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There are many ways to write the same mathematical idea. Some forms are more complex than others, but in general, simpler expressions express ideas more efficiently and are easier to evaluate. This lesson will show how to write an expression as simply as possible.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

How Can These Terms Be Grouped?

The applet shows some terms in a table. Try to group the terms in the table correctly.
Grouping Like Terms
What do these terms have in common? What happens if the terms in the same column are added or subtracted?
Discussion

Like Terms

Two or more terms in an algebraic expression are like terms if they have the same variable(s) with the same exponent(s).
In this expression, there are three sets of like terms — namely, terms, terms, and terms.
Note that all constants are like terms. Like terms in an expression can be combined by using addition or subtraction to rewrite the expression in simplest form.
Pop Quiz

Identifying Like Terms

Determine whether the terms are like terms.

Are the terms like terms
Discussion

Reducing the Number of Terms

Once like terms are identified, it is possible to combine them using addition or subtraction.

Method

Combining Like Terms

In an expression, groups of like terms can be combined to write an equivalent expression with the least possible number of terms. As an example, consider the following expression.
There are three steps to follow to combine like terms.
1
Identify Like Terms
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Start by identifying the like terms in the expression. These are terms with the same variables raised to the same power. Also, all constants are like terms.
Identifying Like Terms
This expression has two terms, two terms, and two terms.
2
Grouping Like Terms
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Now that the like terms are identified, use the Commutative Property of Addition to rearrange the expression so that the like terms are grouped together.
Grouping Like Terms
3
Add and Subtract the Coefficients
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Finally, combine the like terms by adding or subtracting the coefficients of the variables as well as adding or subtracting the constants.
Combining Like Terms
Now that all like terms have been combined, the initial expression has been reduced to its simplest form.
Example

Heichi's Path to the Museum

Heichi got a part-time summer job at a local history museum. He asked his sister for directions from home to the museum. She drew a map for him. She also knows that Heichi is studying algebraic expressions, so she used algebraic expressions to show the relevant distances.

Little map
Write an expression in its simplest form for the total distance from Heichi's house to the museum.

Hint

Add the expressions for each leg.

Determine the like terms.

Solution

Heichi is trying to determine the overall distance from his house to the museum along the route his sister marked for him. The route is broken into three separate sections. Each section has a unique length represented by a unique algebraic expression.

Section Expression
First Leg
Second Leg
Third Leg
These expressions can be added together to find the total distance.
Notice that there are two different terms — and constant terms. These are the like terms.
Now that the like terms have been identified, they can be combined. Use the Commutative Property of Addition to group the like terms together, then add or subtract them.
The simplified expression is
Example

Weekend Visitors to the Museum

On weekends, the museum has special admission prices before noon.

Building

Children pay dollars and adults pay dollars before noon. After noon, the prices go up to and dollars for children and adults, respectively. Heichi kept track of the number of visitors on both Saturday and Sunday in separate tables. The following table shows the number of visitors on Saturday.

Saturday's Visitors
Children Adults
Before Noon
After Noon

The table below shows the number of visitors on Sunday.

Sunday's Visitors
Children Adults
Before Noon
After Noon
Help Heichi figure out how much money the museum took in from admissions on Saturday and Sunday in terms of and Then simplify the total amount.

Hint

Remember that the tickets have different prices depending on the time of day.

Solution

The total amount can be written by adding the revenues from Saturday and Sunday.
The ticket revenue from Saturday can be found by multiplying the number of people that entered by the amount that they have to pay. However, since the prices change depending on the hour, this is better done in two sections. This can be written in a table.
Saturday's Visitors
Children Adults Total
Before Noon
After Noon
The expression for Saturday's revenue can be written by adding the expressions for each part of the day.
The same steps can be followed for Sunday's visitors.
Sunday's Visitors
Children Adults Total
Before Noon
After Noon
Similar to Saturday, the expression for Sunday's revenue can be found by adding the expressions for each part of the day.
Now that the expressions for Saturday and Sunday are written, it is time to add them together. The like terms can be grouped by the Commutative Property of Addition to combine them easily.
This expression represents how much money was made from admissions over the weekend.
Example

A Curious Occurrence in the Souvenir Shop

Heichi's friend Magdalena works at the souvenir shop in the museum. There are three large items that are sold in the shop.

items from the store. Vase has a price a, plus toy a price of b and painting has a price of c

Magdalena told Heichi that three groups of people visited the shop on Saturday.

  • The first group bought three vases, four plush toys, and five paintings.
  • The second group bought six plush toys, returned two vases, and bought one painting.
  • The third group returned one painting and bought one vase.
The curious thing, Magdalena explained, is that the exact same thing happened on Sunday. Suppose that the items returned are fully refunded. Write and simplify an expression for how much money the shop made from these groups of people over the weekend.

Hint

Start by writing an expression for Saturday. Then, multiply that expression by

Solution

Start by writing an expression to find the amount of money made by the store on Saturday. Consider each group separately.

  • The first group bought vases, plush toys, and paintings.
  • The second group bought plush toys, returned vases, and bought painting.
  • The third group returned painting and bought vase.

The money exchanged for each set of items is the product of the number of items sold and its corresponding price. Note that when an item is returned, the prices are negative because money is being taken from the shop and returned to the customer.

Vases Plush Toys Paintings Total
First Group
Second Group
Third Group
The expression for the money made on Saturday is the sum of these expressions.
Magdalena mentioned that the same thing happened on Sunday. This means that the whole expression has to be multiplied by to find the expression for the money made from both days.
This expression can be simplified by combining like terms first and then distributing the or by distributing first. Either process will result in the simplified expression. Here, the terms will be combined first.
This expression represents the total revenue from these groups on both Saturday and Sunday.
Closure

Grouping Like Terms

The beginning of this lesson presented a table to group like terms.
Grouping Like Terms
The column headers identify the type of term. Since all constant terms are like terms, they are grouped in the first column. The terms in the other columns indicate which variables are to be placed there.
These terms can now be combined by adding or subtracting them!