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Chapter 3 Review
3.1 Simplifying Algebraic Equations
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.
+3
3 + (−4 ) + 2
Variable Term
Constant Term
Variable Terms
Constant Term
The number factor of a variable term is called the numerical coefficient. The coefficient of the
term 3 is simply 3. A numerical coefficient of 1 is usually not written. That is, we usually write
, not 1 . Terms that are exactly the same, except that they may have different numerical
coefficients are called like terms.
Like Terms
Unlike Terms
3 ,2
−6 , 2 ,
−3, 4
Distributive Property
2
, −5
If a, b, and c are numbers, then
+
=( + )
3
and
−
5 ,
7 ,7
5, 5
,7
=( − )
A sum or difference of like terms can be simplified using the distributive property.
Here is an example of combining like terms using the distributive property.
7 + 5 = (7 + 5) = 12
An algebraic expression is simplified when all like terms have been combined.
The commutative and associative properties of addition and multiplication help simplify
expressions.
Commutative Properties of Addition and Multiplication
If a, b, and c are numbers, then
+ = +
∙ = ∙
Chapter 3 Review
Commutative Property of Addition
Commutative Property of Multiplication
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The order of adding or multiplying two numbers can be changed without changing their sum or
product.
The grouping of numbers in addition or multiplication can be changed without changing their
sum or product.
Associative Properties of Addition and Multiplication
If a, b, and c are numbers, then
( + )+ =
( ∙ )∙ =
+( + )
∙( ∙ )
Associative Property of Addition
Associative Property of Multiplication
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
We can also use the distributive property to multiply expressions.
The distributive property says that multiplication distributes over addition and subtraction.
2(5 + ) = 2 ∙ 5 + 2 ∙
or
2(5 − ) = 2 ∙ 5 − 2 ∙
= 10 + 2
= 10 − 2
To simplify expressions, use the distributive property first to multiply and then combine any like
terms.
Examples: Simplify each of the following algebraic expressions.
1. 3(5 + ) − 17
2. −19( − 3) + 17
Chapter 3 Review
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3. 9( − 7) + 11
4. −3( + 5) − 21
5. Find the perimeter of the triangle.
6. Find the area of the rectangle.
(2x – 5) meters
7z feet
3z feet
3 meters
9z feet
Helpful Hints:
Perimeter:
• distance around
• measured in units
Chapter 3 Review
Area:
•
•
surface enclosed
measured in square units
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3.2 Solving Linear Equations: Review of Addition and Multiplication
Properties
Examples: Solve each of the following. Don’t forget to check your solution.
1. 2(2 − 3) = 10
2. 2(5 − 3) = 11
3. 21 = 5(4 − 6)
Chapter 3 Review
4.
= −1 − (−8)
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Translate each of the following.
5. The sum of –7 and a number
6. Negative four times a number, increased by 18
7. The product of –6 and the sum of a number and 15
8. Twice the sum of a number and –5
9. The quotient of –20 and a number, decreased by three
Chapter 3 Review
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3.3 Solving Linear Equations in One Variable
3x – 2 = 7 is called a linear equation or first degree equation in one variable. The exponent on
each x is 1 and there is no variable below a fraction bar. It is an equation in one variable because
it contains one variable, x.
Make sure you understand which property to use to solve an equation.
Addition:
Multiplication:
+5=8
3 = 12
To undo addition of 5, we subtract 5 from
To undo multiplication of 3, we divide both
both sides.
sides by 3.
3
12
+5−5=8−5
=
3
3
Using Addition Property of Equality.
Use Multiplication Property of Equality.
=3
=4
Steps for Solving a One Variable Linear 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 to rewrite the equation so that the 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.
Chapter 3 Review
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Examples: Solve each of the following equations.
1. 4 − 11 = −14 − 13
3. 6(5 + ) = 5( − 4)
Translate.
5. The sum of –42 and 16 is –26.
2. 10 + 4 + 6 = 0
4. 6 + 29 + 7 = 0
6. Negative 2 times the sum of 3 and 12 is –30
7. What’s wrong with the following solution?
2(3 − 5) = 5 − 7
6 −5=5 −7
6 −5+5=5 −7+5
6 =5 −2
6 −5 =5 −2−5
=2
Chapter 3 Review
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3.4 Linear Equations in One Variable and Problem Solving
1. UNDERSTAND the problem. During this step, become comfortable with the problem.
Some ways of doing this are:
ü Read and reread the problem.
ü Construct a drawing.
ü 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.
Examples:
1. Five times a number
2. Based on the 2000 Census, Florida has 28 fewer electoral votes
subtracted from 40 is the
for president than California. If the total number of electoral votes for
same as three times the
these two states is 82, find the number for each state.
number. Find the number.
1. UNDERSTAND
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.
Chapter 3 Review
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3. The product of number and 3 is twice the difference
of that number and 8. Find the number.
4. An Xbox 360 game system and several games are sold
for $560. The cost of the Xbox 360 is 3 times as much as
the cost of the games. Find the cost of the Xbox 360 and
the cost of the games.
Chapter 3 Review
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