6. Add and Subtract Linear Expressions
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Chapter 4
6.
Add and Subtract Linear Expressions
This lesson provides an in-depth look at adding and subtracting linear expressions, a fundamental skill in algebra. It uses real-world examples, such as Dominika and Magdalena's competition scores and Ignacio's weekend laundry, to demonstrate how these mathematical operations can be applied in everyday life. The lesson emphasizes the importance of combining like terms and distributing negative signs correctly when performing these operations. It also explains the concept of linear terms and constant terms, which are crucial for understanding linear expressions. The aim is to equip you with the skills needed to simplify and manipulate linear expressions, whether for academic purposes or real-world problem-solving.
| Student Learning Objectives: |
|---|
|
| | 9 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Add and Subtract Linear Expressions
Slide of 9
Linear expressions are one type of common algebraic expression. These expressions are useful to model different situations in real life. This lesson will explore how to identify linear expressions and how to add or subtract them.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Challenge
Buying Equipment for a Competition
Dominika went to the local skate shop to buy some longboard dancing supplies for an upcoming competition.
Dominika used her receipt to write out what she spent as two algebraic expressions. Maybe her math teacher will give her extra credit when she shows it to him!
On her way to the park to practice, Dominika wondered how to combine the expressions to represent the total she spent at the store.
Write an expression for her total expenses.
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Discussion
Linear Expressions
A linear term is an algebraic expression that includes a coefficient multiplied by a variable with an exponent of 1. A linear expression is an expression that includes at least one linear term and any constant terms. No other type of terms may be included.
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Pop Quiz
Identifying Linear Expressions
Determine whether the given expression is a linear expression.
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Discussion
Adding and Subtracting Linear Expressions
The process of adding or subtracting linear expressions is similar to performing those operations on numbers. Consider two linear expressions. Expression One: &2x + 11 Expression Two: &3x - 7 Let's add and subtract these expressions.
Adding
To add two linear expressions, follow the same process as adding two numbers. (2x + 11)+(3x - 7) ⇓ 2x + 11+3x - 7 The result is an expression with two x-terms and two constants. Next, pair up these terms into two sets of like terms. Finally, combine each pair of like terms and simplify.
2x + 11 +3x - 7
CommutativePropAdd
Commutative Property of Addition
2x +3x + 11 - 7
AddTerms
Add terms
5x+11-7
SubTerms
Subtract terms
5x+4
Subtracting
When writing the subtrahend expression, it is important to distribute the negative sign to change each of terms in the subtrahend expression to its additive inverse. (2x + 11)-(3x - 7) ⇓ 2x + 11-3x + 7 Next, group like terms and simplify the expression.
2x + 11 -3x + 7
CommutativePropAdd
Commutative Property of Addition
2x -3x + 11 + 7
SubTerms
Subtract terms
- x+11+7
AddTerms
Add terms
- x + 18
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Example
Longboard Dancing Practice Time
Dominika met up with her friends over the weekend to practice longboard dancing. 
She documented her practice times using linear expressions.
| Day | Practice Time (min) |
|---|---|
| Friday | 4t+32 |
| Saturday | 7t+26 |
| Sunday | 3t-15 |
a
Write a simplified expression for her total practice time on Friday and Saturday.
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b
Dominika practiced longer on Saturday than on Sunday. How much longer did she practice on Saturday than on Sunday? Answer with a simplified expression.
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Hint
a
Add the expressions for the practice times on Friday and Saturday.
b
Subtract the expression representing Sunday's practice time from the expression for Saturday's practice time.
Solution
a
Let's add Dominika's practice times for Friday and Saturday to find the expression for the total time she practiced on those two days.
4t + 32 + 7t + 26 Now, we can simplify the expression by combining the like terms.
Dominika practiced for 11t+58 minutes in total on Friday and Saturday.
b
To find how much longer she practiced on Saturday than Sunday, we can subtract Sunday’s practice time from Saturday’s practice time.
7t+26 - ( 3t-15) Next, we can simplify the expression. To do this, we distribute the minus sign by changing the sign of each term in the parentheses. Then, we combine like terms.
7t+26-( 3t-15)
Distr
Distribute -1
7t+26-3t+15
CommutativePropAdd
Commutative Property of Addition
7t-3t+26+15
SubTerms
Subtract terms
4t+26+15
AddTerms
Add terms
4t+41
Dominika practiced for 4t+41 minutes longer on Saturday than on Sunday.
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Example
Arcade Tickets
Ignacio and Ali went to a local arcade to play games. They earned tickets by winning games that they could exchange for prizes.
The number of tickets Ignacio earned at the end of the evening is represented by the following expression.
3x+2y+12
Ali earned tickets from two different games. 2x+y+7
x+y+6
a
Write a simplified expression for the difference between the number of tickets Ignacio and Ali earned.
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b
Who earned more tickets?
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Hint
a
Subtract the sum of Ali's tickets from the number of tickets Ignacio got.
b
Can we tell who got more tickets?
Solution
a
First, let's simplify the expression for the number of tickets Ali earned. We can do this by combining like terms.
2x+ y+ 7+ x+ y+ 6
CommutativePropAdd
Commutative Property of Addition
2x+ x+ y+ y+ 7+ 6
AddTerms
Add terms
3x+2y+13
Ali earned 3x+2y+13 tickets. Let's subtract this from Ignacio's total tickets. 3x+2y+12 - ( 3x+2y+13) To simplify this expression, first, we distribute the negative sign by changing the sign of each term in the parentheses. Then, we combine like terms.
3x+ 2y+ 12-( 3x+ 2y+ 13)
Distr
Distribute -1
3x+ 2y+ 12- 3x- 2y- 13
CommutativePropAdd
Commutative Property of Addition
3x- 3x+ 2y- 2y+ 12- 13
SubTerms
Subtract terms
-1
The difference between the two expressions is -1.
b
In Part A, we found that subtracting Ali's total tickets from Ignacio's total tickets gave us -1.
Ignacio's Tickets - Ali's Tickets = - 1 This means that Ali earned 1 more ticket than Ignacio.
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Pop Quiz
Practice Adding and Subtracting Linear Expressions
Add or subtract the given linear expressions.
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Closure
How Much Did Dominika Spend on Equipment?
Earlier in this lesson, Dominika wrote a couple of expressions to show how much money she spent buying equipment for the competition.
She wants to combine the expressions to find one that reflects how much money she spent in total. Notice that the expressions she wrote are linear expressions. To add them, group the like terms and combine them.
This means that Dominika spent $8x+9 in total when buying the equipment.
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Add and Subtract Linear Expressions
Exercise 3.1
Kriz is happy because they have a new job. Every week, they divide their paycheck between two bank accounts. The money in the accounts after working for w weeks can be written with algebraic expressions. Account One: & 30w + 120 dollars Account Two: & 15w + 80 dollars Kriz wants to buy a gaming console for $135. How many does weeks Kriz need to work to buy the console if they want to keep $200 between their accounts?
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We know that Kriz wants to buy a gaming console while still keeping $200 in savings. The money in each account is represented by linear expressions. We can add these expressions to find out how much money Kriz has between the accounts after w weeks of working.
We want to determine how many weeks Kriz needs to save up to buy the console for $135 while still keeping $200 between their accounts. In other words, the amount that Kriz needs to save can be written as the sum of 135 and 200. 135+200=335 Kriz has to save $335. This means that Kriz's total savings have to be equal to 335. We can write this as an equation. 45w+200=335 We need the value of w that makes the statement true. We can use the Properties of Equality to isolate w on one side of the equation. Let's do it!
As we can see, when w=3, Kriz's savings are equal to $335. This means that they need to save for 3 weeks to buy the gaming console. We did it!
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Hint & Answer
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Solution
Consider that both the height and the base of a triangle are extended 3 centimeters.
Write and simplify an expression for the area of the shaded region.
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We are asked to find the area of the shaded region.
Let's start by recalling that the area of a triangle is half the product of its base b and its height h. Area of a Triangle = 1/2 bh Notice that the area of the shaded region equals the area of the larger triangle minus the area of the smaller triangle. Let's write an expression that represents the area of the shaded region in square centimeters using this information. Area of the Shaded Region = 1/2 (12+3)(z+3) - 1/2(12)(z) Now we can simplify the expression that we got. We will begin by expanding the expression using the Distributive Property. Then, we will use the Commutative and Associative Properties to reorder and group like terms.
We can see that the expression is a subtraction of linear expressions. Let's simplify it further.
The area of the shaded region is equal to 1.5z+22.5 square centimeters.
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Solution
Add and Subtract Linear Expressions
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