1. 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.
2.1.1: Properties of Equality

2. 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.
2.1.1: Properties of Equality

3. Key Concepts, continued Properties of Equality
Property
In symbols
In words
Reflexive property
of equality
a = a
A number is equal to itself.
Symmetric property
If a = b, then b = a.
If numbers are equal, they will still be equal if the order is changed.
Transitive property
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 property
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.
2.1.1: Properties of Equality

4. Properties of Equality, continued
Key Concepts, continued
Properties of Equality, continued
Property
In symbols
In words
Subtraction
property 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.
Multiplication
If a = b and
c ≠ 0, then
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
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.
2.1.1: Properties of Equality

5. Key Concepts, continued Properties of Equality, continued
Property
In symbols
In words
Substitution
property of equality
If a = b, then b may be
substituted for a in any
expression containing a.
If two numbers are equal, then
substituting one in for another does not change the equality of the equation.
2.1.1: Properties of Equality

6. 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

7. Key Concepts, continued Properties of Operations
Property
General rule
Specific example
Commutative 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 of
multiplication
a • b = b • a
3 • 8 = 8 • 3
Associative property of
(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 • • 2
2.1.1: Properties of Equality

8. 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.
2.1.1: Properties of Equality

9. 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.
2.1.1: Properties of Equality

10. 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.
2.1.1: Properties of Equality

11. 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
2.1.1: Properties of Equality

12. Guided Practice Example 1
Which property of equality is missing in the steps to solve the equation –7x + 22 = 50?
Equation
Steps
–7x + 22 = 50
Original equation
–7x = 28
x = –4
Division property of equality
2.1.1: Properties of Equality

13. Guided Practice: Example 1, continued
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.
2.1.1: Properties of Equality

14. ✔ Guided Practice: Example 1, continued
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.” ✔
2.1.1: Properties of Equality

15. Guided Practice: Example 1, continued
2.1.1: Properties of Equality

16. Guided Practice Example 2
Which property of equality is missing in the steps to
solve the equation ?
Equation
Steps
Original equation
Addition property of equality
–x = 42
x = –42
Division property of equality
2.1.1: Properties of Equality

17. Guided Practice: Example 2, continued
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
2.1.1: Properties of Equality

18. 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 by 6 is –x and the result of multiplying 7 by 6 is 42. This matches the next step in the sequence.
2.1.1: Properties of Equality

19. ✔ Guided Practice: Example 2, continued
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.” ✔
2.1.1: Properties of Equality

20. Guided Practice: Example 2, continued
2.1.1: Properties of Equality