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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Lesson Settings & Tools
| | 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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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.
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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-1/2y+2 | 1/x-2 |
| π x+6y | 5x^2+x-1 |
It is important to keep in mind that before classifying an expression, it must be written in simplest form. 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.
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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
When adding or subtracting linear expressions, the process is similar to performing those operations on numbers. As an example, consider two linear expressions. Expression One: &2x + 11 Expression Two: &3x - 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
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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 (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 |
When the total expression is written, it can be simplified 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 (Minutes) |
|---|---|
| Friday | 4t+32 |
| Saturday | 7t+26 |
| Sunday | 3t-15 |
When subtracting these expressions, remember to distribute the negative sign!
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
Since it is given that the friends practiced longer on Saturday, this means that Dominika and her friends practiced for 4t+41 minutes longer on Saturday than on Sunday.
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Example
Longboard Competition Results
Finally, it is time for the longboard competition!
Each participant's score is given by adding the two best scores from three rounds. After receiving her scores, 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 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
Magdalena's final score is 3x+2y+13. This score will be subtracted 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, 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: & 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 she spent on two different days buying equipment for the competition.
Dominika wanted 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, like terms should be grouped together and then added.
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 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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