Properties of Equality

Below are the key Properties of Equality in mathematics, basic principles that help us to maintain the balance of equations. These rules are important for solving equations and forming justifiable mathematical arguments.

1. Reflexive Property of Equality

Definition:
Every real number is equal to itself.

Mathematical Expression:
 a=a.

Example:
For any number x, x=x.

2. Symmetric Property of Equality

Definition:
If one quantity equals a second, then the second equals the first.

Mathematical Expression:
 If a = b, then b = a.

Example:
If 5 = 3 + 2 then 3 + 2 = 5

3. Transitive Property of Equality

Definition:
When one value is equal to the second value, and that second value is equal to a third, then the first and third values are also equal.

Mathematical Expression:
If a = b and b = c then a = c.

Example:
If 2 + 3 = 5 and 5 = 3 + 2 then 2 + 3 = 3 + 2

Another example:
If 6 + 1 = 7 and 7 = 14 ÷ 2, then 6 + 1 = 14 ÷ 2.

4. Addition Property of Equality

Definition:
If the same number is added to both sides of an equation, the equality is still true.

Mathematical Expression:
If a = b, then a + c = b + c.

Example:
If x = 4, then x + 3 = 4 + 3, so x + 3 = 7.

5. Subtraction Property

Definition: When two values are equal, and then taking away the same amount from each side, so that the equation remains balanced

Mathematical Expression: If a = b, then a − c = b - c.

Example: If y=10, then y−3=10−3, so y−3=7

6. Multiplication Property

Definition: If two quantities are equal, multiplying both sides by the same number maintains equality.

Mathematical Expression:

If a = b,

Then, a·c = b·c

Where c ≠ 0

Example:
 
If 6 = 6

Then 6 × 3 = 6 × 3 ⇒ 18 = 18

If you multiply both sides of an equation by the same number, equality will remain the same.

7. Division Property

Definition: If two quantities are equal, dividing both sides by the same non-zero number maintains equality.

Mathematical Expression:

 If, a = b then a/c=b//c 

where c ≠ 0

Example: If, 12 = 12 then 12/4=12/4 ⇒ 3 = 3

8. Substitution Property

Definition: If two quantities are equal, one can be substituted for the other in any expression.

Mathematical Expression: If a = b, then a can be replaced by b in any expression.

Example: If a = 7, then in the expression a + 3, a can be replaced with 7, resulting in 7 + 3 =10.

9. Square Root Property

Definition: If two quantities are equal, and a variable is squared, you can take the square root of both sides to solve for the variable.
Always remember to include both the positive and negative roots.
 
Mathematical Expression: x² = 16, then √x²   = ±√16, so |x| = ±4

In geometry, the Properties of Equality are important tools for modifying equations and geometric figures without changing their validity. These rules help us to maintain the unity of relationships between shapes during transformations.

For example, the Reflexive Property says that any geometric figure is always compatible with itself. Compatible figures have the same shape and size, which means one can be mapped upon the other using inflexible motions like translations, rotations, or reflections. This relationship is represented by the symbol ≅.

With the help of these properties, equations and geometric statements can become equivalents without compromising the equations.

Properties of equality are basic mathematical rules that help us to protect the balance of equations. They are important in solving algebraic equations and are also useful in our real-life world. Understanding a strong hold of these properties helps us to improve logical thinking and problem-solving qualities.

• Cooking and Recipe Adjustments: For any kind of recipe of food, the Multiplication Property of Equality is widely used. For example, if a recipe is enough for 4 people, but for 8, one just has to double all the ingredients. So the food tastes the same due to the right amount of ingredients. 

• Construction and Measurement: In construction, measurements are crucial. The reflexive property confirms that a measurement is equal to itself, while the symmetric property allows for the interchangeability of measurements, and the transitive property ensures consistency across measurements.

• Scientific Experiments: For any kind of experiment, finding equality is vital. For any kind of chemical solutions or mixing substances, the division property of equality is used to maintain a consistent ratio.