Properties of Equality
2026-02-28 06:21 Diff
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Below are the key Properties of Equality in mathematics, basic principles that help us maintain the balance of equations. These rules are important for solving equations and forming justifiable mathematical arguments.

1. Reflexive Property of Equality

Definition:
Every real number is equal to itself.

Mathematical Expression:
\( a = a\).

Example:
For any number x, \(x = x\)

2. Symmetric Property of Equality

Definition:
If one quantity equals a second, then the second equals the first.

Mathematical Expression:
 If \(a = b\), then b = a.

Example:
If \(5 = 3 + 2 \), then \(3 + 2 = 5\)

3. Transitive Property of Equality

Definition:
When one value is equal to the second value, and that second value is equal to a third, then the first and third values are also equal.

Mathematical Expression:
If \(a = b\) and\( b = c\) then \(a = c\).

Example:
If \(2 + 3 = 5\) and \(5 = 3 + 2 \), then \(2 + 3 = 3 + 2\)

Another example:
If \(6 + 1 = 7\) and \(7 = 14 ÷ 2\), then \( 6 + 1 = 14 ÷ 2\).

4. Addition Property of Equality

Definition:
If the same number is added to both sides of an equation, the equality is still true.

Mathematical Expression:
If \(a = b\), then \(a + c = b + c\).

Example:
If \(x = 4\), then \(x + 3 = 4 + 3\), so \(x + 3 = 7\).

5. Subtraction Property

Definition: When two values are equal, and then taking away the same amount from each side, so that the equation remains balanced


Mathematical Expression: If \(a = b\), then \(a − c = b - c\).


Example: If \( y = 10\), then \(y − 3 = 10 − 3\), so \(y − 3 = 7\)

6. Multiplication Property

Definition: If two quantities are equal, multiplying both sides by the same number maintains equality.

Mathematical Expression:

If \(a = b\),

then \(a × c = b × c\),

where \(c ≠ 0\)

Example:

  If \(6 = 6\)

Then \(6 × 3 = 6 × 3\) ⇒ \(18 = 18\)

If you multiply both sides of an equation by the same number, equality will remain the same.

7. Division Property

Definition: If two quantities are equal, dividing both sides by the same non-zero number maintains equality.

Mathematical Expression:

If \(a = b\),

then \(a ÷ c = b ÷ c\),

where \(c ≠ 0\)

Example: If \(12 = 12\), then \(12 ÷ 4 = 12 ÷ 4 \) ⇒ \(12 ÷ 4 = 12 ÷ 4 \)

8. Substitution Property

Definition: If two quantities are equal, one can be substituted for the other in any expression.


Mathematical Expression: If \(a = b\), then a can be replaced by b in any expression.


Example: If \(a = 7\), then in the expression \(a + 3\), a can be replaced with 7, resulting in \(7 + 3 =10\).

9. Square Root Property

Definition: If two quantities are equal, and a variable is squared, you can take the square root of both sides to solve for the variable.
Always remember to include both the positive and negative roots.

  Mathematical Expression: \(x^2 = 16 \;\;\implies\;\; |x| = 4 \;\;\implies\;\; x = \pm 4 \)