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Additive Property of Equality

Additive Property of Equality

The formal name for the property of equality that allows one to add the same quantity to both sides of an equation. This, along with the multiplicative 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 = 12

Key Formula

If a=b, then a+c=b+c\text{If } a = b, \text{ then } a + c = b + 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 added to both sides

Worked Example

Problem: Solve for x: x − 9 = 14
Step 1: Identify what is being subtracted from x. Here, 9 is subtracted from x.
x−9=14x - 9 = 14
Step 2: Add 9 to both sides of the equation using the Additive Property of Equality.
(x−9)+9=14+9(x - 9) + 9 = 14 + 9
Step 3: Simplify each side. On the left, −9 + 9 cancels to 0, leaving just x. On the right, 14 + 9 = 23.
x=23x = 23
Step 4: Check: substitute x = 23 back into the original equation.
23−9=14✓23 - 9 = 14 \quad \checkmark
Answer: x = 23

Another Example

This example shows that the property also covers subtracting from both sides, since subtracting a number is the same as adding its negative. It also produces a negative answer, which is a common source of sign errors.

Problem: Solve for y: y + 6 = 2
Step 1: Identify what needs to be removed from the variable's side. Here, 6 is added to y, so you need to undo that addition.
y+6=2y + 6 = 2
Step 2: Add −6 (which is the same as subtracting 6) to both sides. This still uses the Additive Property of Equality because you are adding the same quantity, −6, to both sides.
(y+6)+(−6)=2+(−6)(y + 6) + (-6) = 2 + (-6)
Step 3: Simplify each side. On the left, +6 and −6 cancel. On the right, 2 + (−6) = −4.
y=−4y = -4
Step 4: Check: substitute y = −4 into the original equation.
−4+6=2✓-4 + 6 = 2 \quad \checkmark
Answer: y = −4

Frequently Asked Questions

What is the difference between the Additive Property of Equality and the Subtraction Property of Equality?
They are essentially the same property. Subtracting a number is identical to adding its negative. For instance, subtracting 5 from both sides is the same as adding −5 to both sides. Some textbooks list them separately, but mathematically they rely on the same rule: if a = b, then a + c = b + c.
When do you use the Additive Property of Equality?
You use it whenever a number is being added to or subtracted from the variable you are solving for. By adding the opposite of that number to both sides, you isolate the variable. For example, if x − 3 = 10, you add 3 to both sides to get x = 13.
Why do you have to add to both sides and not just one side?
An equation is a statement that two expressions are equal. If you change only one side, the two sides are no longer equal, and the equation becomes false. Adding the same value to both sides preserves the balance, keeping the equation true.

Additive Property of Equality vs. Multiplicative Property of Equality

Additive Property of EqualityMultiplicative Property of Equality
DefinitionAdd the same value to both sidesMultiply both sides by the same nonzero value
FormulaIf a = b, then a + c = b + cIf a = b, then a · c = b · c (c ≠ 0)
When to useTo undo addition or subtraction on the variableTo undo multiplication or division on the variable
Examplex − 5 = 7 → add 5 → x = 123x = 21 → divide by 3 (multiply by 1/3) → x = 7
Restrictionc can be any real numberc cannot be zero (division by zero is undefined)

Why It Matters

The Additive Property of Equality is one of the first algebraic tools you learn, and you will use it in virtually every equation you solve from pre-algebra through calculus. It forms the logical basis for "moving" a term to the other side of an equation. Without it, you would have no justified method for isolating a variable when addition or subtraction is involved.

Common Mistakes

Mistake: Adding a value to one side of the equation but forgetting to add it to the other side.
Correction: Always perform the same operation on both sides. Write the addition step on both the left and right sides before simplifying, so you don't accidentally leave one side unchanged.
Mistake: Adding the same sign instead of the opposite sign to cancel a term. For example, seeing x − 5 = 7 and adding −5 instead of +5.
Correction: To cancel a term, add its opposite (additive inverse). If the term is −5, add +5. If the term is +6, add −6. The goal is for the term and its opposite to sum to zero.

Related Terms