1. 8
Chapter
Chapter 2
Introduction to Algebra

2. Section
8.1
Variable Expressions

3. Objective A
Evaluate Algebraic Expressions for Given Replacement Values for the Variables.

4. Algebraic Expressions
Chapter 1 / Whole Numbers and Introduction to Algebra
Algebraic Expressions
A combination of operations on letters (variables) and numbers is called an algebraic expression.
Algebraic Expressions
5 + x  y  y – 4 + x
4x means 4  x
and
xy means x  y

5. Algebraic Expressions
Chapter 1 / Whole Numbers and Introduction to Algebra
Algebraic Expressions
Replacing a variable in an expression by a number and then finding the value of the expression is called evaluating the expression for the variable.

6. Example
Evaluate x – 2 if x is 7. Replace x with 7 in the expression. x – 2 = 7 – 2 = 5

7. Example
Evaluate y(x – 3) for x = 8 and y = 4. Replace x with 8 and y with 4. y(x – 3) = 4(8 – 3) = 4(5) = 20

8. Example
Evaluate for x = 6 and y = 18.

9. Example
Evaluate 25 – z3 + x for z = 2 and x = 1.

10. Use Properties of Numbers to Combine Like Terms.
Objective B
Use Properties of Numbers to Combine Like Terms.

11. Chapter 1 / Whole Numbers and Introduction to Algebra
Constant and Variable Terms
A term that is only a number is called a constant term, or simply a constant. A term that contains a variable is called a variable term.
3y2 + (–4y) + 2
x + 3
Constant
terms
Variable
terms

12. Chapter 1 / Whole Numbers and Introduction to Algebra
Coefficients
The number factor of a variable term is called the numerical coefficient. A numerical coefficient of 1 is usually not written.
5x x or 1x –7y y 2
Numerical coefficient is 5.
Numerical coefficient is –7.
Understood numerical coefficient is 1.
Numerical coefficient is 3.

13. Chapter 1 / Whole Numbers and Introduction to Algebra
Like Terms
Terms that are exactly the same, except that they may have different numerical coefficients are called like terms.
Like Terms
Unlike Terms
3x, 2x
–6y, 2y, y
–3, 4
5x, x 2
7x, 7y
5y, 5
6a, ab
2ab2, –5b 2a
The order of the variables
does not have to be the same.

14. Chapter 1 / Whole Numbers and Introduction to Algebra
Distributive Property
A sum or difference of like terms can be simplified using the distributive property.
Distributive Property
If a, b, and c are numbers, then
ac + bc = (a + b)c
Also,
ac – bc = (a – b)c

15. Distributive Property
Chapter 1 / Whole Numbers and Introduction to Algebra
Distributive Property
By the distributive property,
7x + 5x = (7 + 5)x
= 12x
This is an example of combining like terms.
An algebraic expression is simplified when all like terms have been combined.

16. Example
Simplify each expression by combining like terms. a. 8y + 4y c. 4x2 + 6x2 – 3 b. y – 6y

17. Chapter 1 / Whole Numbers and Introduction to Algebra
Addition and Multiplication Properties
The commutative and associative properties of addition and multiplication help simplify expressions.
Properties of Addition and Multiplication
If a, b, and c are numbers, then
Commutative Property of Addition
a + b = b + a
Commutative Property of Multiplication
a ∙ b = b ∙ a
The order of adding or multiplying two numbers can be changed without changing their sum or product.

18. Chapter 1 / Whole Numbers and Introduction to Algebra
Associative Properties
The grouping of numbers in addition or multiplication can be changed without changing their sum or product.
Associative Property of Addition
(a + b) + c = a + (b + c)
Associative Property of Multiplication
(a ∙ b) ∙ c = a ∙ (b ∙ c)

19. Helpful Hint
Examples of Commutative and Associative Properties of Addition and Multiplication
4 + 3 = 3 + 4
6 ∙ 9 = 9 ∙ 6
(3 + 5) + 2 = 3 + (5 + 2)
(7 ∙ 1) ∙ 8 = 7 ∙ (1 ∙ 8)
Commutative Property of Addition
Commutative Property of Multiplication
Associative Property of Addition
Associative Property of Multiplication

20. Example
Simplify: 6y y – 3 We begin by writing subtraction as the opposite of addition. 6y y – 3 = 6y y + (–3) = 6y + 4y (–3) = (6 + 4)y (–3) = 10y + 2

21. Example (cont)
Simplify each expression by combining like terms. c. –7y + 2 – 2y – 9x + 12 – x

22. Example
Simplify each expression by combining like terms. a. 6y + 12y – 6 = b. 7y – 5 + y + 8 =

23. Use Properties of Numbers to Multiply Expressions.
Objective C
Use Properties of Numbers to Multiply Expressions.

24. We can also use the distributive property to multiply expressions.
Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplying Expressions
We can also use the distributive property to multiply expressions.
The distributive property says that multiplication distributes over addition and subtraction.
2(5 + x) = 2 ∙ ∙ x = x or
2(5 – x) = 2 ∙ 5 – 2 ∙ x = 10 – 2x

25. Example
Multiply. a. 5(5y) b. –8(9x)

26. Example
Use the distributive property to multiply: 6(x + 5) 6(x + 5) =

27. Example
Multiply: –5(4a + 3) –5(4a + 3) =

28. Simplify Expressions by Multiplying and Then Combining Like Terms.
Objective D
Simplify Expressions by Multiplying and Then Combining Like Terms.

29. Simplifying Expressions
To simply expressions, use the distributive property first to multiply and then combine any like terms.
Simplify: 3(5 + x) – 17
Apply the Distributive Property.
3(5 + x) – 17 =
3 ∙ ∙ x + (–17)
= x + (–17)
Multiply.
= 3x + (–2) or 3x – 2
Combine like terms.
Note: 3 is not distributed to the –17 since –17 is not within the parentheses.

30. Example
Simplify: –7(x – 1) + 5(2x + 3) Use the distributive property to remove parentheses. –7(x – 1) + 5(2x + 3)

31. Find the Perimeter and Area of Figures.
Objective E
Find the Perimeter and Area of Figures.

32. Chapter 1 / Whole Numbers and Introduction to Algebra
Finding Perimeter
7z feet
3z feet
9z feet
Perimeter is the distance around the figure.
Perimeter = 3z + 7z + 9z
= 19z feet
Don’t forget to insert proper units.

33. Finding Area (2x – 5) meters 3 meters A = length ∙ width = 3(2x – 5)
= 6x – 15 square meters
Don’t forget to insert proper units.