Introduction
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Transcript of Introduction
IntroductionEquations are mathematical sentences that state two expressions are equal. In order to solve equations in algebra, you must perform operations that maintain equality on both sides of the equation using the properties of equality. These properties are rules that allow you to balance, manipulate, and solve equations.
2.1.1: Properties of Equality
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Key Concepts• In mathematics, it is important to follow the rules when
solving equations, but it is also necessary to justify, or prove that the steps we are following to solve problems are correct and allowed.
• The following table summarizes some of these rules.
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2.1.1: Properties of Equality
Key Concepts, continued
Properties of Equality
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2.1.1: Properties of Equality
Property In symbols In words
Reflexive propertyof equality
a = a A number is equal to itself.
Symmetric propertyof equality
If a = b, then b = a.
If numbers are equal, they will still be equal if the order is changed.
Transitive propertyof equality
If a = b and b = c, then a = c.
If numbers are equal to the same number, then they are equal to each other.
Addition propertyof equality
If a = b, then a + c = b + c.
Adding the same number to both sides of an equation does not change the equality of the equation.
Key Concepts, continued
Properties of Equality, continued
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2.1.1: Properties of Equality
Property In symbols In words
Subtractionproperty of equality
If a = b, then a – c = b – c.
Subtracting the same number from both sides of an equation does not change the equality of the equation.
Multiplicationproperty of equality
If a = b and c ≠ 0, thena • c = b • c.
Multiplying both sides of the equation by the same number, other than 0, does not change the equality of the equation.
Division propertyof equality
If a = b and c ≠ 0, then a ÷ c = b ÷ c.
Dividing both sides of the equation bythe same number, other than 0, does not change the equality of the equation.
Key Concepts, continued
Properties of Equality, continued
2.1.1: Properties of Equality
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Property In symbols In words
Substitutionproperty of equality
If a = b, then b may besubstituted for a in anyexpression containing a.
If two numbers are equal, thensubstituting one in for another does not change the equality of the equation.
Key Concepts, continued
• You may remember from other classes the properties of operations that explain the effect that the operations of addition, subtraction, multiplication, and division have on equations. The following table describes some of those properties.
2.1.1: Properties of Equality
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Key Concepts, continuedProperties of Operations
2.1.1: Properties of Equality
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Property General rule Specific exampleCommutative property of addition
a + b = b + a 3 + 8 = 8 + 3
Associative property of addition
(a + b) + c = a + (b + c)
(3 + 8) + 2 = 3 + (8 + 2)
Commutative property ofmultiplication
a • b = b • a 3 • 8 = 8 • 3
Associative property ofmultiplication
(a • b) • c = a • (b • c) (3 • 8) • 2 = 3 • (8 • 2)
Distributive property ofmultiplication over addition
a • (b + c) = a • b + a • c
3 • (8 + 2) = 3 • 8 + 3 • 2
Key Concepts, continued• When we solve an equation, we are rewriting it into a
simpler, equivalent equation that helps us find the unknown value.
• When solving an equation that contains parentheses, apply the properties of operations and perform the operation that’s in the parentheses first.
• The properties of equality, as well as the properties of operations, not only justify our reasoning, but also help us to understand our own thinking.
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Key Concepts, continued• When identifying which step is being used, it helps to
review each step in the sequence and make note of what operation was performed, and whether it was done to one side of the equation or both. (What changed and where?)
• When operations are performed on one side of the equation, the properties of operations are generally followed.
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Key Concepts, continued• When an operation is performed on both sides of the
equation, the properties of equality are generally followed.
• Once you have noted which steps were taken, match them to the properties listed in the tables.
• If a step being taken can’t be justified, then the step shouldn’t be done.
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Common Errors/Misconceptions• incorrectly identifying operations
• incorrectly identifying properties
• performing a step that is not justifiable or does not follow the properties of equality and/or the properties of operations
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2.1.1: Properties of Equality
Guided Practice
Example 1Which property of equality is missing in the steps to solve the equation –7x + 22 = 50?
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2.1.1: Properties of Equality
Equation Steps–7x + 22 = 50 Original equation
–7x = 28
x = –4 Division property of equality
Guided Practice: Example 1, continued
1. Observe the differences between the original equation and the next equation in the sequence. What has changed?Notice that 22 has been taken away from both expressions, –7x + 22 and 50.
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2.1.1: Properties of Equality
Guided Practice: Example 1, continued
2. Refer to the table of Properties of Equality.The subtraction property of equality tells us that when we subtract a number from both sides of the equation, the expressions remain equal.
The missing step is “Subtraction property of equality.”
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2.1.1: Properties of Equality
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Guided Practice: Example 1, continued
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2.1.1: Properties of Equality
Guided Practice
Example 2Which property of equality is missing in the steps to
solve the equation ?
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2.1.1: Properties of Equality
Equation Steps
Original equation
Addition property of equality
–x = 42
x = –42 Division property of equality
Guided Practice: Example 2, continued
1. Observe the differences between the original equation and the next equation in the sequence. What has changed?
Notice that 3 has been added to both expressions,
and 4. The result of this step is .
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2.1.1: Properties of Equality
Guided Practice: Example 2, continuedIn order to move to the next step, the division of 6 has been undone.
The inverse operation of the division of 6 is the multiplication of 6.
The result of multiplying by 6 is –x and the result
of multiplying 7 by 6 is 42. This matches the next
step in the sequence.
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Guided Practice: Example 2, continued
2. Refer to the table of Properties of Equality.The multiplication property of equality tells us that when we multiply both sides of the equation by a number, the expressions remain equal.
The missing step is “Multiplication property of equality.”
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2.1.1: Properties of Equality
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Guided Practice: Example 2, continued
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2.1.1: Properties of Equality