345

CHAPTER 5: ALGEBRA

CHAPTER 5 CONTENTS

5.1 Introduction to Algebra

5.2 Algebraic Properties

5.3 Distributive Property

5.4 Solving Equations Using the Addition Property of

Equality

5.5 Solving Equations Using the Multiplication

Property of Equality

5.6 Solving Equations Using the Addition and

Multiplication Properties of Equality

5.7 Translating English Sentences into Mathematical

Equations and Solving

Image from www.coolmath.com

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

346

5.1 Introduction to Algebra

When you think of algebra, you probably think about solving equations for x, where x is some

unknown number. Solving equations is indeed a big part of algebra, but algebra is much more

than that. It is the branch of mathematics that uses symbols (like x) to represent unknown values

and also includes performing operations like adding, subtracting, multiply and dividing on these

symbols. We will start with the basics. Below are definitions of terms that you will use as you

continue to study algebra.

Variable: A variable is a letter that represents an unknown quantity.

Term: A term is a quantity that is added. A term consists of a number or a product of a number

and variables (where the variables may be raised to a power).

Algebraic Expression: An algebraic expression contains at least one variable and is formed by

connecting numbers and variables using operations of addition, subtraction, multiplication,

division, raising to powers, and/or taking roots (but has no equal sign). We will study one type

of algebraic expression in this chapter called a polynomial.

Polynomial: A polynomial contains at least one variable and is a sum of one or more different

terms. The following are examples of algebraic expressions that are polynomials.

5x – 3y + 7 can be written as a sum of terms 5x + (-3y) +7

213.8 7

2x x can be written as a sum of terms

213.8 ( 7)

2x x

3x

2 – 5y + xy – x – 7 can be written as a sum of terms 3x

2 + (-5y) + xy + (-x) + (-7)

Note: In the algebraic expression 3x2 – 5y + xy – x – 7, there are five terms: 3x

2, -5y, xy, -x, -7.

Variable term: A variable term is a term that contains a variable.

The variable terms in 3x2 – 5y + xy – x – 7 are 3x

2, -5y, xy, and -x.

Constant term: A constant term is a term that does not contain a variable.

The constant term in 3x2 – 5y + xy – x – 7 is -7.

Coefficient: A coefficient is a number multiplied with a variable. When writing terms, the

coefficient is placed to the left of the variable part.

In the algebraic expression 3x2 – 5y + xy – x – 7 = 3x

2 + (-5y) + xy + (-x) + (-7)

3 is the coefficient of the term 3x2.

-5 is the coefficient of the term -5y.

1 is the coefficient of the term xy.

-1 is the coefficient of the term -x.

Note: When there is no number multiplied with a variable term, the coefficient is 1 or -1.

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

347

Example 1: For the polynomial, 23 2 5x x xy , answer the following questions.

First rewrite the polynomial as a sum of terms. 2 23 2 5 3 2 ( ) 5x x xy x x xy

What is/are the variable(s)? x and y

What are the terms? 23x , 2x, -xy, and 5

What is the constant term? 5

What is/are the variable term(s)? 23x , 2x and -xy

What is the coefficient of the term –xy? -1

Practice 1: For the polynomial,32 4 5x y yz , answer the following questions.

a. Write the polynomial as a sum of terms.

b. What is/are the variable(s)?

c. What are the terms?

d. What is the constant term?

e. What is/are the variable term(s)?

f. What is the coefficient of the term yz?

Answers: a. 32 ( 4 ) 5x y yz b. x, y, z c. 32 , 4 , , 5x y yz

d. 5 e. 32 , 4 , x y yz f. 1

Watch It: http://youtu.be/EP_RQZPYSVw

Like Terms: Two terms are called like terms if the variable parts of the terms are the same. Like

terms are terms that have the same variable(s) raised to the same power(s).

Note: 2x is the same as 2x1.

Example 2: Determine if the following pairs of terms are like terms.

2a and 3a like terms (The variable parts are the same.)

2x and 3y unlike terms (The variables are not the same.)

2x and 3 unlike terms (One is an x-term: the other a constant.)

x2 and x unlike terms (The exponents are not the same.)

x2

and 0.5x2 like terms (The variable parts are the same.)

x and 1

3x like terms (The variable parts are the same.)

4 and 2 like terms (Constant terms are also like terms.)

3ab and 2b unlike terms (The variable parts are not the same.)

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

348

Practice 2: Determine if the following pairs of terms are like terms.

a. 4x and 6x

b. 2a and 3a

c. 3b and 3

d. 1

2y and 1.2y

e. 2x y and 3xy

f. 3 and 4

Answers: a. like terms b. unlike terms c. unlike terms d. like terms

e. unlike terms f. like terms

Watch It: http://youtu.be/Bs9Dbcrlu7c

Combining like terms: When you ―combine‖ like terms, you add the coefficients of the like

terms and the answer is a like term—the exact same kind of term. When you combine like terms,

the variable part of the term will stay the same.

For Examples 3 – 7, simplify, if possible, by combining like terms.

Example 3: 2x + 3x These are like terms.

= (2+3)x Simplify by adding the coefficients.

= 5x

Practice 3: 8y – 3y Answer: 5y

Watch It: http://youtu.be/k3RuUkrVAr4

Example 4: 2y + 5 These two terms are unlike, so they cannot be

combined into a single term.

Practice 4: 9a + 6 Answer: 9a + 6 (not like terms)

Watch It: http://youtu.be/AnLoNH5HMUI

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

349

Example 5: 3xy – 4xy – 1y + 5y

= (3 – 4)xy + (-1 +5)y Simplify by adding the coefficients of like terms.

= -1xy + 4y

= -xy + 4y

Note: -1xy + 4y is equivalent to -xy + 4y. Either answer is acceptable, but the second is

preferred.

Practice 5: 2x + 7x – 5 + 4 Answer: 9x – 1

Watch It: http://youtu.be/dcbOLjFZ6RA

Example 6: 1 3 3

12 4 2

a a

1 3 3

12 4 2

a

Add coefficients of like terms.

2 3 2 3

4 4 2 2a

Determine common denominators to add fractions.

1 5

4 2a Simplify.

Note: We could write 5

2 as the mixed number

12

2, but when doing algebra problems, simplified

improper fractions are preferred.

Practice 6: 2 1 4

23 6 5

x x Answer: 1 14

2 5x

Watch It: http://youtu.be/ZUGqU9zAfRg

Example 7: 0.8 5.1 2 0.3x x y y

(0.8 5.1) (2 0.3)x y Add coefficients of like terms.

4.3 2.3x y Simplify.

Practice 7: 3.2 0.4 8 0.5a a b b Answer: 3.6a + 8.5b

Watch It: http://youtu.be/4npoLFqbGwk

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

350

Evaluating Expressions

Consider the following expression: 2x + 5.

What does the expression mean? The variable x holds the place for an unknown number. So, the

expression above says to ―multiply 2 times an unknown number named x and then add 5.‖

Note: Although there is no operation symbol between the 2 and the x, the operation between the

2 and the x is multiplication.

Now, suppose that we are told that x is the number 3. What is the value of 2x + 5 if x is 3? By

changing the x to a 3 we get: Remember the order of operations!!

2x +5

= 2(3) + 5 Substitute 3 for x.

= 6 + 5 Multiply first.

= 11 Add.

So, if x = 3, 2x + 5 = 11. This process is evaluating the expression 2x + 5 if x = 3.

Example 8: Evaluate 6x + 7 if x = 2.

By replacing the x with 2, we get:

6x + 7

= 6(2) + 7 Substitute 2 for x.

= 12 + 7 Multiply first.

= 19 Add.

Practice 8: Evaluate 4x – 7 if x = 3. Answer: 5

Watch It: http://youtu.be/KkNrmiSJMfs

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

351

Example 9: Evaluate 1

( 4)2

x if x = 8.

By replacing the x with 8, we get:

1( 4)

2

1( 4)

2

1(12)

2

1 12

2 1

1

2

x

1

8

12

1

6

6

Substitute 8 for x.

Simplify inside the parentheses.

Write 12 as a fraction in order to multiply.

Divide out common factors in the numerator

and denominator.

Multiply.

Practice 9: Evaluate 1

( 4)3

x if x = 2. Answer: 2

Watch It: http://youtu.be/cgLyYJKr9QI

Example 10: Evaluate (x + 3) + 10 if x = -5.

By replacing the x with -5, we get:

(x + 3) + 10

= (-5 + 3) + 10 Substitute -5 for x.

= (-2) + 10 First, simplify inside the parentheses.

= 8 Add.

Practice 10: Evaluate (a – 3) + 21 when a = – 6 Answer: 12

Watch It: http://youtu.be/dTHWdGgYT-8

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

352

Example 11: Evaluate 0.4(2 + x) if x = -1.2.

By replacing the x with -1.2, we get:

0.4(2 + x)

= 0.4(2 + (-1.2)) Substitute -1.2 for x.

=0.4(0.8) First, simplify inside the parentheses.

= 0.32

Practice 11: Evaluate 0.5(2 + x) if x = -2.4. Answer: -0.2

Watch It: http://youtu.be/bfjLSbjOfwc

Example 12: Evaluate x(2 + x) if x = 4.

By replacing each x with 4, we get:

x(2 + x)

= 4(2 + 4) Substitute 4 for x.

= 4(6) Simplify inside the parentheses.

= 24 Multiply.

Practice 12: Evaluate y(y –7) if y = 3. Answer: -12

Watch It: http://youtu.be/DdPBaUf7IsI

Let’s look at another expression: 3x + 4y. This expression is different from the other

expressions we’ve looked at because it has an x and y variable. This expression says to ―multiply

3 with an unknown number called x and multiply 4 with another unknown number called y, then

add those two products.‖

Example 13: Evaluate 3x + 4y if x = 2 and y = 5.

By replacing the x with 2 and the y with 5, we get:

3x + 4y

= 3(2) + 4(5) Substitute 2 for x and 5 for y.

= 6 + 20 Multiply.

= 26

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

353

Practice 13: Evaluate 4x + 2y if x = 6 and y = 5. Answer: 34

Watch It: http://youtu.be/F4WyI17O8PA

Example 14: Evaluate 5x + 2y if x = -2 and y = -3.

By replacing the x with -2 and the y with -3, we get:

5x + 2y

= 5(-2) + 2(-3) Substitute.

= -10 + (-6) Multiply.

= -16 Add.

Practice 14: Evaluate x + 6y if x = – 4 and y = – 7. Answer: -46

Watch It: http://youtu.be/VLdBcoIAjo4

Example 15: Evaluate 3x + 8y if x = 2

3 and y = -1.

By replacing the x with 2

3and the y with -1, we get:

3 8

3 8( )

3 2( 8)

1 3

3

x y

1

2

3-1

2

1 3 ( 8)

2 ( 8)

6

1

Substitute.

Write 3 as the fraction 3

1 in order to multiply.

Divide out common factors in the numerator

and denominator.

Multiply.

Add.

Practice 15: Evaluate 4x + 3y if x = 1

2 and y = -1. Answer: -1

Watch It: http://youtu.be/6u1gl3zO6Ls

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

354

Example 16: Evaluate 1

(3 5 )2

x y if x = 4 and y = 6.

By replacing the x with 4 and the y with 6, we get:

1 (3 5 )

2

1(3( ) 5( ))

2

1(12 30)

2

1(42)

2

21

x y

4 6

Substitute.

Simplify inside the parentheses.

Multiply.

Practice 16: Evaluate 2

(2 6 )3

x y if x = 3 and y = 1. Answer: 8

Watch It: http://youtu.be/K15AFdV95QU

Example 17: Evaluate 2x if x = -3.

By replacing the x with -3, we get:

x2

= (x)2 Rewrite with parentheses around the variable.

= (-3)2 Substitute -3 for x inside the parentheses.

= (-3)(-3) Write in expanded form.

= 9 Multiply.

Practice 17: Evaluate 2x if x = -5. Answer: 25

Watch It: http://youtu.be/dVtyFrTiivY

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

355

Example 18: Evaluate 2( )x y if x = -6 and y = 4.

By replacing the x with -6 and the y with 4, we get:

(x + y)2

= (-6 + 4)2 Substitute -6 for x and 4 for y.

= (-2)2 Simplify inside the parentheses.

= (-2)(-2) Write in expanded form.

= 4 Multiply.

Practice 18: Evaluate 2(2 )x y if x = -2 and y = 5. Answer: 81

Watch It: http://youtu.be/o9-FssTlUF0

Watch All: http://youtu.be/n6-i4ldbZTY

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

356

5.1 Introduction Exercises

For problems 1 – 9, simplify by combining like terms.

1. 8x – 5x + 7 – 3

2. 3x – 7x – 5 – 9

3. x – 11x + 2x + 12 + 10 – 11

4. 16x – 10x + 13y + 22y

5. -x + 4x + 11x – 1 + 4 – 6

6. 1.2x + 0.7x – 1 – 3.5

7. 0.4ab + 1.2ab – 3a + 0.1a

8. 1 3 1 1

5 5 3 2x x

9. 2 23 1 1

4 2 3x x x x

10. Evaluate 3 + 5x if x = 1.

11. Evaluate 14 + (3 + x) if x = -5.

12. Evaluate x(x + 7) if x = -2.

13. Evaluate 8(x + 2) if x = 6.

14. Evaluate 4x + 5y if x = 3 and y = 2.

15. Evaluate 3x + 7y if x = -2 and y = -1.

16. Evaluate 4x if x = -1.

17. Evaluate 2(4 )x y if x = -2 and y = 4.

18. Evaluate 2x – 3y if 3

2x and

1

3y .

19. Evaluate 10x + 3y if x = 0.1 and y = - 0.4.

20. Evaluate 1

( 4)2

x if x = 2.

CCBC Math 081 Introduction to Algebra Section 5.1 Third Edition 12 pages

357

5.1 Introduction Exercises Answers

1. 3x + 4

2. -4x – 14

3. -8x + 11

4. 6x + 35y

5. 14x – 3

6. 1.9x – 4.5

7. 1.6ab – 2.9a

8. 2 5

5 6x

9. 25 2

4 3x x

10. 8

11. 12

12. -10

13. 64

14. 22

15. -13

16. 1

17. 16

18. 2

19. -0.2

20. 3