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Properties of Equality

Properties of Equality
Equation Rules

Equivalence properties and algebra rules for manipulating equations are listed below.

 

Definitions      

1. a = b means a is equal to b.
2. a ≠ b means a does not equal b.

 
Operations 1. Addition: If a = b then a + c = b + c.
2. Subtraction: If a = b then a – c = b– c.
3. Multiplication: If a = b then ac = bc.
4. Division: If a = b and c ≠ 0 then a/c = b/c.

 

Reflexive Property       a = a
Symmetric Property If a = b then b = a.
Transitive Property If a = b and b = c then a = c.

 

See also

Inequality rules, trichotomy, side of an equation

Key Formula

Addition:a=b  ⟹  a+c=b+cSubtraction:a=b  ⟹  a−c=b−cMultiplication:a=b  ⟹  ac=bcDivision:a=b,  c≠0  ⟹  ac=bcReflexive:a=aSymmetric:a=b  ⟹  b=aTransitive:a=b,  b=c  ⟹  a=c\begin{aligned} &\textbf{Addition:} & a = b &\implies a + c = b + c \\ &\textbf{Subtraction:} & a = b &\implies a - c = b - c \\ &\textbf{Multiplication:} & a = b &\implies ac = bc \\ &\textbf{Division:} & a = b,\; c \neq 0 &\implies \frac{a}{c} = \frac{b}{c} \\ &\textbf{Reflexive:} & a &= a \\ &\textbf{Symmetric:} & a = b &\implies b = a \\ &\textbf{Transitive:} & a = b,\; b = c &\implies a = c \end{aligned}
Where:
  • a,ba, b = Any real numbers or expressions that are equal
  • cc = Any real number or expression used to perform the same operation on both sides (must be nonzero for division)

Worked Example

Problem: Solve for x: 3x + 5 = 20. Identify which property of equality you use at each step.
Step 1: Use the Subtraction Property of Equality to subtract 5 from both sides.
3x+5−5=20−5  ⟹  3x=153x + 5 - 5 = 20 - 5 \implies 3x = 15
Step 2: Use the Division Property of Equality to divide both sides by 3 (since 3 ≠ 0).
3x3=153  ⟹  x=5\frac{3x}{3} = \frac{15}{3} \implies x = 5
Step 3: Check: substitute x = 5 back into the original equation using the Reflexive Property (a value equals itself).
3(5)+5=15+5=20  ✓3(5) + 5 = 15 + 5 = 20 \;\checkmark
Answer: x = 5. The Subtraction Property and Division Property of Equality were used to isolate x.

Another Example

This example focuses on the reflexive, symmetric, and transitive properties rather than the arithmetic operation properties used in the first example. It shows how equality chains work without performing addition, subtraction, multiplication, or division.

Problem: Given that x = 2y + 1 and 2y + 1 = 9, determine the value of x without solving for y. Identify the property used.
Step 1: You know x = 2y + 1 (first equation) and 2y + 1 = 9 (second equation). Notice that both x and 9 equal the same expression, 2y + 1.
x=2y+1and2y+1=9x = 2y + 1 \quad \text{and} \quad 2y + 1 = 9
Step 2: Apply the Transitive Property of Equality: if x = 2y + 1 and 2y + 1 = 9, then x = 9.
x=9x = 9
Step 3: By the Symmetric Property, you could also write 9 = x — both forms are valid.
9=x9 = x
Answer: x = 9, found using the Transitive Property of Equality.

Frequently Asked Questions

What is the difference between Properties of Equality and Properties of Inequality?
Properties of Equality let you add, subtract, multiply, or divide both sides of an equation by the same value and keep the equation true. Properties of Inequality follow similar rules, but with one critical difference: when you multiply or divide both sides of an inequality by a negative number, the inequality sign reverses direction. Equality signs never flip.
Why can't you divide both sides of an equation by zero?
Division by zero is undefined in mathematics. If you allowed it, you could "prove" absurd results like 1 = 2. The Division Property of Equality explicitly requires the divisor c ≠ 0 to prevent this logical breakdown.
When do you use the Transitive Property of Equality?
You use the Transitive Property whenever two expressions are each equal to the same third expression. For example, if angle A = angle B and angle B = angle C, the Transitive Property lets you conclude angle A = angle C. This property appears frequently in geometry proofs and in substitution during algebra.

Properties of Equality vs. Properties of Inequality

Properties of EqualityProperties of Inequality
Relation symbolUses = (equals)Uses <, >, ≤, or ≥
Addition/SubtractionAdd or subtract the same value on both sides; equality is preservedAdd or subtract the same value on both sides; inequality direction is preserved
Multiplication/Division by a positive numberMultiply or divide both sides; equality is preservedMultiply or divide both sides; inequality direction is preserved
Multiplication/Division by a negative numberMultiply or divide both sides; equality is preservedMultiply or divide both sides; inequality direction reverses
Symmetric PropertyIf a = b then b = aNot symmetric; if a < b then b > a (the symbol flips)

Why It Matters

The Properties of Equality form the logical backbone of every equation you solve in algebra, geometry proofs, and beyond. Every time you "do the same thing to both sides" of an equation, you are applying one of these properties. Understanding them by name is especially important in two-column proofs, where you must justify each algebraic step with a specific property.

Common Mistakes

Mistake: Performing an operation on only one side of the equation.
Correction: The operation properties require you to apply the same operation to both sides. If you subtract 4 from the left side, you must also subtract 4 from the right side to keep the equation balanced.
Mistake: Dividing both sides by a variable expression without checking whether it could be zero.
Correction: The Division Property requires c ≠ 0. If you divide both sides by an expression containing a variable (like x), you must note that the result is only valid when that expression is nonzero. Otherwise, you may lose solutions.

Related Terms