Division Property of Equality — Formula & Examples
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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.