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Division Property of Equality — Definition, Formula & Examples

The Division Property of Equality is a rule that says if you divide both sides of an equation by the same nonzero number, the two sides remain equal. It is one of the key properties used to isolate a variable when solving equations.

For all real numbers aa, bb, and cc where c≠0c \neq 0: if a=ba = b, then ac=bc\dfrac{a}{c} = \dfrac{b}{c}.

Key Formula

If a=b and c≠0, then ac=bc\text{If } a = b \text{ and } c \neq 0, \text{ then } \frac{a}{c} = \frac{b}{c}
Where:
  • aa = The expression on the left side of the equation
  • bb = The expression on the right side of the equation
  • cc = The nonzero number you divide both sides by

How It Works

When a variable is multiplied by a coefficient, you use this property to undo the multiplication. Divide both sides of the equation by that coefficient, and the variable stands alone. The restriction c≠0c \neq 0 is essential because division by zero is undefined.

Worked Example

Problem: Solve for x: 6x = 42
Identify the coefficient: The variable x is multiplied by 6, so divide both sides by 6.
6x6=426\frac{6x}{6} = \frac{42}{6}
Simplify: On the left, the 6s cancel. On the right, 42 divided by 6 equals 7.
x=7x = 7
Answer: x=7x = 7

Why It Matters

You rely on this property every time you solve a one-step or multi-step equation that involves multiplication. It appears constantly in algebra courses and standardized tests, and it extends directly to solving formulas in science, such as rearranging d=rtd = rt to find rr.

Common Mistakes

Mistake: Dividing by zero. Students sometimes divide both sides by a variable or expression that could equal zero.
Correction: Always confirm the divisor is nonzero. If you divide by a variable, note the restriction that the variable cannot be zero.

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