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Properties of Equalities

Algebra - a vast and a very useful branch of mathematics - is the study of variables and constants. It introduces expressions to the students for the first time. Expressions are the set of one or more variables usually multiplied together with constant terms. We do study about equalities and inequalities also in algebra. Equalities are the expressions with equal (=) sign. While inequalities are the expressions which have inequality (<, >, $\leq$, $\geq$) signs in them.

Here, in this article, we are going to study about equalities, especially the properties of equalities. Before discussing about the properties of equalities in math, we need to know about equality first. Equality should be the state of being quantitatively similar. Few examples for equalities are :
2a + b = 4
3x$^{2}$ + 4y = 0

We are having various properties for the equalities. Here, we will see about the math properties of equality used in arithmetic operations and various mathematical operations. We often perform algebraic operation on equality expressions. While solving an equality, one needs to follow the rule of PEMDAS which means that the algebraic operations are to be performed in the following order - parenthesis, exponent, multiply, divide, add, subtract. The knowledge of these properties are very useful in order to solve those. Let us go ahead and learn about the properties of inequalities and their applications.

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List of Properties of Equalities

The properties of equalities in math are listed below :

• Subtraction property
• Multiplication property
• Division property
• Commutative property
• Associative property
• Distributive property
• Transitive property

Identity and Equality Properties

The brief explanation for math properties of equality is as follows:

Let us consider a, b and c as any numbers.

• If a = b, then a + c = b + c

When we add the same number with both side of equation, that should not affect the result of the equation.

For example, if a - 9 = 7

Then, a - 9 + 9 = 7 + 9

a = 16

Subtraction Property of Equality

• If a = b, then a - c = b - c

When we subtract the same number with both side of equation, that should not affect the result of the equation.

Subtraction Property of Equality Example

If a + 3 = 6

Then, a + 3 - 3 = 6 - 3

a = 3

Multiplication Property of Equality

• If a = b, then a × c = b × c

When we multiply the same number in the both side of the equation, that should not affect the result of the equation.

Multiplication Property of Equality Examples

If $\frac{x}{5}$ = 7

then, $\frac{x}{5}$ * 5 = 7 * 5

=> x = 35.

Division Property of Equality

• If a = b, then $\frac{a}{c}$ = $\frac{b}{c}$

When we divide the same number with both side of the equation, that should not affect the result of the equation.

Division Property of Equality Examples

If 4x = 24

Then, $\frac{4x}{4}$ = $\frac{24}{4}$

x = 6

Commutative Property of Equality

For real number a and b

1. a + b = b + a
2. a × b = b × a

Example:

1. 2 + 3 = 5 = 3 + 2
2. 4 × 5 = 20 = 5 × 4

Associative Property of Equality

For real numbers a, b and c

1. (a + b) + c = a + (b + c)
2. (a × b) × c = a × (b × c)

Example:

1. (6 + 2) + 3 = 8 + 3 = 11 and 6 + (2 + 3) = 6 + 5 = 11
2. (3 × 1) × 5 = 3 × 5 = 15 and 3 × (1 × 5) = 3 × 5 = 15

Distributive Property of Equality

For any real numbers a, b and c

a × (b + c) = a × b + a × c

Example:

If a = 4, b = 2 and c = 7, then,

a × (b + c) = 4 × (2 + 7) = 4 × 9 = 36

a × b + a × c = 4 × 2 + 4 × 7 = 8 + 28 = 36

Transitive Property:

For real numbers a, b and c. If a = b and b = c, then a = c.

For any real number a,

a + 0 = 0 + a = a

Here, '0' is the additive identity.

Multiplicative Identity:

For any real number a,

a * 1 = 1 * a = a

Here, '1' is the multiplicative identity.

Substitution Property of Equality

Substitution property states that if two values are equivalent, then we can substitute one for another in an expression. If x = y, x can replace y or y can replace x in any expression.

Substitution Property of Equality Examples

Given below are some of the examples on substitution property.

Solved Examples

Question 1: Evaluate 5m + 6 if m = -1
Solution:
5m + 6 = 5(-1) + 6
= -5 + 6
= 1

Question 2: Solve 2x - 6 for x = 3
Solution:
2x - 6 = 2(3) - 6
= 6 - 6
= 0

Reflexive Property of Equality

Reflexive property of equality is one of the equivalence properties of equality. Any number is equal to itself is the reflexive property of the equality.

If a is any real number, then a = a.

Reflexive Property of Equality Examples

For example, 4 = 4 and -2 = -2 satisfies the reflexive property.

Symmetric Property of Equality

Symmetric property of equality states that if first number is equal to second number, then second number is equal to first number.

For real numbers, x and y

If x = y, then y = x.

Symmetric Property of Equality Examples

Given below are some of the examples on symmetric property of equality.
1. If 7 = 3 + 4, then 3 + 4 = 7
2. If 7 = n - 3, then n - 3 = 7

Transitive Property of Equality

If first number is equal to second and second number is equal to third, then first number is equal to third. The transitive property of equality for any real numbers a, b, and c is as follows:

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

Transitive Property of Equality Examples

Given below are some of the examples on transitive property.

Solved Example

Question: Prove the transitive property of equality for any real numbers a, b, and c.
Solution:

Let a = 2x - 1, b = 3x - 2 and c = 3x + 6

According to transitive property of equality

If a = b and b = c, then a = c => 2x - 1 = 3x - 2 and 3x - 2 = 3x + 6, then

2x - 1 = 3x + 6

Take -3x on both sides

2x - 1 - 3x = 3x + 6 - 3x

=> - x -1 = 6

Adding 1 on both the sides, we get

=> -x - 1 + 1 = 6 + 1

=> -x = 7

x = -7

Properties of Equality and Congruence

Properties of congruence are similar to the equivalence properties of equality.

For any real numbers x, y and z

Reflexive Property of congruence

Anything is congruent to itself.

=> x $\cong$ x

Symmetric Property of Congruence

If first number is congruence to second, then second number is congruence to first.

If x $\cong$ y, then y $\cong$ x

Transitive Property of Congruence

If first number is congruence to second and second is congruence to third, then the first number is congruence to third.

If x $\cong$ y and y $\cong$ z, then x $\cong$ z
 More topics in Properties of Equality Transitive Property of Equality Division Property of Equality Addition Property of Equality
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