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# Division Property of Equality — Formula & Examples

## TL;DR
The division property of equality states that dividing both sides of a true equation by the same non-zero number keeps it true. This article gives the formula, six worked examples, why the "non-zero" rule exists, and the errors students make most.

## What Division Property of Equality Means
The **division property of equality** states that if two quantities are equal, then dividing both sides of the equation by the same **non-zero** number keeps the two sides equal. In symbols, if a=b and c≠0, then \( \frac{a}{c} = \frac{b}{c} \). It is the rule that lets you undo a multiplication and isolate a variable.

## The Formula
The property is written compactly as a conditional statement.

If a=b and c≠0, then \( \frac{a}{c} = \frac{b}{c} \)

Read it in plain English: whatever you divide the left side by, you must divide the right side by the same thing — and that thing cannot be zero.

## The Properties of Equality
The division property is one of eight properties of equality. Below is the list:

| Property      | What it says (given a=b)                                    |
|---------------|-----------------------------------------------------------|
| Reflexive     | a=a; any quantity equals itself.                           |
| Symmetric     | If a=b, then b=a.                                         |
| Transitive    | If a=b and b=c, then a=c.                                 |
| Substitution   | If a=b, then a may replace b in any expression.          |
| Addition      | If a=b, then a+c=b+c.                                    |
| Subtraction   | If a=b, then a−c=b−c.                                    |
| Multiplication| If a=b, then ac=bc.                                      |
| Division      | If a=b and c≠0, then \( \frac{a}{c} = \frac{b}{c} \). |

## Examples of the Division Property of Equality
### Example 1
**Solve 5x=25.**

5x=25  
\( \frac{5x}{5} = \frac{25}{5} \)  
x=5

### Example 2
**Solve −3x=21.**

−3x=21  
\( \frac{-3x}{-3} = \frac{21}{-3} \)  
x=−7

### Example 3
**Solve 7y=3.**

7y=3  
\( \frac{7y}{7} = \frac{3}{7} \)  
y=\( \frac{3}{7} \)

### Example 4
**Solve \( \frac{2}{3}x = 8 \).**  
\( \frac{2}{3}x = 8 \)  
x=12

### Example 5
**A geometry use: ∠ABC is bisected into two equal angles that together measure 80°.**  
Let each half be x:  
2x=80°  
x=40°

### Example 6
**A word problem: four identical notebooks cost $52 in total.**  
Let one notebook cost n:  
4n=52  
n=13

## Why Does the Non-Zero Rule Matter?
The single most important part of the property is the phrase c≠0. Division by zero is undefined — there is no number that answers "what times zero gives a non-zero result?" Therefore, the property cannot apply when the divisor is zero.

## Key Takeaways
- The **division property of equality** says: if a=b and c≠0, then \( \frac{a}{c} = \frac{b}{c} \).
- You must divide both sides by the same number, and that number can never be zero.
- It works with fractions and negatives as long as the divisor is non-zero.