Property of Equality Rules and Solved Equation Examples
In Mathematics, two things are equal if and only if they are exactly similar in every way i.e. they have the same Mathematical value and Mathematical properties. Mathematics uses the equality sign (=) to represent equality. For example, the statement A = B implies that A and B are equal.
What is the Property of Equality?
Property of equality is the fact or truth that is applied to two or more quantities related by an equal sign. Let us learn different properties of equality with examples.
Addition Property of Equality
The Addition property of equality states that when the same number is added from both sides of the equation, the equation still holds true. In other words, it states that if,
X = Y,
then X + Z = Y + Z
Example:
We know that, 4 + 3 = 7
Here, if we add 5 to both sides of the equation, the equation still holds true.
i.e. 4 + 3 + 5 = 7 + 5
12 = 12
LHS = RHS
Subtraction Property of Equality
The subtraction property of equality states that when the same number is subtracted from both sides of the equation, the equation still holds. In other words it states that if,
X = Y, then X - Z = Y - Z
Example:
We know that, 9 - 3 = 6
Here, if we subtract 2 to both sides of the equation, the equation still holds
9 - 3 - 2 = 6 - 2
4 = 4
LHS = RHS
Multiplication Property of Equality
The multiplication property of equality states that when both sides of an equation are multiplied by the same number, the two sides remain equal.
In other words, it states that if x, y, and z are real numbers such that x = y, then
a x c = b x c .
Example, If a = 10, b = 10, and c = 12, then
a x c = b x c
10 x 12 = 10 x 12
120 = 120
Therefore, LHS = RHS
Division Property of Equality
The division property of equality states that when both sides of an equation are divided by the same non-zero number, two sides remain equal. In other words, it states that if a = b, and c ≠ 0, then a ÷ c = b ÷ c
Example:
If a = 10, b = 10, and c = 5, then
a ÷ c = b ÷ c
10 ÷ 5 = 10 ÷ 5
2 = 2
Therefore, LHS = RHS
Symmetric Property of Equality
The symmetric property of equality states that if we interchange the sides of an equation, the equation still holds. In other words, it states that if X = Y, then Y = X.
Example: If 3 + 5 = 8, then 8 =?
3 + 5 = 8 or
8 = 5 + 3
Reflexive Property of Equality
The reflexive property of equality states that a number is always equal to itself. In other words, it states that if x is a number, then x = x.
Example:
2 = 2
Transitive Property of Equality
The transitive property of equality states that when given 3 numbers a, b, and c then,
a = b, b= c, then a = c.
Example:
If a = b, and b = 3, then c is also equal to 3 according to the transitive property of equality.
Substitution Property of Equality
The substitution property of equality states that if two quantities are equal, then one can replace the other in any Mathematical equation or expression.
In other words, it states that if a = b, then b can be substituted for a, in any Mathematical expression.
Example:
If a = 2 and a + 3 = 5, then 2 can be substituted in a + 3 = 5 to get 2 + 3 = 5.
Conclusion
In short, the property of equality is the truth about any quantity that is related by an equal sign. By understanding the property of equality thoroughly, we are able to simplify, manipulate, balance, and solve Mathematical equations easily. Also, we are able to draw conclusions supported by valid reasons.
FAQs on Understanding the Property of Equality in Algebra
1. What is the property of equality in math?
The property of equality states that if two expressions are equal, performing the same operation on both sides keeps them equal. In algebra, this principle allows you to solve equations without changing their truth value.
- If a = b, then a + c = b + c
- If a = b, then a − c = b − c
- If a = b, then ac = bc
- If a = b and c ≠ 0, then a/c = b/c
2. What are the main types of properties of equality?
The main types of properties of equality are addition, subtraction, multiplication, division, reflexive, symmetric, and transitive properties. Each property describes how equality behaves under specific operations.
- Addition Property: If a = b, then a + c = b + c
- Subtraction Property: If a = b, then a − c = b − c
- Multiplication Property: If a = b, then ac = bc
- Division Property: If a = b and c ≠ 0, then a/c = b/c
- Reflexive: a = a
- Symmetric: If a = b, then b = a
- Transitive: If a = b and b = c, then a = c
3. How do you use the addition property of equality?
The addition property of equality says you can add the same number to both sides of an equation without changing its solution. This is commonly used to isolate variables.
- Example: Solve x − 5 = 9
- Add 5 to both sides: x − 5 + 5 = 9 + 5
- Simplify: x = 14
4. What is the multiplication property of equality?
The multiplication property of equality states that multiplying both sides of an equation by the same number keeps the equation true. If a = b, then ac = bc.
- Example: Solve x/4 = 3
- Multiply both sides by 4: 4(x/4) = 3 × 4
- Simplify: x = 12
5. What is the division property of equality?
The division property of equality states that dividing both sides of an equation by the same nonzero number keeps the equation true. If a = b and c ≠ 0, then a/c = b/c.
- Example: Solve 5x = 20
- Divide both sides by 5: 5x/5 = 20/5
- Simplify: x = 4
6. What is the reflexive property of equality?
The reflexive property of equality states that any number or expression is equal to itself. In symbolic form, a = a.
- Example: 7 = 7
- Example: x + 3 = x + 3
7. What is the symmetric property of equality?
The symmetric property of equality states that if a = b, then b = a. It allows you to reverse the sides of an equation.
- Example: If x = 10, then 10 = x
8. What is the transitive property of equality?
The transitive property of equality states that if a = b and b = c, then a = c. It connects two equal relationships into one conclusion.
- Example: If x = 5 and 5 = y, then x = y
9. How do properties of equality help in solving equations?
The properties of equality help solve equations by keeping both sides balanced while isolating the variable. Each step maintains equality.
- Example: Solve 2x + 3 = 11
- Subtract 3 from both sides (subtraction property): 2x = 8
- Divide both sides by 2 (division property): x = 4
10. What is the difference between properties of equality and properties of inequality?
The key difference is that properties of equality keep equations equal, while properties of inequality may reverse the inequality sign when multiplying or dividing by a negative number. For example:
- Equality: If a = b, then ac = bc
- Inequality: If a < b and c is negative, then ac > bc
