1. Solving Equations: The Addition and Multiplication Properties
Module 2
Solving Equations: The Addition and Multiplication Properties

2. Chapter 1 / Whole Numbers and Introduction to Algebra
Equations
Statements like = 7 are called equations.
An equation is of the form expression = expression.
An equation can be labeled as
Equal sign
x = 9
left side
right side

3. Solutions of Equations
Chapter 1 / Whole Numbers and Introduction to Algebra
Solutions of Equations
When an equation contains a variable, deciding which values of the variable make an equation a true statement is called solving an equation for the variable.
A solution of an equation is a value for the variable that makes an equation a true statement.

4. Chapter 1 / Whole Numbers and Introduction to Algebra
Solving Equations
To solve an equation, we will use properties of equality to write simpler equations, all equivalent to the original equation, until the final equation has the form
x = number or number = x
Equivalent equations have the same solution.
The word “number” above represents the solution of the original equation.

5. Addition Property of Equality
Chapter 1 / Whole Numbers and Introduction to Algebra
Addition Property of Equality
Let a, b, and c represent numbers.
If a = b, then
a + c = b + c
and
a – c = b - c
In other words, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation.

6. Helpful Hint
Remember to check the solution in the original equation to see that it makes the equation a true statement.

7. Helpful Hint
Remember that we can get the variable alone on either side of the equation. For example, the equations
x = 3 and 3 = x
both have a solution of 3.

8. Multiplication Property of Equality
Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplication Property of Equality
Let a, b, and c represent numbers and let c  0. If a = b, then
a ∙ c = b ∙ c and
In other words, both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution of the equation.

9. Simplifying Algebraic Expressions

10. 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

11. 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.

12. 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.

13. 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

14. 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.

15. 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.

16. 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)

17. 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

18. 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

19. 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.

20. 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.

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

22. Solving Equations: Review of Addition and Multiplication Properties

23. Equation vs. Expression
Chapter 1 / Whole Numbers and Introduction to Algebra
Equation vs. Expression
Statements like = 7 are called equations.
An equation is of the form expression = expression.
An equation can be labeled as
Equal sign
x = 9
left side
right side

24. Addition Property of Equality
Chapter 1 / Whole Numbers and Introduction to Algebra
Addition Property of Equality
Let a, b, and c represent numbers.
If a = b, then
a + c = b + c
and
a – c = b - c
In other words, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation.

25. Multiplication Property of Equality
Chapter 1 / Whole Numbers and Introduction to Algebra
Multiplication Property of Equality
Let a, b, and c represent numbers and let c  0. If a = b, then
a ∙ c = b ∙ c and
In other words, both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution of the equation.

26. Addition Subtraction Multiplication Division sum difference product
quotient
plus
minus
times
divided
added to
subtracted from
multiply
shared equally among
more than
less than
twice
per
increased by
decrease by of
divided by
total
less
twice/double/ triple
divided into

27. Solving Linear Equations in One Variable

28. Steps for Solving an Equation
Step 1: If parentheses are present, use the distributive property. Step 2: Combine any like terms on each side of the equation. Step 3: Use the addition property of equality to rewrite the equation so that variable terms are on one side of the equation and constant terms are on the other side. Step 4: Use the multiplication property of equality to divide both sides by the numerical coefficient of the variable to solve. Step 5: Check the solution in the original equation.

29. Writing Sentences as Equations
Key Words or Phrases
Examples
Symbols
equals
3 equals 2 plus 1
3 = 2 + 1
gives
the quotient of 10 and –5 gives –2
is/was
17 minus 12 is 5
17 – 12 = 5
yields
11 plus 2 yields 13
= 13
amounts to
twice –15 amounts to –30
2(–15) = –30
is equal to
24 is equal to 2 times –12
–24 = 2(–12)
Objective A

30. Solving Linear Equations in One Variable and Problem Solving

31. Addition Subtraction Multiplication Division sum difference product
quotient
plus
minus
times
divided
added to
subtracted from
multiply
shared equally among
more than
less than
twice
per
increased by
decrease by of
divided by
total
less
twice/double/ triple
divided into

32. Using Problem-Solving Steps to Solve Problems
1. UNDERSTAND the problem. During this step, become comfortable with the problem. Some ways of doing this are:
• Read and reread the problem.
• Construct a drawing.
• Propose a solution and check. Pay careful attention to how you check your proposed solution. This will help when writing an equation to model the problem.
Choose a variable to represent the unknown. Use this variable to represent any other unknowns.
2. TRANSLATE the problem into an equation.
3. SOLVE the equation.
4. INTERPRET the results: Check the proposed solution in the stated problem and state your conclusion.
Objective A

33. Introduction to Fractions and Mixed Numbers

34. Identifying Numerators and Denominators
In a fraction, the top number is called the numerator and the bottom number is called the denominator. The bar between the numbers is called the fraction bar.
Names
Fraction
Meaning
Numerator number of parts being considered
Denominator number of equal parts in the whole
Objective A 34

35. Identifying Proper Fractions, Improper Fractions and Mixed Numbers
Proper Fraction: A fraction whose numerator is less than its denominator.
Numerator
Denominator
Improper Fraction: A fraction whose numerator is greater than or equal to its denominator.
Objective A 35

36. Fractions on Number Lines
Chapter 1 / Whole Numbers and Introduction to Algebra
Fractions on Number Lines
Another way to visualize fractions is to graph them on a number line. 3
5 equal parts 1

37. Division Properties of 0 and 1
Chapter 1 / Whole Numbers and Introduction to Algebra
Division Properties of 0 and 1
If n is any integer other than 0, then
If n is any integer, then

38. Writing a Mixed Number as an Improper Fraction
To write a mixed number as an improper fraction: Step 1: Multiply the denominator of the fraction by the whole number. Step 2: Add the numerator of the fraction to the product from Step 1. Step 3: Write the sum from Step 2 as the numerator of the improper fraction over the original denominator.

39. Writing an Improper Fraction as a Mixed Number or a Whole Number
To write an improper fraction as a mixed number or a whole number: Step 1: Divide the denominator into the numerator. Step 2: The whole number part of the mixed number is the quotient. The fraction part of the mixed number is the remainder over the original denominator.

40. Factors and Simplest Form

41. Finding the Factors of Numbers
To perform many operations, it is necessary to be able to factor a number.
Since 7 · 9 = 63, both 7 and 9 are factors of 63, and 7 · 9 is called a factorization of 63.
Objective A 41

42. Prime and Composite Numbers
Prime Numbers
A prime number is a natural number greater than 1 whose only factors are 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, … .
Composite Numbers
A composite number is a natural number, greater than 1, that is not prime.
Objective A 42

43. Prime Factorization Prime Factorization
The prime factorization of a number is the factorization in which all the factors are prime numbers.
Every whole number greater than 1 has exactly one prime factorization.
Objective A 43

44. Divisibility Tests
Objective A 44

45. Writing Fractions in Simplest Form
Fractions that represent the same portion of a whole are called equivalent fractions.
There are many equivalent forms of a fraction. A special form of a fraction is called simplest form.
Simplest Form of a Fraction
A fraction is written in simplest form or lowest terms when the numerator and denominator have no common factors other than 1.
Objective A 45

46. Simplest Form Writing a Fraction in Simplest Form
To write a fraction in simplest form, write the prime factorization of the numerator and the denominator and then divide both by all common factors.
Objective A 46

47. Equality of Fractions
Objective A 47

48. Multiplying and Dividing Fractions

49. Multiplying Fractions
To multiply two fractions, multiply the numerators and multiply the denominators.
If a, b, c, and d represent positive whole numbers we have
Objective A 49

50. Finding Reciprocals of Fractions
Definition
Example
Reciprocals: Two numbers are reciprocals of each other if their product is 1.
Reciprocal of a fraction: To find the reciprocal of a fraction, interchange its numerator and denominator.
Objective A 50

51. Dividing Fractions
Objective A 51