Theory

Properties of Equality

Rule

Addition Property of Equality

Adding the same number to both sides of a given equation yields an equivalent equation. Let a, b, and c be real numbers.

If a=b, then a+c=b+c.

The Addition Property of Equality is an axiom, so it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider the following example.

By adding 3 to both sides of the equation, the variable x was isolated and the solution of the equation was found. Note that the Addition Property of Equality also holds true if a, b, and c are complex numbers.

Rule

Subtraction Property of Equality

Given an equation, subtracting the same number from each side of the equation yields an equivalent equation. Let a, b, and c be real numbers.

If a=b, then a−c=b−c.

The Subtraction Property of Equality is an axiom, so it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider the following example.

By subtracting 2 from each side of the equation, the variable x was isolated and the solution of the equation was found. Note that the Subtraction Property of Equality also holds true if a, b, and c are complex numbers.

Rule

Multiplication Property of Equality

Given an equation, multiplying each side of the equation by the same number yields an equivalent equation. Let a, b, and c be real numbers.

If a=b, then a×c=b×c.

The Multiplication Property of Equality is an axiom, so it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider the following example.

Here, by multiplying both sides of the equation by 4, the variable x was isolated and the solution of the equation was found. Note that the Multiplication Property of Equality also holds true if a, b, and c are complex numbers.

Rule

Division Property of Equality

Dividing each side of an equation by the same nonzero number yields an equivalent equation. Let a, b, and c be real numbers.

If a=b and c≠0, then $a \div c = b \div c .$

The Division Property of Equality is an axiom, so it does not need a proof to be accepted as true. This property is one of the Properties of Equality that can be used when solving equations.

As can be observed, by dividing both sides of the equation by 5, the variable x was isolated and the solution of the equation was found. Note that the Division Property of Equality also holds true if a, b, and c are complex numbers.

Rule

Reflexive Property of Equality

For any real number, the number is equal to itself.

a=a

This property is an axiom, so it does not need a proof. This property is used to solve equations. For example, consider the equation below.

The sum on the left-hand side can be interpreted as a number, so this sum has to equal 8. This implies that x must be equal to 5.

Rule

Symmetric Property of Equality

For all real numbers, the order of an equality does not matter. Let a and b be real numbers.

If a=b, then b=a.

This property can be used together with other Properties of Equality to solve equations or to isolate a variable and express it in terms of another variable. In the example below, the Symmetric Property of Equality is used to express x in terms of y.

This property is an axiom, so it does not need a proof to be accepted as true. The Symmetric Property of Equality also holds true if a and b are complex numbers.

Rule

Transitive Property of Equality

For all real numbers, if two numbers are equal to the same number, then they are equal to each other. Let a, b, and c be real numbers.

If a=b and b=c, then a=c.

This property can be used together with other Properties of Equality to solve equations.

Since this property is an axiom, it does not need a proof to be accepted as true. The Transitive Property of Equality also holds true if a, b, and c are complex numbers.

Rule

Substitution Property of Equality

If two real numbers are equal, then one can be substituted for another in any expression.

If a=b, then a can be substituted for b in any expression.

Since the Substitution Property of Equality is an axiom, it does not need a proof. This property is one of the Properties of Equality that can be used when solving equations. Consider the following example.

{2 x + 1 = 20 y (2 x + 1) = 40 (I) (II)

Solve by substitution

Substitute

(II): 2x+1=20

{2 x + 1 = 20 y (20) = 40

DivEqn

(II): $LHS / 20 = RHS / 20$

{2 x + 1 = 20 y = 2

By substituting 2x+1 with 20 in Equation (II), the value of y was obtained. Note that the Substitution Property of Equality also holds true if a and b are complex numbers.