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Properties of Equality

We have mainly nine properties of equality, namely addition property, subtraction property, multiplication property, division property, reflexive property, symmetric property, transitive property, substitution property, and square root property of equality. Properties of equality give the relation between two quantities that are equal and how the equation remains balanced after applying an operation. When an operation (addition, subtraction, multiplication, and division) is applied on both sides of an equation, the equation still holds true.

In this article, we will explore the concept of the properties of equality with explanations and examples. We will list the various properties of equality along with examples for a better understanding of the concept. We shall also discuss the applications of these properties and provide a summary of the properties in a table for a quick review.

1. What are Properties of Equality?
2. List of Properties of Equality
3. Properties of Equality Table
4. Applications of Properties of Equality
5. FAQs on Properties of Equality

What are Properties of Equality?

Properties of equality describe the relation between two equal quantities and if an operation is applied on one side of the equation, then it must be applied on the other side to keep the equation balanced. We have mainly nine properties of equality - addition, subtraction, multiplication, division, reflexive, symmetric, transitive, substitution, and square root properties. The addition, subtraction, multiplication, and division properties of equality help to solve algebraic equations involving real numbers. The reflexive, symmetric, and transitive properties of equality together define the equivalence relation.

Properties of Equality Definition

The properties that do not change the truth value of an equation, that is, the properties that do not impact the equality of two or more quantities are called the properties of equality. Such properties of equality help us to solve various algebraic equations and define an equivalence relation.

properties of equality

List of Properties of Equality

We will focus on nine properties of equality. Let us list them below and define each one of them to understand these properties:

We will now go through each of these properties in detail to understand them better.

Addition Property of Equality

The addition property of equality is defined as "When the same amount is added to both sides of an equation, the equation still holds true". We can express this property mathematically as, for real numbers a, b, and c, if a = b, then a + c = b + c. This property can be used in arithmetic and algebraic equations.

Subtraction Property of Equality

The subtraction property of equality states that if the same real number is subtracted from both sides of an equation, then the equation still holds true. The formula for this property can be written as, for real numbers a, b, c, if a = b, then a - c = b - c. We can use this property to solve algebraic equations.

Multiplication Property of Equality

According to the multiplication property of equality, when the same real number is multiplied by both sides of an equation, then the two sides of the equation remain equal. We can express the formula for this property as, for real numbers a, b, and c, if a = b, then a × c = b × c.

Division Property of Equality

The division property of equality states that when both sides of an equation are divided by the same real number, then equality still holds. Mathematically, we can write this property as, for real numbers a, b, and c, if a = b, then a/c = b/c. This property is used to find the unknown variable in an algebraic equation.

Reflexive Property of Equality

According to the reflexive property of equality, every real number is equal to itself. We can express it mathematically as, for an arbitrary real number x, we have x = x.

Symmetric Property of Equality

The symmetric property of equality states that, when a real number x is equal to a real number y, then we can say that y is equal to x. This property can be expressed as, if x = y, then y = x.

Transitive Property of Equality

The transitive property of equality is defined as, for real numbers x, y, and x, when x is equal to y and y is equal to z, then we can say that x is equal to z. Mathematically, we can express this property of equality as, for real numbers x, y, and x, if x = y and y = z, then we have x = z.

Substitution Property of Equality

According to the substitution property of equality, for real numbers x and y, if we have x = y, then we can substitute y in place of x in any algebraic expression. In other words, we can say that if x = y, then y can be substituted for x in any algebraic expression to find the value of the unknown variable. We can express the substitution property as, for real numbers x, y, and z, if x = y and x = z, then we can write y = z

Square Root Property of Equality

The square root property of equality states that if a real number x is equal to a real number y, then the square root of x is equal to the square root of y. We can write this property mathematically as, for real numbers x and y, if x = y, then √x = √y.

Properties of Equality Table

Now, we have understood the various properties of equality in the previous section. Let us now summarize these properties in a table given below along with their meanings for a quick review.

Property of Equality Meaning
Addition Property

For real numbers x, y, and z,

If x = y, then x + z = y + z
Subtraction Property

For real numbers x, y, and z,

If x = y, then x - z = y - z
Multiplication Property

For real numbers x, y, and z,

If x = y, then x × z = y × z
Division Property

For real numbers x, y, and z,

If x = y, then x ÷ z = y ÷ z
Reflexive Property

Every real number is equal to itself. For a real number x,

x = x,
Symmetric Property

Order of equality does not matter. For real numbers x and y,

If x = y, then y = x
Transitive Property

Numbers equal to the same number are equal to each other. For real numbers x, y, and z,

If x = y and y = z, then x = z
Substitution Property

Any two real numbers equal to each other can be substituted for one another in any expression. For real numbers x and y,

If x = y, then y can be substituted for x.
Square Root Property

Square Roots of Equal Numbers are equal. For real numbers x and y,

If x = y, then √x = √y

Applications of Properties of Equality

Now that we have understood the meaning of the different properties of equality, let us now solve a few examples based on these properties to understand the application of the properties.

Example 1: Solve x - 3 = 8

Solution: To find the value of x, we will use the addition property of equality.

Add 3 to both sides of the equation. So, we have

x - 3 = 8

⇒ x - 3 + 3 = 8 + 3

⇒ x = 8 + 3

⇒ x = 11

Example 2: Find the value of the expression x2 + 3x - 4 if x = 2.

Solution: To find the value of the given expression, we will use the substitution property of equality. Since x = 2, we will substitute 2 in place of x in the expression x2 + 3x - 4.

x2 + 3x - 4

= 22 + 3(2) - 4

= 4 + 6 - 4

= 6

So, the value of the expression x2 + 3x - 4 is equal to 6 when x = 2.

Important Notes on Properties of Equality

  • We have mainly nine properties of equality:
    • 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
  • These properties help in solving algebraic equations, finding the value of expressions, and defining the equivalence relation.

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Properties of Equality Examples

  1. Example 1: Solve the algebraic equation 2y + 4 = 16 using the properties of equality.

    Solution: To solve the given equation, we will use the subtraction and division properties of equality.

    Subtract 4 from both sides of the equation.

    2y + 4 = 16

    ⇒ 2y + 4 - 4 = 16 - 4

    ⇒ 2y = 12

    Now, divide both sides of the above equation by 2.

    2y/2 = 12/2

    ⇒ y = 6

    Answer: y = 6

  2. Example 2: Show that 'Is equal to' is an equivalence relation.

    Solution: 'Is equal to (=)' is an equivalence relation on any set of numbers A.

    For all real numbers a, b, c ∈ R, we have

    • a = a, --- (Using Reflexive Property of Equality)
    • a = b ⇒ b = a, --- (Using Symmetric Property of Equality)
    • a = b, b = c ⇒ a = c --- (Using Transitive Property of Equality)

    This implies (=) is reflexive, symmetric, and transitive.

    Answer: Hence, we have shown 'Is equal to (=)' is an equivalence relation.

  3. Example 3: Find the square root of x, given that x = 9 using the properties of equality.

    Solution: To find the square root of x, we will use the square root property of equality. We know that if x = y, then √x = √y. So, we have

    x = 9

    ⇒ √x = √9

    ⇒ √x = 3 --- (Because the square root of 9 is 3)

    Answer: The square root of x is equal to 3.

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FAQs on Properties of Equality

What are the Properties of Equality in Algebra?

Properties of equality describe the relation between two equal quantities and if an operation is applied on one side of the equation, then it must be applied on the other side to keep the equation balanced.

How Many Properties of Equality Are There?

We have mainly nine properties of equality:

  • 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

What is the Difference Between Properties of Equality and Properties of Inequality?

The main difference between the properties of equality and the properties of inequality is that if we multiply or divide both sides of an equation by the same negative real number, the equation remains the same, but if we multiply or divide both sides of an inequality by the same real negative number, the inequality reverses.

How to Solve an Equation Using the Properties of Equality?

We can solve an equation using the four properties of equality - Addition, Subtraction, Multiplication, and Division. We can simply add, subtract, multiply or divide both sides of an equation to find the value of the unknown variable.

What is the Difference Between the Properties of Equality and the Properties of Congruence?

The main difference between the properties of equality and the properties of congruence is that the properties of equality are based on algebra whereas the properties of congruence are based on geometry.

What is the Distributive Property of Equality?

According to the distributive property of equality, for real numbers a, b, and c, we have (a + b)c = ab + bc.

Why Do We Use Properties of Equality?

We use the properties of equality to solve different algebraic equations and find the value of the unknown variable. We can also use these properties to define the equivalence relation.

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