Download Properties of Equality

Introduction
Equations 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.
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2.1.1: Properties of Equality
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
Property
In symbols
In words
Reflexive property
a=a
of equality
A number is equal to itself.
Symmetric
property
of equality
If a = b, then
b = a.
If numbers are equal, they will still
be equal if the order is changed.
Transitive
property
of equality
If numbers are equal to the same
If a = b and b = c,
number, then they are equal to
then a = c.
each other.
Addition property
of 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.
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2.1.1: Properties of Equality
Key Concepts, continued
Properties of Equality, continued
Property
In symbols
In words
Subtraction
If a = b, then
property of equality a – c = b – c.
Subtracting the same number from
both sides of an equation does not
change the equality of the equation.
If a = b and
Multiplication
c ≠ 0, then
property of equality
a • 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 property
of equality
If a = b and
c ≠ 0, then
a ÷ c = b ÷ c.
Dividing both sides of the equation by
the same number, other than 0, does
not change the equality of the
equation.
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2.1.1: Properties of Equality
Key Concepts, continued
Properties of Equality, continued
Property
In symbols
If a = b, then b
may be
Substitution
substituted for
property of equality a in any
expression
containing a.
In words
If two numbers are equal, then
substituting one in for another does
not change the equality of the
equation.
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2.1.1: Properties of Equality
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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2.1.1: Properties of Equality
Key Concepts, continued
Properties of Operations
Property
General rule
Commutative property of
a+b=b+a
addition
Associative property of
addition
Specific example
3+8=8+3
(a + b) + c = a + (b + c) (3 + 8) + 2 = 3 + (8 + 2)
Commutative property of
a•b=b•a
multiplication
3•8=8•3
Associative property of
multiplication
(a • b) • c = a • (b • c)
(3 • 8) • 2 = 3 • (8 • 2)
Distributive property of
multiplication over
addition
a • (b + c) = a • b + a • c 3 • (8 + 2) = 3 • 8 + 3 • 2
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2.1.1: Properties of Equality
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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2.1.1: Properties of Equality
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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2.1.1: Properties of Equality
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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2.1.1: Properties of Equality
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 1
Which property of equality is missing in the steps
to solve the equation –7x + 22 = 50?
Equation
–7x + 22 = 50
Steps
Original equation
–7x = 28
x = –4
Division property of equality
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2.1.1: Properties 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
Guided Practice: Example 1, continued
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2.1.1: Properties of Equality
Guided Practice
Example 2
Which property of equality is missing in the steps to
x
solve the equation -3 = 4?
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Equation
x
-3 =4
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x
- =7
6
Steps
Original equation
Addition property of equality
–x = 42
x = –42
Division property of equality
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2.1.1: Properties 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,
x
x
-3 - and 4. The result of this step is - = 7.
6
6
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2.1.1: Properties of Equality
Guided Practice: Example 2, continued
In 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 -
x
by 6 is –x and the result
6
of multiplying 7 by 6 is 42. This matches the next
step in the sequence.
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2.1.1: Properties of Equality
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
Guided Practice: Example 2, continued
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2.1.1: Properties of Equality