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Multiplicative Property of Equality

Multiplicative Property of Equality

The formal name for the property of equality that allows one to multiply the same quantity by both sides of an equation. This, along with the additive property of equality, is one of the most commonly used properties for solving equations.

 

Property: If  a = b  then  a·c = b·c.
Example:      x/5 = 7
(x/5)·5 = 7·5
        x = 35

Key Formula

If a=b, then a⋅c=b⋅c\text{If } a = b, \text{ then } a \cdot c = b \cdot c
Where:
  • aa = The expression on the left side of the equation
  • bb = The expression on the right side of the equation
  • cc = Any real number you multiply both sides by (can be any value except 0 if you want to preserve equivalence in reverse)

Worked Example

Problem: Solve for x: x / 4 = 9
Step 1: Identify the operation acting on the variable. Here x is being divided by 4.
x4=9\frac{x}{4} = 9
Step 2: To undo division by 4, multiply both sides of the equation by 4. This is the Multiplicative Property of Equality.
x4⋅4=9⋅4\frac{x}{4} \cdot 4 = 9 \cdot 4
Step 3: Simplify both sides. On the left, dividing by 4 and then multiplying by 4 cancel each other out.
x=36x = 36
Step 4: Check: substitute x = 36 back into the original equation.
364=9✓\frac{36}{4} = 9 \quad \checkmark
Answer: x = 36

Another Example

This example shows that dividing both sides by a number is the same as multiplying both sides by its reciprocal — both actions use the Multiplicative Property of Equality. The first example undid division; this one undoes multiplication.

Problem: Solve for y: 3y = 24
Step 1: The variable y is being multiplied by 3. To isolate y, multiply both sides by the reciprocal of 3, which is 1/3. This is still the Multiplicative Property of Equality — you are multiplying both sides by the same quantity.
3y⋅13=24⋅133y \cdot \frac{1}{3} = 24 \cdot \frac{1}{3}
Step 2: Simplify both sides. On the left, 3 times 1/3 equals 1, leaving just y.
y=8y = 8
Step 3: Check: substitute y = 8 back into the original equation.
3(8)=24✓3(8) = 24 \quad \checkmark
Answer: y = 8

Frequently Asked Questions

What is the difference between the Multiplicative Property of Equality and the Additive Property of Equality?
The Multiplicative Property of Equality says you can multiply both sides of an equation by the same number. The Additive Property of Equality says you can add the same number to both sides. You use the multiplicative property to undo multiplication or division on a variable, while you use the additive property to undo addition or subtraction.
Can you multiply both sides of an equation by zero?
Technically, the property allows it — if a = b, then a · 0 = b · 0 is true (both sides become 0). However, this destroys the information in the equation and makes it impossible to solve, so multiplying by zero is never useful when solving equations. The resulting equation 0 = 0 tells you nothing about the variable.
Is dividing both sides of an equation also the Multiplicative Property of Equality?
Yes. Dividing both sides by a number c is the same as multiplying both sides by 1/c. So when you divide both sides of 5x = 30 by 5, you are really applying the Multiplicative Property of Equality with c = 1/5. Some textbooks call this the Division Property of Equality, but it follows directly from the multiplicative property.

Multiplicative Property of Equality vs. Additive Property of Equality

Multiplicative Property of EqualityAdditive Property of Equality
DefinitionIf a = b, then a · c = b · cIf a = b, then a + c = b + c
OperationMultiply both sides by the same numberAdd the same number to both sides
When to useWhen the variable is multiplied or divided by a number (e.g., 3x = 12 or x/5 = 7)When a number is added to or subtracted from the variable (e.g., x + 3 = 10 or x − 4 = 6)
UndoesMultiplication and divisionAddition and subtraction

Why It Matters

The Multiplicative Property of Equality is one of the two core tools you use to solve nearly every algebraic equation. Whenever a variable is trapped inside a fraction or multiplied by a coefficient, this property is how you isolate it. You will rely on it from pre-algebra through calculus — for example, when solving formulas like d = rt for r, you multiply both sides by 1/t.

Common Mistakes

Mistake: Multiplying only one side of the equation instead of both sides.
Correction: Whatever you do to one side, you must do to the other. If you multiply the left side by 3, you must also multiply the right side by 3, or the equation is no longer balanced.
Mistake: Forgetting to multiply every term on each side when the equation has more than one term.
Correction: If you have (x/2) + 3 = 10 and you multiply both sides by 2, every term must be multiplied: x + 6 = 20, not x + 3 = 20. Use the distributive property to ensure every term is included.

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