Properties of Equality — List, Definitions, Examples

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Properties of Equality — List, Definitions, Examples

TL;DR

The properties of equality are the nine algebraic rules that say if you do the same thing to both sides of an equation, the equation stays true. This article lists all nine, gives three worked examples from Quick to Stretch, walks through the mistakes that cost marks, and credits the mathematicians who codified the axioms.

The Nine Properties — Definitions and Symbolic Statements

1. Reflexive Property of Equality.

Any quantity equals itself.

a = a

2. Symmetric Property of Equality.

If a = b, then b = a.

3. Transitive Property of Equality.

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

4. Addition Property of Equality.

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

5. Subtraction Property of Equality.

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

6. Multiplication Property of Equality.

If a = b, then ac = bc.

7. Division Property of Equality.

If a = b and c ≠ 0, then a/c = b/c.

8. Substitution Property of Equality.

If a = b, then a can replace b in any expression.

9. Square Root Property of Equality.

If a² = b², then a = ±b (for real numbers).

Three Problems, One Method

Quick.

Solve x + 5 = 12 and name the property used.

Subtract 5 from both sides:

x = 7
Final answer: x = 7, by the Subtraction Property of Equality.

Standard (Wrong-Path-First).

Solve 3x + 7 = 22.

Step 1 — subtract 7 from both sides:

3x = 15

Step 2 — divide both sides by 3:

x = 5
Final answer: x = 5.

Stretch.

Use the properties of equality to prove: if 2x + 3 = 5x - 9, then x = 4.

Step 1 — Subtract 2x from both sides:

3 = 3x - 9

Step 2 — Add 9 to both sides:

12 = 3x

Step 3 — Divide both sides by 3:

x = 4
Final answer: x = 4.

Common Errors When Applying Equality Properties

Mistake 1: Applying a property to only one side of the equation.

Correct way: every property of equality says "do the operation to both sides".

Mistake 2: Dividing by zero or by an unknown that could be zero.

Correct way: the Division Property has a non-zero clause.

Mistake 3: Forgetting the ± when applying the Square Root Property.

Correct way: the Square Root Property has two solutions: x = 4 or x = -4.

Properties of Equality — Full Comparison Table

# Property Symbolic Statement What It Lets You Do Worked Snippet
1 Reflexive a = a State that any quantity equals itself. 5 = 5; x + 3 = x + 3.
2 Symmetric If a = b, then b = a Reverse an equation without changing it. If x = 7, then 7 = x.
3 Transitive If a = b and b = c, then a = c Chain equalities through a middle term. If x = y and y = 4, then x = 4.
4 Addition If a = b, then a + c = b + c Add the same number to both sides. x - 3 = 5 ⇒ x = 8.
5 Subtraction If a = b, then a - c = b - c Subtract the same number from both sides. x + 4 = 11 ⇒ x = 7.
6 Multiplication If a = b, then ac = bc Multiply both sides by the same number. x² = 6 ⇒ x = 12.
7 Division If a = b and c ≠ 0, then a/c = b/c Divide both sides by the same non-zero number. 3x = 15 ⇒ x = 5.
8 Substitution If a = b, replace a with b "Plug in" a known value anywhere it appears. If y = x + 3 and x = 4, then y = 7.
9 Square Root If a² = b², then a = ±b (reals) Take the square root of both sides. x² = 49 ⇒ x = ±7.

Key Takeaways