# Properties of Equality

Here is a quick summary of the Properties of Equality.

1) **Reflexive Property of Equality**

For any number a, a=a.

\Rightarrow It states that any quantity is equal to itself.

Examples:

2=2 \\

1+4=1+4 \\

3x^2=3x^2 \\

2) **Symmetric Property of Equality**

For any numbers a and b, if a=b then b=a.

\Rightarrow If one quantity equals a second, then the second quantity equals the first.

Examples:

If 1+2=3 then 3=1+2.

If 2y = x + y then x + y = 2y.

If {x^2} - {y^2} = \sqrt 2 then \sqrt 2 = {x^2} - {y^2}.

3) **Transitive Property of Equality**

For any numbers a, b, and c, if a=b and b=c, then a=c.

\Rightarrow If the first quantity is equal to the second quantity, and the second quantity is equal to the third quantity, then the first quantity must be equal to the third quantity.

Examples:

If 7 + 8 = 12 + 3 and 12 + 3 = 15, then 7 + 8 = 15.

If 3 \times 4 = 2 \times 6 and 2 \times 6 = 12, then 3 \times 4 = 12.

If m - n = {k^2} and {k^2} = p + {d^3}, then m - n = p + {d^3}.

4) **Addition Property of Equality**

For any numbers a, b, and c, if a=b, then a+c=b+c.

\Rightarrow A true equation will remain true and unchanged when the same or common value is **added** to each side.

Example 1:

Example 2:

5) **Subtraction Property of Equality**

For any numbers a, b, and c, if a=b, then a-c=b-c.

\Rightarrow A true equation will remain true and unchanged when the same or common value is **subtracted** from each side.

Example 1:

Example 2:

6) **Multiplication Property of Equality**

For any numbers a, b, and c, if a=b, then a \times c = b \times c.

\Rightarrow As long as the same value or quantity is** multiplied** on both sides of the equation, the new equation remains true and the same in meaning.

Example 1:

Example 2:

7) **Division Property of Equality**

For any numbers a, b, and c, if a=b, then a \div c = b \div c.

\Rightarrow As long as the same value or quantity is** divided** from both sides of the equation, the new equation remains true and the same in meaning.

Example 1:

Example 2:

8) **Substitution Property of Equality**

If a=b then b can be substituted for a in any expression or equation.

\Rightarrow A quantity can be substituted or replaced by another quantity of equal or the same value in any expression or equation.

Example 1:

If x=y and x+2, then y+2.

Example 2:

If m=5 and m-2, then 5-2.

Example 3:

If x=3 and 5+x=8, then 5+3=8.

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