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.
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| | 9 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
One common type of expressions called linear expressions. These expressions are useful to model different situations in real life. Because of this, 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.
Challenge
Buying Equipment for a Competition
Dominika really enjoys longboard dancing. She is excited to participate in a competition soon. However, she needs some equipment, so she decided to go to the local skate shop to buy some stuff.
Dominika used her receipt to write out what she spent as an algebraic expression. Maybe her math teacher would give her extra credit when she showed it to him!
While preparing her board for practice, Dominika noticed that she forgot to buy a couple of things, so she had to go back to the store. She wrote another expression for her new purchase.
Now she is ready to practice. 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 one. 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. The most common form of a linear expression is given below.
In this expression, a and b are real numbers, with a=0. To completely understand the definition of a linear expression, some important concepts will be be broken down. Consider the example linear expression x−5y+2.
| x−5y+2 | ||
|---|---|---|
| Concept | Explanation | Example |
| Term | Parts of an expression separated by a +or −sign. |
x, -5y, 2 |
| Coefficient | A constant that multiplies a variable. If a coefficient is 1, it does not need to be written due to the Identity Property of Multiplication. | 1, -5 |
| Linear Term | A term that contains exactly one variable whose exponent is 1. | x, -5y |
| Constant Term | A term that contains no variables. It consists only of a number with its corresponding sign. | 2 |
The following table shows some examples of linear and non-linear expressions.
| Linear Expressions | Non-linear Expressions |
|---|---|
| 3x | 5 |
| -5y+1 | 2xy−3 |
| 3x−21y+2 | x1−2 |
| πx+6y | 5x2+x−1 |
3x+4−2x+9−x=13
The expression 3x+4−2x+9−x looks like a linear expression, but the result after simplifying is 13, which is a constant and therefore not a linear expression.Pop Quiz
Identifying Linear Expressions
Determine whether the given expression is a linear expression.
Discussion
Adding and Subtracting Linear Expressions
When adding or subtracting linear expressions, the process is similar to performing those operations on numbers. As an example, consider two linear expressions.
Adding two linear expressions can result in either a new linear expression or a constant, depending on whether the linear terms cancel each other out. In summary, when adding and subtracting linear expressions, simplify the resulting expression by combining like terms.
Expression One:Expression Two:2x+113x−7
Next, these expressions will be added and subtracted. 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
Subtracting linear expressions is similar to adding them. However, when writing the expression of the subtraction, 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
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 (minutes) |
|---|---|
| Friday | 4t+32 |
| Saturday | 7t+26 |
| Sunday | 3t−15 |
a Write an expression for the total practice time on Friday and Saturday.
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b Dominika and her friends practiced more on Saturday than on Sunday. Write the difference between the practice times on these days.
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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 (Minutes) |
|---|---|
| Friday | 4t+32 |
| Saturday | 7t+26 |
| Sunday | 3t−15 |
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
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 (Minutes) |
|---|---|
| Friday | 4t+32 |
| Saturday | 7t+26 |
| Sunday | 3t−15 |
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
Example
Longboard Competition Results
Finally, it is time for the longboard competition!
Dominika’s Final Score3x+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 will win. She then wrote down Magdalena's two best scores. | 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 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 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, the sum of two linear expressions is subtracted 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
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
b In Part A, it was 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: Second Place: MagdalenaDominika
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!
Pop Quiz
Practice Adding and Subtracting Linear Expressions
Add or subtract the given linear expressions.
Closure
How Much Did Dominika Spend on Equipment?
Earlier in this lesson, Dominika wrote a couple of expressions to show how much she spent on two different days buying equipment for the competition.
Add the following linear expressions.
m+11m−7
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We can write a new expression by adding the two given linear expressions. (m+11)+(m-7) ⇓ m+11+m-7 Now we need to simplify this expression. The first step in simplifying this expression is to identify which, if any, terms can be combined. Remember, only like terms — constant terms or terms with the same variable — can be combined. m + 11 + m - 7 In this case, we can see two m-terms and two constants. Both of these pairs can be combined. We will apply the Commutative Property of Addition to group the like terms together and then combine them by adding or subtracting.
The expression is now simplified. Good job!
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Consider the following linear expressions.
6−7n-16n+3
Subtract the second expression from the first expression.
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We can write a new expression by subtracting the two given linear expressions. Remember that we need to distribute the negative sign to replace every term of the subtrahend expression by its corresponding additive inverse. (6-7n)-(-16n+3) ⇓ 6-7n + 16n - 3 Now we need to simplify this expression. The first step in simplifying this expression is to identify which, if any, terms can be combined. Remember, only like terms — constant terms or terms with the same variable — can be combined. 6 - 7n + 16n - 3 In this case, we can see two n-terms and two constants. Both of these pairs can be combined. We will apply the Commutative Property of Addition to group the like terms together and then combine them by adding or subtracting.
Now the expression is simplified. Good job! Remember that we can apply the Commutative Property of Addition again to rewrite this expression if we want. 3+9n ⇓ 9n+3
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Ignacio is almost out of clean clothes. Because of this, he decided to do the laundry during the weekend.
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We want to find how many pounds of laundry Ignacio washed over the weekend. We can find this by adding the pounds washed from Saturday and Sunday. Saturday's Laundry + Sunday's Laundry We are given these quantities as linear expressions. Saturday's Laundry + Sunday's Laundry ⇓ 3c+1 + 5c-2 If we add them together, we have an expression for the total laundry. However, we can simplify it to write it more compactly. First, let's identify the like terms. 3c + 1 + 5c - 2 We can see that there are two c-terms and two constants. Both of these pairs of like terms can be combined. Let's apply the Commutative Property of Addition to group the terms, then combine them by adding or subtracting.
This means that Ignacio did 8c-1 pounds of laundry during the weekend. We did it!
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Izabella has been training for a marathon, but she wants to be more consistent with her training. She ran different distances on Friday and on Saturday.
Friday Miles: Saturday Miles: 3m+12m+5
Write and simplify an expression for the difference between the distances that Izabella ran on Friday and on Saturday.
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We want to find the difference between the distances Izabella ran on Friday and Saturday. This difference is found by subtracting the miles run from each day. Friday Miles- Saturday Miles We are given these miles as linear expressions. Let's substitute them to write the expression for the difference. Friday Miles- Saturday Miles ⇓ 3m+1 - ( 2m+5 ) Now we have an expression for the difference. This is great progress! Let's simplify it. To start, we will distribute the negative sign and identify the like terms — terms with the same variables raised to the same power. 3m+1 - (2m+5 ) ⇓ 3m + 1 - 2m - 5 We can see that there are two m-terms and two constants. These terms can be grouped using the Commutative Property of Addition and combined by adding or subtracting.
The difference between the distances Izabella ran on Friday and Saturday was m-4 miles.
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Tiffaniqua followed these three steps to subtract two linear expressions.
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Let's subtract the linear expressions ourselves to identify whether Tiffaniqua made a mistake. We can begin by distributing the negative sign to replace every term in the expression being subtracted with its additive inverse. (x+1)-(x+3) = x+1 -x - 3 After distributing the negative sign, we can see that the 3 is being subtracted. When Tiffaniqua did the subtraction, she added the 3. This means that Tiffaniqua made a mistake in Step 1. Let's continue solving to see the real result.
The correct result of the subtraction is -2.
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Add and Subtract Linear Expressions
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