5. Simplify Algebraic Expressions
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Chapter 4
5.
Simplify Algebraic Expressions
| Student Learning Objectives: |
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Why
This lesson focuses on simplifying algebraic expressions, a fundamental skill in algebra. It teaches how to identify like terms in an expression and combine them to make the expression simpler. This skill is particularly useful in various real-world scenarios, such as calculating total costs or distances. For example, if someone is trying to figure out the total distance for a multi-leg journey, they can use these techniques to combine the distances into a single, simpler expression. Understanding how to simplify algebraic expressions can make problem-solving quicker and more efficient, whether you're a student or a professional dealing with data analysis.
Lesson Settings & Tools
| | 9 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Simplify Algebraic Expressions
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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.
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Explore
How Can These Terms Be Grouped?
The applet shows some terms in a table. Try to group the terms in the table correctly.
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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). 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.
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Pop Quiz
Identifying Like Terms
Determine whether the terms are like terms.
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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. 7xy+2x-3xy-2x^2+x+5x^2 There are three steps to follow to combine like terms.
1
Identify Like Terms
expand_more 
2
Grouping Like Terms
expand_more 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
expand_more 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
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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.
Write an expression in its simplest form for the total distance from Heichi's house to the museum.
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Hint
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 | x |
| Second Leg | 3x+7 |
| Third Leg | 4x-6 |
These expressions can be added together to find the total distance. 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
The simplified expression is 8x+1.
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Example
Weekend Visitors to the Museum
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 |
Help Heichi figure out how much money the museum took in from admissions on Saturday and Sunday in terms of x and y. Then simplify the total amount.
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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. 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 ($) | |
| Before Noon | 7x | 12x^2 | 7x+ 12x^2 |
| After Noon | 12y | 20y^2 | 12y+ 20y^2 |
The expression for Saturday's revenue can be written by adding the expressions for each part of the day. Saturday's Revenue ($) 7x + 12x^2 + 12y + 20y^2 The same steps can be followed for Sunday's visitors.
| Sunday's Visitors | |||
|---|---|---|---|
| Children ($) | Adults ($) | Total ($) | |
| Before Noon | 8x | 15x^2 | 8x+ 15x^2 |
| After Noon | 3y | 4y^2 | 3y+ 4y^2 |
Similar to Saturday, the expression for Sunday's revenue can be found by adding the expressions for each part of the day. Sunday's Revenue ($) 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
This expression represents how much money was made from admissions over the weekend.
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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.
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.
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Hint
Start by writing an expression for Saturday. Then, multiply that expression by 2.
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 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 |
The expression for the money made on Saturday is the sum of these expressions. 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
This expression represents the total revenue from these groups on both Saturday and Sunday.
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Closure
Grouping Like Terms
The beginning of this lesson presented a table to group like terms.
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Simplify Algebraic Expressions
Exercise 2.1
Davontay and his friends want to get ice cream.
Their favorite place, the Ice Cream Home, has the following prices.
| Ice Cream Home Menu | |
|---|---|
| Item | Price ($) |
| Medium Ice Cream Sundae | x |
| Large Ice Cream Sundae | 2x |
| add 1 topping | add $ 1.00 |
| add 2 toppings | add $ 1.50 |
| add 3 toppings | add $ 2.00 |
The friends placed the following order.
- Two medium ice cream sundaes with 3 toppings
- One large ice cream sundae with 1 topping
- Four medium ice cream sundaes with 2 toppings
- Three medium ice cream sundaes without toppings
Write and simplify an expression for the total amount of money that Davontay's group will pay.
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We are given the menu for ice cream sundaes at the Ice Cream Home. Prices vary depending on the size and number of toppings on the sundae.
| Ice Cream Home Menu | |
|---|---|
| Item | Price ($) |
| Medium Ice Cream Sundae | x |
| Large Ice Cream Sundae | 2x |
| add 1 topping | add $ 1.00 |
| add 2 toppings | add $ 1.50 |
| add 3 toppings | add $ 2.00 |
We want to find the an expression for how much money the friends paid for their ice cream. Let's review their order.
- Two medium ice cream sundaes with 3 toppings
- One large ice cream sundae with 1 topping
- Four medium ice cream sundaes with 2 toppings
- Three medium ice cream sundaes without toppings
The friends can be divided into four groups based on their orders. Let's write the price of a single order from each group and then multiply these by the number of people who made that order. We can organize the information in a table.
| Price of a Single Order ($) | Total Group Price ($) | |
|---|---|---|
| Group 1 | x+2.00 | 2(x+2) |
| Group 2 | 2x+1.00 | 1(2x+1) |
| Group 3 | x+1.50 | 4(x+1.5) |
| Group 4 | x | 3x |
Now that we have an expression for each different order, we can find the total by adding all the expressions. 2(x+2)+ 2x+1 + 4(x+1.5) + 3x Let's simplify this expression. Remember to use the Distributive Property to simplify the terms between parentheses first.
Now that we have used the Distributive Property, we need to group like terms and add them. Let's do it!
The group paid a total of $11x+11 for the entire order. We did it!
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Hint & Answer
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Solution
Ali wants to create on a short film for a festival. He decides to rent the equipment he needs. The cost to rent the equipment is shown below.
| Item | Daily Rental Cost ($) | Down Payment ($) |
|---|---|---|
| Camera Kit | 300 | 150 |
| Audio Kit | 250 | 100 |
| Lighting Kit | 100 | 30 |
Write an expression for the total amount of money that Ali needs to rent everything he wants for d days.
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We are told that Ali needs to rent equipment for d days to shoot a short film. We are asked to write an expression that can be used to represent the total cost. Let's take a look at the cost of each piece of equipment.
| Item | Daily Rental Cost ($) | Down Payment ($) |
|---|---|---|
| Camera Kit | 300 | 150 |
| Audio Kit | 250 | 100 |
| Lighting Kit | 100 | 30 |
Let's start by finding the total cost of renting the camera kit for d days. The rental fee is $ 300 per day. If Ali wants to rent the camera kit for d days, he will pay the product of d and $ 300. 300 d However, that is not the whole fee. Ali will also need to pay an additional down payment of 150. Let's add this to our expression. 300 d + 150 This is the total cost to rent the camera kit for d days. Let's write the costs of the other pieces of equipment in a similar way. We can organize this information in the table.
| Item | Daily Rental Cost ($) | Down Payment ($) | Cost for d Days ($) |
|---|---|---|---|
| Camera Kit | 300 | 150 | 300 d + 150 |
| Audio Kit | 250 | 100 | 250 d+ 100 |
| Lighting Kit | 100 | 30 | 100 d+ 30 |
Finally, let's add the costs of each piece of equipment to write an expression for the total cost of the rental. Remember to group the like terms together to combine them and simplify the expression.
Ali will have to pay $650d+280 to be able to shoot his dream project. Good job!
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Solution
Consider the following rectangle.
Write and simplify an expression for the perimeter of the rectangle.
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We are asked to write an algebraic expression for the perimeter of the given rectangle. Let's take a look at the figure.
The perimeter of a figure is the sum of all its side lengths. We are only given the length of two sides, but remember that a rectangle has two pairs of sides of equal length. Let's fill in the rest of the diagram.
We can multiply each both lengths by 2 and add the products to find the perimeter. Remember to use the Distributive Property.
The perimeter of the rectantle is 9z+2. We did it!
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