Algebra

1. Introduction to Algebra

Algebra is a branch of mathematics where we use letters and symbols to represent numbers. These letters, usually called variables, help us form general rules and solve problems without knowing all the values at once.

2. Use of Letters in Mathematics

We use letters to represent unknown quantities, write rules or generalizations, and simplify long expressions. For example: Area = l × b

3. Variables and Constants

Variables: Symbols that represent unknown or changeable values (e.g., x, y, n).

Constants: Fixed numbers that do not change (e.g., 2, 5, 100).

4. Algebraic Expressions

Combinations of constants, variables, and operations. Examples: x + 5, 3y - 7.

5. Formation of Algebraic Expressions

Verbal Statement	Algebraic Expression
Five more than a number x	x + 5
Three times a number y	3y
Ten less than a number a	a - 10
n multiplied by 2 and then added 3	2n + 3

6. Solving Expressions by Substitution

If x = 4, then 3x + 2 = 3(4) + 2 = 14

7. Equation and Its Solution

An equation states that two expressions are equal, e.g., x + 3 = 7. Solve by isolating the variable.

8. Some Common Rules and Patterns

• Even numbers: 2n
• Odd numbers: 2n + 1
• Sum of consecutive numbers: n + (n + 1) = 2n + 1
• Perimeter of square: 4s
• Area of rectangle: l × b

9. Use of Algebra in Geometry

Write formulas for shapes using variables:

• Perimeter of rectangle: 2(l + b)
• Area of square: s²

10. Advantages of Using Algebra

• Saves time
• Generalizes problems
• Helps in logical reasoning
• Used in various fields

11. Matchstick Patterns

e.g., Squares in a row:

Number of matchsticks = 3n + 1

12. Word Problems in Algebra

• Rani is 4 years younger than her sister: x - 4
• 3 phones at ₹p each: 3p

13. Simple Equations

Solve 5x = 20 ⇒ x = 4

14. Important Points to Remember

• Variables can take many values
• Equations have equal signs
• Use inverse operations to solve

15. Practice Questions

1. Write expressions:
   • 6 added to x: x + 6
   • 3n − 2: 3n - 2
   • 2p − 7: 2p - 7
2. Evaluate for x = 3, y = 2:
   • x + y = 5
   • 3x − y = 7
   • x² + y² = 13
3. Solve:
   • x − 4 = 6 ⇒ x = 10
   • 2x = 10 ⇒ x = 5
   • x/3 = 5 ⇒ x = 15

16. Real-Life Applications of Algebra

• Shopping calculations
• Budget planning
• Construction and design
• Scientific formulas
• Programming logic

17. Summary

• Use letters to represent numbers
• Expressions vs. Equations
• Solve equations step-by-step
• Real-world usage of algebra

18. Fun Fact!

The word "Algebra" comes from the Arabic word "Al-Jabr", introduced by Al-Khwarizmi in the 9th century.