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.
Show less Show more expand_more| Student Learning Objectives: |
|---|
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| | 9 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Add and Subtract Linear Expressions
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Linear expressions are one type of common algebraic expressions. These expressions are useful to model different situations in real life. It is sometimes necessary to perform mathematical operations with these expressions. 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 really enjoys longboard dancing. She decided to go to the local skate shop to buy some 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 for the competition.
She documented her practice times using linear expressions.
| Day | Practice Time (min) |
|---|---|
| Friday | 4t+32 |
| Saturday | 7t+26 |
| Sunday | 3t-15 |
a
Write an expression for her total practice time on Friday and Saturday.
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b
Dominika and her friends practiced longer on Saturday than on Sunday. How much longer did Dominika and her friends practice on Saturday than on Sunday?
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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
Add the practice time from each day to write the total practice time for Friday and Saturday.
Friday's Practice Time + Saturday's Practice Time The corresponding expressions can be found in the given table.
| Day | Practice Time (min) |
|---|---|
| Friday | 4t+32 |
| Saturday | 7t+26 |
| Sunday | 3t-15 |
After finding the sum, we will simplify the expression by combining the like terms.
Friday's Practice Time + Saturday's Practice Time
SubstituteExpressions
Substitute expressions
4t+32+ 7t+26
CommutativePropAdd
Commutative Property of Addition
4t+7t+32+26
AddTerms
Add terms
11t+58
Dominika and her friends practiced for 11t+58 minutes in total on Friday and Saturday.
b
This time Sunday's practice time will be subtracted from Saturday's.
Saturday's Practice Time - Sunday's Practice Time The corresponding expressions can be found in the given table.
| Day | Practice Time (min) |
|---|---|
| Friday | 4t+32 |
| Saturday | 7t+26 |
| Sunday | 3t-15 |
When subtracting, remember to distribute the negative sign before combining the expressions!
Saturday's Practice Time - Sunday's Practice Time
SubstituteExpressions
Substitute expressions
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 and her friends practiced for 4t+41 minutes longer on Saturday than on Sunday.
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Example
Longboard Competition Results
It's finally time for the longboard dance competition!
Each participant's score is given by adding the two best scores from three rounds. Dominika wrote her final score as an expression. Dominika's Final Score 3x+2y+12 Dominika is really proud of her score and she thinks that she can win. After seeing her friend Magdalena perform, though, Dominika starts thinking that maybe Magdalena might win.
| Magdalena's Two Best Scores | |
|---|---|
| 2x+y+7 | x+y+6 |
In the end, Dominika and Magdalena placed in the top two positions of the competition.
a
Write a simplified expression for the difference between Dominika's and Magdalena's scores.
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b
Who won first place?
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Hint
a
Subtract the sum of Magdalena's scores from Dominika's score.
b
Can we tell who has the higher score?
Solution
a
The difference between Dominika's and Magdalena's final scores can be found by subtracting Magdalena's final score from Dominika's score.
Dominika's Final Score - Magdalena's Final Score Magdalena's final score is not given but we know it can be found by adding her two best scores. Then the difference between the girls' scores can be written as an expression. 3x+2y+12 - ( 2x+y+7 + x+y+6) Here we are subtracting the sum of two linear expressions from another linear expression. Remember that the order of operations must be followed — this means that the expressions between parentheses are simplified first by combining the 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
Magdalena's final score is 3x+2y+13. Let's subtract this score from Dominika's final score.
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 their scores is -1. They are really close!
b
In Part A, we found that subtracting Magdalena's final score from Dominika's score results in -1.
Dominika's Final Score - Magdalena's Final Score ⇓ -1 This means that Magdalena's score is slightly higher than Dominika's. Since the girls won the top two places, Magdalena won the first place. First Place: & Magdalena Second Place: & Dominika Dominika did not win first place in the competition, but she is really proud of herself for winning second place and being so close. Now she is motivated to practice and get even better!
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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.
Dominika wants to combine the expressions to find one that reflected 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. She is really dedicated to longboard dancing!
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Add and Subtract Linear Expressions
Exercise 2.1
Consider that a triangle has a perimeter of 19x+13.
Write and simplify an expression for the third side of the triangle.
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We want to determine the length of the missing side of the triangle. Let's start by looking at the given diagram!
The side lengths are given as linear expressions. We know two side lengths, 8x-14 and 7x+19. We also know that the perimeter of the triangle, 19x+13, is equal to the sum of all the side lengths of the triangle. This means that to calculate the missing side length, we can subtract the known lengths of the triangle from 19x+13. 19x+13- (8x-14)- (7x+19) We can remove the parentheses by distributing the negative signs to the expressions being subtracted. When doing this, remember that every term is replaced by its additive inverse. Then, we can use the properties of operations to group like terms and simplify.
The missing side of the triangle is 4x + 8.
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Hint & Answer
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Solution
Fill in the blank.
|
Adding two linear expressions results in a linear expression. |
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We want to know what happens when two linear expressions so we can fill in the blank. Let's experiment with a few different examples. First, consider two linear expressions. Expression One: & 5x + 4 Expression Two: & 7x + 3 Let's add the expressions to examine the result.
We can see that the result is another linear expression. This means that adding two linear expressions at least sometimes results in a linear expression. To discard the always option, consider the following linear expressions where the variable terms have opposite coefficients. Expression One: & 5x + 4 Expression Two: & -5x + 6 Let's add these expressions together and see what happens!
A linear expression needs to have at least one linear term, which means that 10 is not a linear expression. We can see that there are cases where adding two linear expressions results in a linear expression and some cases where it gives us only a constant. Therefore, the answer is sometimes.
Adding two linear expressions sometimes results in a linear expression.
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Solution
A right triangle has the following side lengths.
- The height has a length of a.
- The base is 3 more than two times the length of the height.
- The final side has a length that is 0.5 longer than the base.
Write and simplify an expression for the perimeter of the triangle.
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We want to write and simplify an expression for the perimeter of a triangle. Recall that the perimeter of a triangle equals the sum of its sides. Since the given triangle is a right triangle, the height is one of its sides. Height +Base+Last Side Now let's examine the given information to determine the length of each side. We are told that the height has a length of a. Height: a We also know that the base is 3 more than 2 times the length of the height. We can write this as an expression. Base: ( 2a+ 3) Finally, the last side is 0.5 longer than to the base of the triangle. Let's write the expression. Last Side: ( 2a+ 3+ 0.5) Now, we are ready to write the expression for the perimeter of the triangle by substituting the linear expressions for each side. Height +Base+Last Side ⇓ a+( 2a+ 3)+( 2a+ 3+ 0.5) Finally, we will use the Associative Property of Addition and the Commutative Property of Addition to simplify the expression we created.
The perimeter of the triangle equals 5a+6.5. Good job!
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Add and Subtract Linear Expressions
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