evaluating algebraic expressions · evaluate algebraic expressions for given replacement values of...

Section 1.1 Evaluating Algebraic Expressions 3

Evaluating Algebraic Expressions1.1

Essential QuestionEssential Question How can you evaluate algebraic expressions for

given replacement values of the variables?

Evaluating Expressions with Two Unknown Values

Work with a partner. Write an expression for

the measure of ∠EHG. Then complete the table.

• Choose values of y and z to complete the fi rst

two columns.

• For the third column, write the expressions for

the sums of the angle measures.

• For the last column, evaluate the expression to

fi nd the measure of ∠EHG.

a.

b.

c.

Communicate Your AnswerCommunicate Your Answer 3. How can you evaluate algebraic expressions for given replacement values

of the variables?

4. The expression x + y represents the sum of two angle measures. Choose values

for x and y. Then use a protractor to draw a diagram that shows each angle

measure and the sum of the angle measures.

F

H

G

E y°

2y° + z°

m∠EHF m∠FHG m∠EHF + m∠FHG m∠EHG

Evaluating Expressions with One Unknown Value

Work with a partner. Write an expression for

the measure of ∠ADC. Then complete the table.

• Choose three values of x to complete the

fi rst column.

• For the third column, write the expressions for

the sums of the angle measures.

• For the last column, evaluate the expression to

fi nd the measure of ∠ADC.

a.

b.

c.

A

CD

B

45°x°

m∠ADB m∠BDC m∠ADB + m∠BDC m∠ADC

45°

45°

45°

Ch 1 book.indb 3Ch 1 book.indb 3 7/11/17 8:32 AM7/11/17 8:32 AM

4 Chapter 1 Solving Linear Equations

1.1 Lesson What You Will LearnWhat You Will Learn Evaluate algebraic expressions for given replacement values

of the variables.

Evaluate expressions to solve real-life problems.

Evaluating Algebraic ExpressionsAn algebraic expression is an expression that may contain numbers, operations, and

one or more variables. To evaluate an algebraic expression for given values of the

variables, substitute the values into the expression and simplify.

Evaluating Expressions with One Variable

Evaluate each expression for x = 8.

a. ∣ 4 − x ∣ + 2 b. √—

2x − 3

SOLUTION

a. ∣ 4 − x ∣ + 2 = ∣ 4 − 8 ∣ + 2 Substitute 8 for x.

= ∣ −4 ∣ + 2 Subtract.

= 4 + 2 Evaluate the absolute value.

= 6 Add.

b. √—

2x − 3 = √—

2 ⋅ 8 − 3 Substitute 8 for x.

= √—

16 − 3 Multiply.

= 4 − 3 Take the square root.

= 1 Subtract.

Evaluating Expressions with Two Variables

Evaluate each expression for x = −3 and y = 4.

a. 2x + 3y b. x + y —

x2

SOLUTION

a. 2x + 3y = 2(−3) + 3(4) Substitute −3 for x and 4 for y.

= −6 + 12 Multiply.

= 6 Add.

b. x + y

—x2

= −3 + 4

— (−3)2

Substitute −3 for x and 4 for y.

= −3 + 4

— 9 Evaluate the exponent.

= 1 —

9 Add.

algebraic expression, p. 4

Previousexpressionvariable

Core VocabularyCore Vocabullarry



Evaluating Expressions

Evaluate each expression for a = 2 — 3 , b = 9, and c = −4.

a. 2(9a2 − 2b + 3c) b. √—

b − 3a + 2c

SOLUTION

a. 2(9a2 − 2b + 3c) = 2 ( 9 ( 2 — 3 ) 2 − 2(9) + 3(−4) ) Substitute for variables.

= 2 ( 9 ( 4 — 9 ) − 2(9) + 3(−4) ) Evaluate exponent.

= 2(4 − 18 − 12) Multiply.

= 2(−26) Add.

= −52 Multiply.

b. √—

b − 3a + 2c = √—

9 − 3 ( 2 — 3 ) + 2(−4) Substitute for variables.

= 3 − 3 ( 2 — 3 ) + 2(−4) Evaluate square root.

= 3 − 2 + (−8) Multiply.

= 1 + (−8) Subtract.

= −7 Add.

Monitoring ProgressMonitoring ProgressEvaluate the algebraic expression.

1. 2 √—

m + 31 ; m = −6 2. 4s − t; s = 3 —

4 and t =

1 —

2

3. 1 − ∣ a ∣ ÷ b; a = − 2 —

3 , b = 2 4. 5s − √—

r ÷ 6; r = 9, s = 1.5

Solving a Real-Life Problem

You want to tile a 10 foot square fl oor with tiles that are twice as long as they are wide.

Determine how many tiles you need to cover the fl oor when the tile length is 2 feet.

SOLUTION

Let x represent the tile length. The number N of tiles needed is given by the area of the

fl oor, 10 ⋅ 10 = 100 ft2, divided by the area of one tile, x ⋅ x —

2 . So, N =

100 —

x ⋅ x —

2 .

N = 100

— 2 ⋅

2 —

2

Substitute 2 for x.

= 100

— 2 Simplify.

= 50 Divide.

You need 50 tiles.

Monitoring ProgressMonitoring Progress

5. In Example 4, determine how many tiles the students will need to cover the fl oor

when the tile length is 1 —

2 foot.

REMEMBERThe order of operations is:

1. (Grouping Symbols)

2. Exponents

3. Multiplication and Division

4. Addition and Subtraction



Non-Standard OperationsThe mathematical operations you have learned so far, such as the standard operations

addition and subtraction, give you instructions for what to do with two numbers. For

instance, addition, denoted by the “+” symbol, instructs you to fi nd the sum of two

numbers. You can also defi ne non-standard operations using combinations of standard

operations. For example, you could defi ne the following operation.

a ⊡ b = 4a —

b

Evaluating Non-Standard Operations

Evaluate the expressions using the non-standard operations a ⊙ b = 2a + b and

a ⊕ b = 2a

— b .

a. 2 ⊙ 4 b. (3 ⊙ 5) ⊙ 2 c. (3 ⊙ 5) ⊕ 2

SOLUTION

a. 2 ⊙ 4 = 2(2) + 4 Substitute 2 for a and 4 for b in a ⊙ b.

= 4 + 4 Multiply.

= 8 Add.

b. (3 ⊙ 5) ⊙ 2 = (2(3) + 5) ⊙ 2 Substitute 3 for a and 5 for b in a ⊙ b.

= (6 + 5) ⊙ 2 Multiply.

= 11 ⊙ 2 Add.

= 2(11) + 2 Substitute 11 for a and 2 for b in a ⊙ b.

= 22 + 2 Multiply.

= 24 Add.

c. (3 ⊙ 5) ⊕ 2 = (2(3) + 5) ⊕ 2 Substitute 3 for a and 5 for b in a ⊙ b.

= (6 + 5) ⊕ 2 Multiply.

= 11 ⊕ 2 Add.

= 2(11)

— 2 Substitute 11 for a and 2 for b in a ⊕ b.

= 22

— 2 Multiply.

= 11 Add.

Monitoring ProgressMonitoring Progress

Evaluate the expression using the non-standard operations a ⊙ b = a − 3b and a ⊕ b = a2 + b.

6. 1 ⊙ 3 7. 2 ⊕ 7

8. ( 4 ⊙ 1 —

2 ) ⊙ 2 9. (8 ⊙ 5) ⊕ 4



Dynamic Solutions available at BigIdeasMath.comExercises1.1

Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics

Vocabulary and Core Concept Check 1. VOCABULARY What can be included in an algebraic expression?

2. WRITING Describe how to evaluate the expression 5v − √—

4w for v = 2 and w = 9.

3. WHICH ONE DOESN'T BELONG? Which expression does not belong with the other three?

Explain your reasoning.

√—

16 x + y 3 —

5 m + 4n a +

4 —

7 b √

— 25 + 13

In Exercises 4–9, evaluate the expression for the given value of the variable. (See Example 1.)

4. −x + 8; x = 5 5. ∣ 2x + 7 ∣ ; x = −4

6. x − 6; x = 7 — 2

7. √—

x + 2; x = 49

8. ∣ x ∣ + 2.7; x = −1.5

9. √—

−6 ÷ x ; x = − 2 — 3

In Exercises 10–15, evaluate the given expression for

x = 3 — 4 and y = −2. (See Example 2.)

10. 4x + 8y 11. 2x − 6y

12. √—

−2y − 2 —

3 x 13. 2x + ∣ 3y ∣

14. x + 2y — 3x − 4

15. 8x − 4y — 4x

− 2y

— 12x

In Exercises 16–19, evaluate the algebraic expression. (See Example 3.)

16. 2 × (x2 + y2 − 5z); x = 3, y = −4, z = 6

17. ∣ m ÷ n ∣ − 3ℓ; ℓ = 9, m = −12, n = 1

18. √—

2a2 + c − (b − 2a); a = 5, b = 7, c = 14

19. ∣ 4r ∣ − √—

p + (q + 6p)2; p = 1 —

4 , q =

1 —

2 , r = −

3 —

2

20. MODELING WITH MATHEMATICS The cost for a

adults, c children, and s seniors to attend a performance

is given by the expression 9.50a + 5.25c + 7.00s.

What is the cost for a group of 4 adults, 6 children,

and 2 seniors to attend the performance?

21. MODELING WITH MATHEMATICS A party planner

is comparing prices for catering a large event. The

expression 75(r + d ) + 25(v + d ) represents the

total cost, where r is the price of a regular meal, v is

the price of a vegetarian meal, and d is the price of a

drink. Find the total cost for each caterer in the table.

Which caterer is the least expensive?

Caterer r ($) v ($) d ($)

Kim's Kitchen 15.75 14.90 1.25

Savory Eats 13.89 13.89 1.75

Bon Appetit 16.20 12.15 1.35

ERROR ANALYSIS In Exercises 22 and 23, describe and correct the error in evaluating the expression.

22. (x − 2.9)2; x = 1.8 (x − 2.9)2 = (1.8 − 2.9)2

= (−1.1)2

= −1.21

✗

23. a2 − b ; a = 1 — 4

, b = 7 — 2

a2 − b = ( 7 — 2

) 2

− 1 — 4

= 49 — 4

− 1 — 4

= 12

✗

ppppy ppppppppppp



Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyUse the order of operations to simplify the expression.

32. 2 + 4(3 − 8)2 − 7 33. 5 − 8 —

8 − 5 + (7 − 22) 34. 6 ÷ 3 × 2 + 3

Write a related addition equation for the subtraction equation.

35. 12 − 7 = 5 36. 15 − 4 = 11

37. 5 — 4 −

1 —

4 = 1 38. 7.85 − 2.34 = 5.51

Write a related multiplication equation for the division equation.

39. 12 ÷ 3 = 4 40. 25 ÷ 5 = 5

41. 2 — 3 ÷

5 —

3 =

2 —

5 42. 2.7 ÷ 1.2 = 2.25

Reviewing what you learned in previous grades and lessons

In Exercises 24–27, evaluate the expression using the non-standard operations a ⊙ b = 2a + b and a ⊕ b = a ÷ b.

24. 5 ⊙ 3

25. 34 ⊕ 2

26. 2 ⊙ (3 ⊙ 4)

27. (3 ⊙ 2) ⊕ 4

28. ANALYZING RELATIONSHIPS Choose values for

x and y, and complete the fi rst row of the table

below. What do you think might be true about the

expressions? Complete the table for different values

of x and y, including negative values. Do you think

your conjecture is true? Explain.

x y (x − y)(x + y) x2 − y2

29. MODELING WITH MATHEMATICS In a weighted

average, some values are worth more than others in

computing an average value. A teacher gives a quiz,

a test, and a fi nal exam. The expression q + 3t + 5f

— 9

gives the weighted average of the scores, where q

is the quiz score, t is the test score, and f is the fi nal

exam score. What is the weighted average for a

student who scores 86 on the quiz, 75 on the test,

and 80 on the fi nal exam?

30. PROBLEM SOLVING The surface area of a square

pyramid with a base edge length of b and a height of

h is given by the expression b2 + 2b √— ( b — 2 )

2

+ h2 .

h

b

b

a. A gift box in the shape of a square pyramid

has a base edge length of 4.8 inches. Use a

calculator to complete the table for each of the

possible heights shown.

Height (in.) Surface Area (in.2)

3.5

3.6

3.7

3.8

3.9

b. Use the table to estimate the height of the box

for a surface area of 67 square inches.

31. COMPARING METHODS Evaluate 4(x + 3y) for

x = 5 —

2 and y =

5 —

6 in two ways: (a) distribute the 4

and then substitute the values of the variables, and

(b) substitute the values of the variables and then

simplify. Which method do you prefer? Explain.


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