Reference

Properties of Equality

Rule

Addition Property of Equality

Adding the same number to both sides of an equation results in 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 an example.

x - 3 = 5

By adding

3

to both sides of the equation, the variable

x

can be isolated and the solution to the equation can be found.

x - 3 = 5

AddEqn

$LHS + 3 = RHS + 3$

x - 3 + 3 = 5 + 3

Simplify left-hand side

x - 3 + 3 = 5 + 3

AddTerms

Add terms

x = 8

The Addition Property of Equality also holds true if

a,

b,

and

c

are complex numbers.

Rule

Subtraction Property of Equality

Subtracting the same number from both sides of an equation results in 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 an example.

x + 2 = 7

By subtracting

2

from both sides of the equation, the variable

x

can be isolated and the solution to the equation can be found.

x + 2 = 7

SubEqn

$LHS - 2 = RHS - 2$

x + 2 - 2 = 7 - 2

Simplify left-hand side

x + 2 - 2 = 7 - 2

SubTerms

Subtract terms

x = 5

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 $a \times c = b \times c .$

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.

x \div 4 x \div 4 \times 4 x = 2 = 2 \times 4 = 8

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 $c \neq = 0,$ 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.

5 x 5 x \div 5 x = 10 = 10 \div 5 = 2

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.

x + 3 = 8

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.

{x = 5 y - 30 5 y - 30 = 20 \Rightarrow x = 20

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

$(II):$ $2 x + 1 = 20$

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

DivEqn

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

{2 x + 1 = 20 y = 2

By substituting

2 x + 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.

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