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“Descartes, if you solve for in the equation, what do you get?”
“I can’t find my algebra tiles, so I am painting some of my dog biscuits.”
“Now I will be able to solve the equation 2 x (-2) = 2.”
Expressions and
Equations
an’t find my algebra tiles, so I am inting some of my dog biscuits.”
“Now I will be able to solve the equation 2 x (-2)^ = 2.”
3.1 Algebraic Expressions
3.2 Adding and Subtracting Linear Expressions
3.3 Solving Equations Using Addition3.3 Solving Equations Using Addition
or Subtractionor Subtraction
3.4 Solving Equations Using Multiplication3.4 Solving Equations Using Multiplication
or Divisionor Division
3.5 Solving Two-Step Equations3.5 Solving Two-Step Equations
3.13.1 Algebra3.1 Algebra3. 333 .1 Algebra. 11 11 AlgebraAAAAAAAAlll lll gggg ggeeeeeeeebbbbbbbb bbrr rraaa
Example 1 Evaluate 6 x + 2 y when
x = − 3 and y = 5.
6 x + 2 y = 6(−3) + 2(5) Substitute −3 forx and 5 fory. = − 18 + 10 Using order of operations, multiply 6 and −3, and 2 and 5. = − 8 Add −18 and 10.
Example 2 Evaluate 6x 2 − 3(y + 2) + 8 whenx = − 2 andy = 4.
6 x^2 − 3( y + 2) + 8 = 6(−2)^2 − 3(4 + 2) + 8 Substitute −2 forx and 4 fory. = 6(−2)^2 − 3(6) + 8 Using order of operations, evaluate within the parentheses. = 6(4) − 3(6) + 8 Using order of operations, evaluate the exponent. = 24 − 18 + 8 Using order of operations, multiply 6 and 4, and 3 and 6. = 14 Subtract 18 from 24. Add the result to 8.
Evaluate the expression whenx = −
— 4
andy = 3.
1. 2 xy 2. 12 x − 3 y 3. − 4 x − y + 4 4. 8 x − y^2 − 3
Writing Algebraic Expressions
Example 3 Write the phrase as an algebraic expression.
a. the sum of twice a number m and four b. eight less than three times a number x 2 m + 4 3 x − 8
Write the phrase as an algebraic expression.
5. five more than three times a number q 6. nine less than a number n 7. the product of a number p and six 8. the quotient of eight and a number h 9. four more than three times a number t 10. two less than seven times a number c
What You
Learned Before
“Hey, Descartes ... The expressions are equivalent.”^ True or False:
Section 3.1 Algebraic Expressions 81
Use what you learned about simplifying algebraic expressions to complete Exercises 12–14 on page 84.
3. IN YOUR OWN WORDS How can you simplify an algebraic expression? Give an example that demonstrates your procedure. 4. REASONING Why would you want to simplify an algebraic expression? Discuss several reasons.
Work with a partner. Use your results from Activity 1 to write a lesson on simplifying an algebraic expression.
22 ACTIVITY: Writing a Math Lesson
Simplifying an Algebraic Expression
Examples
Exercises
Key Idea (^) Use the following steps to simplify an algebraic expression.
Simplify the expression.
a.
b.
c.
Describe steps you can use to simplify an expression.
Write 3 examples. Use expressions from Activity 1.
Write 3 exercises. Use expressions different from the ones in Activity 1.
Communicate Precisely What can you do to make sure that you are communicating exactly what is needed in the Key Idea?
Math Practice
3.1 Lesson
82 Chapter 3 Expressions and Equations
Simplify (^3) — 4
y + 12 −^1 — 2
y − 6.
3 — 4
y + 12 − (^) —^1 2
y − 6 = (^3) — 4
y + 12 + (^) ( − —^1 2
y (^) ) + (−6) Rewrite as a sum.
= —^3
4
y + (^) ( −^1 — 2
y (^) ) + 12 + (−6) Commutative Property of Addition = (^) [ 3 — 4
y + 12 + (−6) Distributive Property
1 — 4 y + 6 Combine like terms.
Identify the terms and like terms in the expression.
1. y + 10 − (^) —^3 2
y 2. 2 r^2 + 7 r − r^2 − 9 3. 7 + 4 p − 5 + p + 2 q
Simplify the expression.
4. 14 − 3 z + 8 + z 5. 2_._ 5 x + 4.3 x − 5 6. (^) —^3 8
b − (^) —^3 4
b
EXAMPLE 22 Simplifying an Algebraic Expression
Identify the terms and like terms in each expression. a. 9 x − 2 + 7 − x b. z^2 + 5 z − 3 z^2 + z Rewrite as a sum of terms. Rewrite as a sum of terms. 9 x +(−2) + 7 + (− x ) z^2 + 5 z + (− 3 z^2 ) + z
Terms: 9 x , −2, 7, − x Terms: z^2 , 5 z , − 3 z^2 , z Like terms: 9 x and − x , −2 and 7 Like terms: z^2 and − 3 z^2 , 5 z and z
EXAMPLE 11 Identifying Terms and Like Terms
Exercises 5– and 12–
Key Vocabulary like terms, p. 82 simplest form, p. 82
Parts of an algebraic expression are called terms. Like terms are terms that have the same variables raised to the same exponents. Constant terms are also like terms. To identify terms and like terms in an expression, first write the expression as a sum of its terms.
An algebraic expression is in simplest form when it has no like terms and no parentheses. To combine like terms that have variables, use the Distributive Property to add or subtract the coefficients.
Study Tip To subtract a variable term, add the term with the opposite coefficient.
Lesson Tutorials
84 Chapter 3 Expressions and Equations
3.1 Exercises
1. WRITING Explain how to identify the terms of 3 y − 4 − 5 y. 2. WRITING Describe how to combine like terms in the expression 3 n + 4 n − 2. 3. VOCABULARY Is the expression 3 x + 2 x − 4 in simplest form? Explain. 4. REASONING Which algebraic expression is in simplest form? Explain.
5 x − 4 + 6 y 4 x + 8 − x
3(7 + y ) 12 n − n
9 +(-6)=3 3 +(-3)= 4 +(-9)= 9 +(-1)=
Identify the terms and like terms in the expression.
5. t + 8 + 3 t 6. 3 z + 4 + 2 + 4 z 7. 2 n − n − 4 + 7 n 8. − x − 9 x^2 + 12 x^2 + 7 9. 1.4 y + 5 − 4.2 − 5 y^2 + z 10. 1 — 2
s − 4 + 3 — 4
s + 1 — 8
− s^3
11. ERROR ANALYSIS Describe and correct the error in identifying the like terms in the expression.
Simplify the expression.
12. 12 g + 9 g 13. 11 x + 9 − 7 14. 8 s − 11 s + 6 15. 4.2 v − 5 − 6.5 v 16. 8 + 4 a + 6.2 − 9 a 17. 2 — 5
y − 4 + 7 − —^9 10
y
18. 4( b − 6) + 19 19. 4 p − 5( p + 6) 20. − —^2 3
(12 c − 9) + 14 c
21. HIKING On a hike, each hiker carries the items shown. Write an expression in simplest form that represents the weight carried by x hikers. Interpret the expression.
Help with Homework
3 x − 5 + 2 x^2 + 9 x = 3 x + 2 x^2 + 9 x − 5 Like Terms: 3 x , 2 x^2 , and 9 x
✗
e t
2.2 lb
4.6 lb
3.4 lb
Section 3.1 Algebraic Expressions 85
22. STRUCTURE Evaluate the expression − 8 x + 5 − 2 x − 4 + 5 x when x = 2 before and after simplifying. Which method do you prefer? Explain. 23. REASONING Are the expressions 8 x^2 + 3( x^2 + y (^) ) and 7 x^2 + 7 y + 4 x^2 − 4 y equivalent? Explain your reasoning. 24. CRITICAL THINKING Which solution shows a correct way of simplifying 6 − 4(2 − 5 x )? Explain the errors made in the other solutions.
A 6 − 4(2 − 5 x ) = 6 − 4(−3 x ) = 6 + 12 x B 6 − 4(2 − 5 x ) = 6 − 8 + 20 x = −2 + 20 x
C 6 − 4(2 − 5 x ) = 2(2 − 5 x ) = 4 − 10 x D 6 − 4(2 − 5 x ) = 6 − 8 − 20 x = −2 − 20 x
25. BANNER Write an expression in simplest form that represents the area of the banner. 26. CAR WASH Write an expression in simplest form that represents the earnings for washing and waxing x cars and y trucks.
MODELING Draw a diagram that shows how the expression can represent the area of a figure. Then simplify the expression.
27. 5(2 + x + 3) 28. (4 + 1)( x + 2 x ) 29. You apply gold foil to a piece of red poster board to make the design shown. a. Write an expression in simplest form that represents the area of the gold foil. b. Find the area of the gold foil when x = 3. c. The pattern at the right is called “St. George’s Cross.” Find a country that uses this pattern as its flag.
Order the lengths from least to greatest. (Skills Review Handbook)
30. 15 in., 14.8 in., 15.8 in., 14.5 in., 15.3 in. 31. 0.65 m, 0.6 m, 0.52 m, 0.55 m, 0.545 m 32. MULTIPLE CHOICE A bird’s nest is 12 feet above the ground. A mole’s den is 12 inches below the ground. What is the difference in height of these two positions? (Section 1.3) A 24 in. B 11 ft C 13 ft D 24 ft
Car Truck^26
Wash $8 $
Wax $12 $
3 ft
(3(3 àà xxx ) ft) ft
We’re #
12 in.
20 in.
x in.
x in.
Section 3.2 Adding and Subtracting Linear Expressions 87
Use what you learned about adding and subtracting algebraic expressions to complete Exercises 6 and 7 on page 90.
5. IN YOUR OWN WORDS How can you use algebra tiles to add or subtract algebraic expressions? 6. Write the difference of two algebraic expressions modeled by the algebra tiles. Then use the algebra tiles to simplify the expression.
(^)
Work with a partner. Write the difference of two algebraic expressions modeled by the algebra tiles. Then use the algebra tiles to simplify the expression.
a.
b.
c. ^
d.
^
Work with a partner. Use algebra tiles to model the sum or difference. Then use the algebra tiles to simplify the expression. a. (2 x + 1) + ( x − 1) b. (2 x − 6) + (3 x + 2) c. (2 x + 4) − ( x + 2) d. (4 x + 3) − (2 x − 1)
33 ACTIVITY: Subtracting Algebraic Expressions
44 ACTIVITY: Adding and Subtracting Algebraic Expressions
Use Expressions What do the tiles represent? How does this help you write an expression?
Math Practice
3.2 Lesson
88 Chapter 3 Expressions and Equations
Find 2( − 7.5 z + 3) + (5 z − 2).
2(−7.5 z + 3) + (5 z − 2) = − 15 z + 6 + 5 z − 2 Distributive Property
= − 15 z + 5 z + 6 − 2 Commutative Property of Addition = − 10 z + 4 Combine like terms.
Find the sum.
1. ( x + 3) + (2 x − 1) 2. (–8 z + 4) + (8 z − 7) 3. (4 − n ) + 2(–5 n + 3) 4. 1 — 2 ( w − 6) + 1 — 4 ( w + 12)
EXAMPLE 22 Adding Linear Expressions
Find each sum. a. ( x − 2) + (3 x + 8) Vertical method: Align x − 2 like terms vertically and add. + 3 x + 8 4 x + 6 b. (− 4 y + 3) + (11 y − 5) Horizontal method: Use properties of operations to group like terms and simplify.
(− 4 y + 3) + (11 y − 5) = − 4 y + 3 + 11 y − 5 Rewrite the sum.
= − 4 y + 11 y + 3 − 5 Commutative Property of Addition = (− 4 y + 11 y ) + (3 − 5) Group like terms. = 7 y − 2 Combine like terms.
EXAMPLE
d h
11 Adding Linear Expressions
Exercises 8–
Key Vocabulary linear expression, p. 88
A linear expression is an algebraic expression in which the exponent of the variable is 1.
Linear Expressions (^) − 4 x 3 x + (^5 5) − —^1 6 x
Nonlinear Expressions x^2 − 7 x^3 + x x^5 + 1
You can use a vertical or a horizontal method to add linear expressions.
Lesson Tutorials
90 Chapter 3 Expressions and Equations
3.2 Exercises
VOCABULARY Determine whether the algebraic expression is a linear expression. Explain.
1. x^2 + x + 1 2. − 2 x − 8 3. x − x^4 4. WRITING Describe two methods for adding or subtracting linear expressions. 5. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.
Subtract x from 3 x − 1. Find 3 x − 1 decreased by x.
What is x more than 3 x − 1? (^) What is the difference of 3 x − 1 and x?
9 +(-6)=3 3 +(-3)= 4 +(-9)= 9 +(-1)=
Write the sum or difference of two algebraic expressions modeled by the algebra tiles. Then use the algebra tiles to simplify the expression.
6. ^
Find the sum.
8. ( n + 8) + ( n − 12) 9. (7 − b ) + (3 b + 2) 10. (2 w − 9) + (− 4 w − 5) 11. (2 x − 6) + 4( x − 3) 12. 5(−3.4 k − 7) + (3 k + 21) 13. (1 − 5 q ) + 2(2.5 q + 8) 14. 3(2 – 0.9 h ) + (−1.3 h − 4) 15.^1 — 3
(9 − 6 m ) + (^1) — 4
(12 m − 8) 16. − —^1 2
(7 z + 4) + (^) —^1 5
(5 z − 15)
17. BANKING You start a new job. After w weeks, you have (10 w + 120) dollars in your savings account and (45 w + 25) dollars in your checking account. Write an expression that represents the total in both accounts. 18. FIREFLIES While catching fireflies, you and a friend decide to have a competition. After m minutes, you have (3 m + 13) firefl ies and your friend has (4 m + 6) fi reflies. a. Write an expression that represents the number of firefl ies you and your friend caught together. b. The competition ends after 5 minutes. Who has more fi refl ies?
Help with Homework
Section 3.2 Adding and Subtracting Linear Expressions 91
Find the difference.
19. (− 2 g + 7) − ( g + 11) 20. (6 d + 5) − (2 − 3 d ) 21. (4 − 5 y ) − 2(3.5 y − 8) 22. (2 n − 9) − 5(−2.4 n + 4) 23.^1 — 8
(− 8 c + 16) − (^) —^1 3
(6 + 3 c ) 24.^3 — 4
(3 x + 6) − (^1) — 4
(5 x − 24)
25. ERROR ANALYSIS Describe and correct the error in finding the difference.
( 4 m + 9) − 3(2 m − 5) = 4 m + 9 − 6 m − 15 = 4 m − 6 m + 9 − 15 = − 2 m − 6
✗
26. STRUCTURE Refer to the expressions in Exercise 18. a. How many fireflies are caught each minute during the competition? b. How many fireflies are caught before the competition starts? 27. LOGIC Your friend says the sum of two linear expressions is always a linear expression. Is your friend correct? Explain. 28. GEOMETRY The expression 17 n + 11 represents the perimeter (in feet) of the triangle. Write an expression that represents the measure of the third side. 29. TAXI Taxi Express charges $2.60 plus $3.65 per mile, and Cab Cruiser charges $2.75 plus $3.90 per mile. Write an expression that represents how much more Cab Cruiser charges than Taxi Express. 30. MODELING A rectangular room is 10 feet longer than it is wide. One-foot-by-one-foot tiles cover the entire floor. Write an expression that represents the number of tiles along the outside of the room. 31. Write an expression in simplest form that represents the vertical distance between the two lines shown. What is the distance when x = 3? when x = −3?
Evaluate the expression when x = − (^4) — 5 and y = 1 — 3
. (Section 2.2) 32. x + y 33. 2 x + 6 y 34. − x + 4 y 35. MULTIPLE CHOICE What is the surface area of a cube that has a side length of 5 feet? (Skills Review Handbook) A 25 ft 2 B 75 ft 2 C 125 ft 2 D 150 ft^2
5 n á 6 4 n á 5
x
y 3
4
2 1
Ź 3 Ź 4
Ź 5 Ź 4 Ź 3 Ź 2 Ź 1 1 3 4
y â 2 x Ź 4
y â x Ź 1
Extension 3.2 Factoring Expressions 93
Factor − 2 out of − 4 p + 10.
Write each term as a product of −2 and another factor.
− 4 p = − 2 ⋅ 2 p Think: − 4 p is −2 times what?
10 = − 2 ⋅ (−5) Think: 10 is −2 times what?
Use the Distributive Property to factor out −2.
− 4 p + 10 = − 2 ⋅ 2 p + (−2) ⋅ (−5) Rewrite the expression.
= −2[2 p + (−5)] Distributive Property = −2(2 p − 5) Simplify.
So, − 4 p + 10 = −2(2 p − 5).
EXAMPLE 33 Factoring Out a Negative Number
Factor the expression using the GCF.
1. 9 + 21 2. 32 − 48 3. 8 x + 2 4. 3 y − 24 5. 20 z − 8 6. 15 w + 65 7. 36 a + 16 b 8. 21 m − 49 n
Factor out the coefficient of the variable.
9. 1 — 3
b − 1 — 3
3 — 8
d + 3 — 4
11. 2.2 x + 4.4 12. 4 h – 3 13. Factor − 1 — 2
out of − 1 — 2
x + 6. 14. Factor − 1 — 4
out of − 1 — 2
x − 5 — 4
y.
15. WRESTLING A square wrestling mat has a perimeter of (12 x − 32) feet. Write an expression that represents the side length of the mat (in feet). 16. MAKING A DIAGRAM A table is 6 feet long and 3 feet wide. You extend the table by inserting two identical table leaves. The longest side length of each rectangular leaf is 3 feet. The extended table is rectangular with an area of (18 + 6 x ) square feet. a. Make a diagram of the table and leaves. b. Write an expression that represents the length of the extended table. What does x represent? 17. STRUCTURE The area of the trapezoid is ( 3 — 4
x − 1 — 4 ) square centimeters. Write two different pairs of expressions that represent possible lengths of the bases.
1 2 cm
View as Components How does rewriting each term as a product help you see the common factor?
Math Practice
94 Chapter 3 Expressions and Equations
3 Study Help
Make four squares to help you study these topics.
1. simplest form 2. linear expression 3. factoring expressions
After you complete this chapter, make four squares for the following topics.
4. equivalent equations 5. solving equations using addition or subtraction 6. solving equations using multiplication or division 7. solving two-step equations
You can use a four square to organize information about a topic. Each of the four squares can be a category, such as defi nition , vocabulary , example , non-example , words , algebra , table , numbers , visual , graph , or equation. Here is an example of a four square for like terms.
Definition Terms that have the same variables raised to the same exponents
Words To combine like terms that have variables, use the Distributive Property to add or subtract the coefficients.
Examples 2 and −3, 3 x and −7 x , x^2 and 6 x^2
Non-Examples y and 4, 3 x and −4 y , 6 x^2 and 2 x
Like Terms
Graphic Organizer
“My four square shows that my new red skateboard is faster than my old blue skateboard.”
Solving Equations Using
Addition or Subtraction
96 Chapter 3 Expressions and Equations
Work with a partner. Use algebra tiles to model and solve the equation. a. x − 3 = − 4
Model the equation x − 3 = −4. â
To get the variable tile by itself, remove â the tiles on the left side by adding tiles to each side.
How many zero pairs can you remove from each side? â Circle them.
The remaining tile shows the value of x. â
So, x =.
b. z − 6 = 2 c. p − 7 = − 3 d. − 15 = t − 5
How can you use algebra tiles to solve addition or subtraction equations?
Work with a partner. Use algebra tiles to model and solve the equation. a. − 5 = n + 2
Model the equation − 5 = n + 2. â
Remove the tiles on the right side by adding tiles to each side.^ â
How many zero pairs can you remove from the right side?^ â Circle them.
The remaining tiles show the value of n. â
So, n =.
b. y + 10 = − 5 c. 7 + b = − 1 d. 8 = 12 + z
22 ACTIVITY: Solving Equations
11 ACTIVITY: Solving Equations
Solving Equations In this lesson, you will ● (^) write simple equations. ● (^) solve equations using addition or subtraction. ● (^) solve real-life problems.
Section 3.3 Solving Equations Using Addition or Subtraction 97
Use what you learned about solving addition or subtraction equations to complete Exercises 5 – 8 on page 100.
Work with a partner. Write an equation shown by the algebra tiles. Then solve. a. â
b. â
c. â
d. â
33 ACTIVITY: Writing and Solving Equations
Work with a partner. The melting point of a solid is the temperature at which the solid melts to become a liquid. The melting point of the element bromine is about 19°F. This is about 57°F more than the melting point of mercury. a. Which of the following equations can you use to find the melting point of mercury? What is the melting point of mercury?
x + 57 = 19 x − 57 = 19 x^ +^19 =^57 x + 19 = − 57
b. CHOOSE TOOLS How can you solve this problem without using an equation? Explain. How are these two methods related?
44 ACTIVITY: Using a Different Method to Find a Solution
5. IN YOUR OWN WORDS How can you use algebra tiles to solve addition or subtraction equations? Give an example of each. 6. STRUCTURE Explain how you could use inverse operations to solve addition or subtraction equations without using algebra tiles. 7. What makes the cartoon funny? 8. The word variable comes from the word vary. For example, the temperature in Maine varies a lot from winter to summer. Write two other English sentences that use the word vary.
“Dear Sir: Yesterday you said x = 2. Today you are saying x = 3. Please make up your mind.”
Interpret Results How can you add tiles to make zero pairs? Explain how this helps you solve the equation.
Math Practice
