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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Key Concepts, continuedProperties of Equality

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Property In symbols In wordsReflexive 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, continuedProperties of Equality, continued

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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, continuedProperties of Equality, continued


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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.


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Key Concepts, continuedProperties of Operations


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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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Guided PracticeExample 1Which property of equality is missing in the steps to solve the equation –7x + 22 = 50?

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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.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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Guided Practice Example 2Which property of equality is missing in the steps to

solve the equation ?

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Equation Steps

Original equation

Addition property of equality

–x = 42

x = –42 Division property of equality


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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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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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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Introduction

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