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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
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Section 1.1 Evaluating Algebraic Expressions 5
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
Ch 1 book.indb 5Ch 1 book.indb 5 7/11/17 8:32 AM7/11/17 8:32 AM
6 Chapter 1 Solving Linear Equations
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
Ch 1 book.indb 6Ch 1 book.indb 6 7/11/17 8:32 AM7/11/17 8:32 AM
Section 1.1 Evaluating Algebraic Expressions 7
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
Ch 1 book.indb 7Ch 1 book.indb 7 7/11/17 8:32 AM7/11/17 8:32 AM
8 Chapter 1 Solving Linear Equations
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.
Ch 1 book.indb 8Ch 1 book.indb 8 7/11/17 8:32 AM7/11/17 8:32 AM
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