Subtraction Property of Equality - Definition & Examples

TL;DR

The subtraction property of equality states that if a = b, then a - c = b - c. This property preserves the balance of an equation when the same number is subtracted from both sides. This article outlines the formula, provides six examples including fractions, and discusses its relationship with the addition property.

Last updated on June 29, 2022. 9 min read

What Is The Subtraction Property of Equality?

The subtraction property of equality states that when the same quantity is subtracted from both sides of an equation, the two sides remain equal. Formally, if a = b, then a - c = b - c, for any numbers a, b, and c.

The key word is both. You must subtract from both sides to preserve the equality.

What Is The Formula For The Subtraction Property of Equality?

The formula is short:

If a = b, then a - c = b - c

Variable glossary:

How Does The Subtraction Property Solve Equations?

The property is how you undo addition. Take x + 8 = 15. To isolate x, subtract 8 from both sides:

x + 8 - 8 = 15 - 8
=> x = 7

Examples of the Subtraction Property of Equality

Example 1

Solve x + 5 = 12.
Subtract 5 from both sides:

x + 5 - 5 = 12 - 5
=> x = 7
Final answer: x = 7. Check: 7 + 5 = 12. Correct.

Example 2

Solve x + 9 = 4. To isolate x, subtract 9 from both sides:

x + 9 - 9 = 4 - 9
=> x = -5
Final answer: x = -5. Check: -5 + 9 = 4. Correct.

Example 3

A two-column proof step:
If AB + BC = AC and BC = 4, show that AB = AC - 4.

Subtract BC from both sides:

AB + BC - BC = AC - BC
=> AB = AC - BC
=> AB = AC - 4.
Final answer: AB = AC - 4.

Example 4

Solve x + \frac{3}{4} = \frac{5}{4}.
Subtract \frac{3}{4} from both sides:

x + \frac{3}{4} - \frac{3}{4} = \frac{5}{4} - \frac{3}{4}
=> x = \frac{1}{2}.
Final answer: x = \frac{1}{2}.

Example 5

Solve \frac{2}{3} - x = \frac{3}{4}. Subtract \frac{2}{3} from both sides:

\frac{2}{3} - x - \frac{2}{3} = \frac{3}{4} - \frac{2}{3}
=> -x = \frac{1}{12}
=> x = -\frac{1}{12}.
Final answer: x = -\frac{1}{12}.

Example 6

A word problem: A bag had some marbles. After 7 were added, it held 20. Let m be the initial count:

m + 7 = 20
Subtract 7 from both sides:

m + 7 - 7 = 20 - 7
=> m = 13
Final answer: 13 marbles.

Why Subtraction And Addition Are One Principle In Two Directions

The subtraction property of equality has a mirror twin: the addition property of equality which states that if a = b, then a + c = b + c. They let you move terms across the equals sign.

Tripping Points To Avoid

Mistake 1: Subtracting from one side only

Don't do this: Avoid deleting terms from one side. The correct way: Subtract from both sides to keep balance.

Mistake 2: Subtracting the wrong term

Don't do this: Avoid taking whichever number looks smaller or easier. The correct way: Subtract the term attached to the variable.

Mistake 3: Sign errors with negative results

Don't do this: Avoid flipping negative answers to positive. The correct way: Let negatives stand; they are valid answers.

Practice Questions on the Subtraction Property of Equality

1. Solve x + 6 = 10.
2. Solve x + 12 = 5.
3. Solve x + \frac{1}{2} = \frac{7}{6}.
4. In a proof with PQ + QR = PR given QR = 3, show PQ = PR - 3, naming the property at each step.
5. **A jar gained 9 coins to hold 22. How many were there to start?
   **

Answers:

1. x = 4. Check: 4 + 6 = 10.
2. x = -7. Check: -7 + 12 = 5.
3. x = \frac{2}{3}.
4. PQ = PR - QR (subtraction property).
5. m = 13 coins.

Key Takeaways

• The subtraction property of equality says if a = b, then a - c = b - c.
• Always subtract from both sides and choose the correct term to isolate variables.
• The property works for any numbers, including fractions and negatives.