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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

The applet shows some terms in a table. Try to group the terms in the table correctly.

What do these terms have in common? What happens if the terms in the same column are added or subtracted?

Discussion

Two or more terms in an algebraic expression are like terms if they have the same variable(s) with the same exponent(s).

$7xy+2x−3xy−2x_{2}+x+5x_{2} $

In this expression, there are three sets of like terms — namely, $xy-$terms, $x-$terms, and $x_{2}-$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

Determine whether the terms are like terms.

Discussion

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

Method

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.
*expand_more*
*expand_more*
*expand_more*

$7xy+2x−3xy−2x_{2}+x+5x_{2} $

There are three steps to follow to combine like terms.
1

Identify Like Terms

2

Grouping Like Terms

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.

3

Add and Subtract the Coefficients

Finally, combine the like terms by adding or subtracting the coefficients of the variables as well as adding or subtracting the constants.

Now that all like terms have been combined, the initial expression has been reduced to its simplest form.

$7xy+2x−3xy−2x_{2}+x+5x_{2}⇕4xy+3x+3x_{2} $

Example

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.

Write an expression in its simplest form for the total distance from Heichi's house to the museum.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["8x+1"]}}

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 | $x$ |

Second Leg | $3x+7$ |

Third Leg | $4x−6$ |

$x+3x+7+4x−6 $

Notice that there are two different terms — $x-terms$ and constant terms. These are the like terms.
$x+3x+7+4x−6 $

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.
$x+3x+7+4x−6$

CommutativePropAdd

Commutative Property of Addition

$x+3x+4x+7−6$

AddTerms

Add terms

$8x+7−6$

SubTerm

Subtract term

$8x+1$

Example

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

Children pay $x$ dollars and adults pay $x_{2}$ dollars before noon. After noon, the prices go up to $y$ and $y_{2}$ 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 | $7$ | $12$ |

After Noon | $12$ | $20$ |

The table below shows the number of visitors on Sunday.

Sunday's Visitors | ||
---|---|---|

Children | Adults | |

Before Noon | $8$ | $15$ |

After Noon | $3$ | $4$ |

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mord\">$<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["15x+27x^2+15y+24y^2"]}}

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

The total amount can be written by adding the revenues from Saturday and Sunday.

The expression for Saturday's revenue can be written by adding the expressions for each part of the day.

Similar to Saturday, the expression for Sunday's revenue can be found by adding the expressions for each part of the day.
This expression represents how much money was made from admissions over the weekend.

$Saturday’s Revenue+Sunday’s Revenue $

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 | $7x$ | $12x_{2}$ | $7x+12x_{2}$ |

After Noon | $12y$ | $20y_{2}$ | $12y+20y_{2}$ |

$Saturday’s Revenue($)7x+12x_{2}+12y+20y_{2} $

The same steps can be followed for Sunday's visitors. Sunday's Visitors | |||
---|---|---|---|

Children $($)$ | Adults $($)$ | Total $($)$ | |

Before Noon | $8x$ | $15x_{2}$ | $8x+15x_{2}$ |

After Noon | $3y$ | $4y_{2}$ | $3y+4y_{2}$ |

$Sunday’s Revenue($)8x+15x_{2}+3y+4y_{2} $

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.
$7x+12x_{2}+12y+20y_{2}+8x+15x_{2}+3y+4y_{2}$

CommutativePropAdd

Commutative Property of Addition

$7x+8x+12y+3y+12x_{2}+15x_{2}+20y_{2}+4y_{2}$

AddTerms

Add terms

$15x+15y+27x_{2}+24y_{2}$

Example

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

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.

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["a","b","c"],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mord\">$<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["4a+20b+10c"]}}

Start by writing an expression for Saturday. Then, multiply that expression by $2.$

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 $three$ 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 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 | $3a$ | $4b$ | $5c$ | $3a+4b+5c$ |

Second Group | $-2a$ | $6b$ | $1c$ | $-2a+6b+1c$ |

Third Group | $-1a$ | $0$ | $-1c$ | $1a−1c$ |

$Saturday’s Revenue($)3a+4b+5c+6b−2a+1c−1c+1a $

Magdalena mentioned that the same thing happened on Sunday. This means that the whole expression has to be multiplied by $2$ to find the expression for the money made from both days. $Weekend Revenue($)2(3a+4b+5c+6b−2a+1c−1c+1a) $

This expression can be simplified by combining like terms first and then distributing the $2$ or by distributing first. Either process will result in the simplified expression. Here, the terms will be combined first.
$2(3a+4b+5c+6b−2a+1c−1c+1a)$

CommutativePropAdd

Commutative Property of Addition

$2(3a−2a+1a+4b+6b+5c+1c−1c)$

AddTerms

Add terms

$2(4a−2a+10b+6c−1c)$

SubTerms

Subtract terms

$2(2a+10b+5c)$

Distr

Distribute $2$

$2⋅2a+2⋅10b+2⋅5c$

Multiply

Multiply

$4a+20b+10c$

Closure

The beginning of this lesson presented a table to group 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!