always a single-step process. Often, complex expressions appear on either side of an equation, and it requires several steps to isolate the variable. It can save time and prevent confusion if you can simplify the expression by combining like terms before attempting to isolate the variable.

A term is a collection of numbers, variables, and powers combined through multiplication. Terms are referred to by their variable and power or exponent.

Term
A collection of numbers, variables, and powers combined through multiplication

Here is an example of an expression with three terms:

  • In this expression, the first term, 6 x squared, is called an x squared term. The 6 in front of the x squared is called a coefficient. The 2 is the exponent or power of the term.
  • The second term, 3x, is called an x term. The coefficient of this term is 3. Here the variable x has no written exponent, so the implied exponent is 1.
x to the power of 1 equals x
  • The last term, 2, is a constant term. Terms without a variable are called constant terms because the value of that term will be the same no matter what the variable is.

Like terms have the same variables, with each variable having the same exponent. In the expression above, there are no like terms, because in each term, the variable x has a different exponent. However, in the following expression, 3x and 5x are like terms, and you can simplify the expression by combining these two terms.

Like Terms
Terms involving the same variables with each variable having the same exponent


2. Combining Like Terms

Combining like terms is a method to simplify expressions before solving the equation using inverse operations. Terms may only be added or subtracted if they are like terms. Like terms are combined by adding or subtracting their coefficients and leaving the variable and exponent the same. A coefficient is the number that appears in front of a variable and acts as a factor or multiplier. Coefficients can be any real number, and terms without a written coefficient have an implied coefficient of 1, as illustrated here:

1 x equals x
Coefficient
The number in front of a variable term that acts as a factor, or multiplier
There are several common mistakes to avoid when combining like terms:

  • The exponent does not change when combining like terms. For example, 4x plus 3x does not equal 7 x squared, because the exponent does not change. Instead, 4x plus 3x means that you are adding 4 xs and 3 more xs, which equals a total of 7 xs. Thus, 4x plus 3x equals 7x.
table attributes columnalign left end attributes row cell 4 x plus 3 x not equal to 7 x squared end cell row cell 4 x plus 3 x equals 7 x end cell end table
  • You cannot combine terms that are not like terms. For example, 2x plus 2 does not equal 4x. Even though both terms have a 2, they are not like terms, because their variable and variable exponent are not the same. Therefore, 2x and 2 cannot be combined or simplified.
2 x plus 2 not equal to 4 x
When combining like terms, it’s useful to remember the commutative property for addition, which states that a plus b is equal to or the same as b plus a. In other words, you can add real numbers in any order. This property is useful when simplifying expressions involving like terms, because it allows the terms to be rearranged when you combine like terms using addition and subtraction.

a plus b equals b plus a

Suppose you are simplifying the following expression. In it, you have two like terms, x and 3x. The term x has an implied coefficient of 1, even though it is not written. To combine the terms, you add the coefficients 1 and 3. 1x plus 3x is 4x, so your expression becomes:
table attributes columnalign left end attributes row cell 4 x squared plus x plus 9 plus 3 x end cell row cell 4 x squared plus 4 x plus 9 end cell end table

How would you combine like terms in a more complex expression with varying terms? Suppose you are simplifying this expression:
2 x cubed plus 3 minus 2 x squared plus 11 plus 7 x squared minus 4
First, you can see that you have two x squared terms. The subtraction sign in front of the 2 x squared makes the term a negative 2 x squared. This can be combined with 7 x squared. Adding your coefficients, negative 2 and 7, equals 5. Therefore, the expression simplifies to the following:

2 x cubed plus 5 x squared plus 3 plus 11 minus 4
Next, you can see that you have three constant terms, 3, 11, and negative 4. Again, you can combine these three like terms: 3 plus 11 minus 4, or 3 plus 11 plus negative 4, which equals 10. The remaining term, 2 x to the third, cannot be combined with any other terms in the expression, so it is left unchanged. Your final expression is:

2 x cubed plus 5 x squared plus 10

A complex expression involving both varying terms and fractions requires several steps in the simplification process.

Suppose you want to simplify the expression:

1 fifth x cubed plus 3 over 4 x minus 1 half x plus 1 fifth x cubed
Step 1: You have two x cubed—or x to the third powered—terms, which can be combined. You need to add their coefficients 1/5 and 1/5. Since their denominators are the same, you can simply add the numerators.

2 over 5 x cubed plus 3 over 4 x minus 1 half x
Step 2: Next, you can combine your two x terms. You need to add their coefficients, 3/4 and negative 1/2. In this case, the denominators are not the same, so you need to find a common denominator. The least common denominator would be 4, so you can leave the fraction 3/4 unchanged and multiply the fraction 1/2 by 2 in the denominator and the numerator, which equals negative 2/4.

table attributes columnalign left end attributes row cell 2 over 5 x cubed plus 3 over 4 x minus fraction numerator 1 cross times 2 over denominator 2 cross times 2 end fraction x end cell row cell 2 over 5 x cubed plus 3 over 4 x minus 2 over 4 x end cell end table
Step 3: Now the denominators are the same, so you have 3/4 plus negative 2/4. You can add the numerators, which equals 1/4, so the two x terms simplify to 1/4x. Your final simplified expression is:

2 over 5 x cubed plus 1 fourth x
Today you learned that a term is a collection of numbers, variables, and powers combined through multiplication. You also learned the definition of like terms, which are terms that have the same variable, with each variable having the same exponent. Lastly, you learned that you can simplify expressions by combining like terms before attempting to isolate the variable, noting that only like terms may be added or subtracted.

Source: This work is adapted from Sophia author Colleen Atakpu.

TERMS TO KNOW
  • Coefficient

    The number in front of a variable term that acts as a factor or multiplier.

  • Terms

    A collection of numbers, variables, and powers combined through multiplication.

  • Like Terms

    Terms involving the same variables with each variable having the same exponent.