Properties of Equality

The properties of equality describe the logical rules that let us manipulate equations while preserving their truth. These properties are foundational to solving algebraic expressions and proving mathematical statements.

Core Properties of Equality

1. Reflexive Property

a = a

Every number is equal to itself.

2. Symmetric Property

a = b \Rightarrow b = a

If one value equals another, then the reverse is also true.

3. Transitive Property

a = b,\ b = c \Rightarrow a = c

If one value equals a second, and that second equals a third, then the first equals the third.

Operational Properties of Equality

These properties show how equations remain valid when operations are applied equally to both sides—or undone under certain conditions.

4. Additive Property

a = b \Rightarrow a + c = b + c

If an equality sign holds, adding the same value c to both sides preserves the equality.

5. Multiplicative Property

a = b \Rightarrow a \times c = b \times c

Multiplying both sides of an equation by the same value c preserves the equality.

6. Cancellation Property (Addition)

a + c = b + c \Rightarrow a = b

In an equation, if both sides have the same additive expression like + c, then that expression can be removed from both sides and the equality will be preserved.

7. Cancellation Property (Multiplication)

a \times c = b \times c,\ c \neq 0 \Rightarrow a = b

In an equation, if both sides have the same multiplicative expression like × c, then that expression can be removed from both sides and the equality will still hold.

Activity: Match The Columns!

Reorder the Definition to match them with the correct Equality Property.

Equality Property

Definition

$\text{Reflexive Property}$

$\text{Symmetric Property}$

$\text{Transitive Property}$

$\text{Additive Property of Equality}$

$\text{Multiplicative Property of Equality}$

$\text{Cancellation Property of Addition}$

$\text{Cancellation Property of Multiplication}$

True or False Activity!!!

Read each statement carefully and decide if it is true or false.

Every real number is equal to itself is the symmetric property

If a = b, then b = a is the transitive property.

If a = b and b = c, then a must equal c.

If a = b, then a + c = b - c.

If $a \times c = b \times c$ and $c \ne 0$ , then a = b.

Build the Mathematical Statements

Build the Correct Equations Activity

Reflexive Property

Symmetric Property

Transitive Property

Additive Property of Equality

Multiplicative Property of Equality

Cancellation Property of Addition

Cancellation Property of Multiplication

End of Lesson