Properties of Equality

Properties of Equality are fundamental rules governing equations in mathematics. We have various types of properties of equality in math to solve algebraic and other mathematical problems.

Some of the common properties for equality are addition property, subtraction property, multiplication property, division property, transitive property, etc. In this article, we will learn the concept of properties of equality with examples as well as explanations.

In this article, we have provided the properties of Equality, which are very useful for equation solving, along with detail explanation of all properties of equality.

What is Equality?

Equality is a fundamental concept in mathematics that represents the idea of two things being the same or having the same value.

In mathematical terms, equality is denoted by the equals sign “=”. When two quantities are equal, they are interchangeable and can be replaced by each other without changing the truth or validity of an equation.

Some examples of equality include:

• 2 + 3 = 5 states that the sum of 2 and 3 is equal to 5.
• x² = 9 asserts that the square of x is equal to 9.
• 1/2 = 0.5 indicates that one-half is equal to 0.5.

Properties of Equality

Properties of Equality

The Properties of Equality in mathematics are fundamental rules that govern equations, ensuring that both sides of an equation remain equal throughout various operations. These properties are essential in solving equations and simplifying expressions.

Properties of equality are used to express the idea that both the sides of a equation are equal and balanced. The properties of equality are applicable in the rational, real or complex number systems.

List of Properties of Equality

List of all properties of equality include:

• Addition Property of Equality
• Subtraction Property of Equality
• Multiplication Property of Equality
• Division Property of Equality
• Reflexive Property of Equality
• Symmetric Property of Equality
• Transitive Property of Equality
• Substitution Property of Equality
• Square Root Property of Equality

Now, we will discuss these properties in detail with explanations for better understanding.

1. Addition Property of Equality

To keep an equation true or unchanged, we add the same value on both sides of the equation. This property is known as addition property of equality. This property is used to solve the arithmetic problems as well as algebraic problems.

Explanation: Let us suppose, a, b and c are the real numbers.

• If a = b, then a + c = b + c.

2. Subtraction Property of Equality

To keep an equation true or unchanged, we subtract the same value from both sides of the equation. This concept is known as subtraction property of equality. This property is used to solve the algebraic problems.

Explanation: Let us suppose, a, b and c are the real numbers.

• If a = b, then, a – c = b – c

3. Multiplication Property of Equality

To keep an equation true or unchanged, we multiply both sides of the equation by the same value. This property is known as multiplication property of equality.

Explanation: Let us suppose, a, b and c are the real numbers.

• If a = b, then a × c = b × c.

4. Division Property of Equality

To keep an equation true or unchanged, we divide both sides of the equation by the same value. This property is called division property of equality.

Explanation: Suppose, a, b and c are the real numbers.

• If a = b, then a/c = b/c

5. Reflexive Property of Equality

Reflexive Property of Equality states that the every real number is equal to itself.

Explanation: If a is a real number, then according to reflexive property of equality, we can write

• a = a

6. Symmetric Property of Equality

Symmetric property of equality can be defined as if a certain real number is equal to another real number, then the another number is also equal to the certain real number.

Explanation: Let us suppose, a and b are the real numbers.

• If a = b, then b = a

7. Transitive Property of Equality

Transitive Property of equality is defined as if a real number (first) is equal to another second real number and the second real number is equal to another third real number, then the third number is equal to first number.

Explanation :

Let us suppose, a , b and c are the real numbers.

If a = b & b = c

Then, according to transitive property of equality, we can write

a = c

8. Substitution Property of Equality

Substitution Property of equality states that if a real number (first number) equal to another real number (second number), then we can substitute the second number in place of first number.

Explanation: Suppose, a, b and x are the real numbers.

• If a = b and x = a

Then , according to substitution property of equality, we can write

• b = x

9. Square Root Property of Equality

Square Root Property of equality states that if two real numbers are equal to each other then their square root is also equal to each other.

Explanation: Let us suppose, a and b are the real numbers.

• If a = b

Then according to Square Root Property of equality, we can write

• √a = √b

Properties of Equality – Table

We already discussed the list of properties of equality in detail. Here we will summaries the properties of equality in a tabular form that helps the students to review it quickly.

Property of Equality	Mathematical Formula
Addition	For the real numbers of a, b and c, If a = b Then a + c = b + c.
Subtraction	For the real numbers of a, b and c , If a = b Then a – c = b – c
Multiplication	For the real numbers of a, b and c, If a = b Then a × c = b × c
Division	For the real numbers of a, b and c, If a = b Then a ÷ c = b ÷ c
Reflective	If a is a real number which is equal to itself, Then a = a
Symmetric	For the real numbers of a and b, If a = b Then b = a
Transitive	For the real numbers of a, b and c, If a = b and b = c Then, a = c
Substitution	For the real numbers of a, b and c, If a = b and a = c Then, b = c
Square Root	For the real numbers of a and b, If a = b Then, √a = √b

Uses of Properties of Equality

The applications of Properties of equality can help us in many ways in mathematics . The applications of Properties of equality are described below.

• To solve equation in mathematics i.e., using properties of equality, we can solve the equations and find the values of unknown variables.
• To simplify the expression.
• By applying the properties of equality, we can get the basic expression of equations.

People Also Read:

• Commutative Property 
• Transitive Property
• Triangle Inequality
• Inequalities: Definition, Properties, Rules, Types

Properties of Equality Examples

To understand the properties of equality in a better way, we will have to solve some examples by applying the properties of equality. Here are the examples –

Example 1. Solve the equation, 5x – 10 = 5.

Solution:

5x – 10 = 5

Apply the addition property of equality in both sides, we get

5x – 10 + 10 = 5 + 10

⇒ 5x = 15

Now, applying the division property of equality, we get

5x/5 = 15/5

⇒ x = 3

Thus, the required answer is 5.

Example 2. Solve the equation : x/3 + 3 = 10.

Solution:

x/3 + 3 = 10

Apply the substitution property of equality in both sides, we get

x/3 + 3 – 3 = 10 – 3

⇒ x/3 = 7

Now, applying the multiplication property of equality, we get

x/3 × 3 = 7 × 3

⇒ x = 21

Thus the required answer is 7.

Example 3: Solve the equation : y² + 4y – 20 if y = 5.

Solution:

y²+ 4y – 20

Applying the substitution property of equality, we get

= 5² + 4 × 5 – 20

= 25 + 20 – 20

= 25

Thus, the required answer is 25.

Practice Problems on Properties of Equality

1. Solve for x given that x + 3 = 10.

2. If 4x = 24, find x.

3. Solve for z, when z/2 = 7.

FAQs on Properties of Equality

What is Equality in Math?

Equality in math means two quantities or expressions having the same value.

What is Sign of Equality?

The sign of equality is “=”.

What does the Equal Sign Represent?

The equal sign “=” represents equivalence or equality between two expressions, indicating that both sides have the same value.

List Properties of Equality.

Some of the common properties of equality are:

• Addition Property of Equality
• Subtraction Property of Equality
• Multiplication Property of Equality
• Division Property of Equality
• Reflexive Property
• Symmetric Property
• Transitive Property
• Substitution Property
• Square Root Property

Does Property of Equality holds for Inequality as well?

No, the properties of equality do not hold for inequalities. Inequalities have their own set of properties and rules distinct from those of equality.

How are the properties of equality useful in math?

The properties of equality are fundamental in mathematics because they form the backbone of logical reasoning, problem-solving, and the manipulation of equations. These properties ensure that equations remain balanced and that the equality between two expressions is maintained under certain operations.