Introduction to Algebra
Think about this puzzle: "I am thinking of a number. If I add 5 to it, I get 12. What is the number?" You can quickly figure out the answer is 7. But what if the puzzle were more complicated? How would you write it down? You could say: "some number + 5 = 12" — and use a letter like x to stand for that unknown number. Then you write: x + 5 = 12.
This is the beginning of algebra. Algebra is the branch of maths where we use letters (like x, y, n, a) to represent numbers we do not know yet. These letters are called variables.
Using algebra, you can write rules and patterns in a short way. Instead of saying "the perimeter of a square is four times its side," you simply write P = 4s. Algebra is like a shorthand for maths — it makes things shorter, cleaner, and easier to work with.
Algebra was developed over centuries. Indian mathematicians like Brahmagupta and Aryabhata were among the first to use algebra to solve problems. The word "algebra" comes from the Arabic word "al-jabr," which means "restoring." Today, algebra is used in science, engineering, computers, and even in everyday situations like calculating shopping bills or splitting expenses among friends.
In this chapter, you will learn the basic ideas of algebra — variables, constants, algebraic expressions, and how to form equations from word problems. This topic is part of the Algebra chapter in Grade 6 Maths (NCERT/CBSE). It forms the foundation for all the algebra you will study in higher classes.
What is Introduction to Algebra - Grade 6 Maths (Algebra)?
Definition: Algebra is a branch of mathematics that uses letters and symbols to represent numbers and quantities in formulas and equations.
Variable:
- A variable is a letter that stands for an unknown or changing number.
- Common letters used: x, y, z, a, b, n, p, etc.
- The value of a variable can change — that is why it is called "variable" (it varies).
- Example: In x + 3 = 10, the letter x is a variable. Its value is 7.
Constant:
- A constant is a number that has a fixed value. It does not change.
- Examples: 5, 100, −3, 0 are all constants.
- In the expression 2x + 7, the number 7 is a constant and 2x is a variable part.
Algebraic Expression:
- An algebraic expression is a combination of variables, constants, and operations (+, −, ×, ÷).
- Examples: 3x + 2, 5y − 8, 2a + 3b, n/4 + 1
Introduction to Algebra Formula
Writing algebraic expressions from statements:
Words → Algebra
Common translations:
- "A number added to 5" → x + 5
- "3 less than a number" → x − 3
- "Twice a number" → 2x
- "A number divided by 4" → x/4
- "5 more than three times a number" → 3x + 5
- "The product of a number and 7" → 7x
Formulas using variables:
Perimeter of square = 4s
Perimeter of rectangle = 2(l + b)
Area of rectangle = l × b
These formulas use variables (s, l, b) to express rules that work for ALL squares and rectangles, not just one specific case.
Derivation and Proof
Before algebra, people had to describe every mathematical relationship in words. This was long and confusing.
Without algebra:
"To find the perimeter of a rectangle, add the length and the breadth, then multiply the result by two."
With algebra:
P = 2(l + b)
See how much shorter and clearer the algebraic version is? That is the power of using letters for numbers.
Why letters?
- Letters stand for numbers that can change. If a square has side 5, its perimeter is 4 × 5 = 20. If the side is 8, the perimeter is 4 × 8 = 32. The formula P = 4s covers ALL cases at once.
- Letters let you solve problems where the answer is unknown. "I have some marbles. My friend gives me 6. Now I have 15. How many did I start with?" → x + 6 = 15, so x = 9.
Rules for writing algebraic expressions:
- We write 2x instead of 2 × x (no multiplication sign needed between a number and a variable).
- We write x × y as xy.
- We write x × x as x².
- The number in front of a variable is called its coefficient. In 5x, the coefficient is 5.
Types and Properties
Type 1: Identify variables and constants
- Given an expression, identify which parts are variables and which are constants.
- Example: In 4y + 9 → y is the variable, 4 is the coefficient, 9 is the constant.
Type 2: Write algebraic expressions from word statements
- Convert an English sentence into an algebraic expression.
- Example: "7 added to a number" → x + 7
Type 3: Write word statements from algebraic expressions
- Convert an algebraic expression back into words.
- Example: 3n − 5 → "5 less than three times a number"
Type 4: Substitute and evaluate
- Put a given value in place of the variable and calculate.
- Example: If x = 4, find 3x + 2 → 3(4) + 2 = 14
Type 5: Write formulas using variables
- Express mathematical rules as formulas.
- Example: "The total cost of n pencils at Rs. 5 each" → Cost = 5n
Type 6: Forming simple equations
- Write an equation from a word problem.
- Example: "A number increased by 8 equals 20" → x + 8 = 20
Solved Examples
Example 1: Example 1: Identifying Variables and Constants
Problem: In the expression 6p + 11, identify: (a) the variable (b) the coefficient of p (c) the constant.
Solution:
- (a) The variable is p.
- (b) The coefficient of p is 6.
- (c) The constant is 11.
Example 2: Example 2: Writing Expressions from Words
Problem: Write algebraic expressions for: (a) 8 added to a number (b) A number multiplied by 3 (c) 10 subtracted from a number (d) A number divided by 5
Solution:
- (a) 8 added to a number → x + 8
- (b) A number multiplied by 3 → 3x
- (c) 10 subtracted from a number → x − 10
- (d) A number divided by 5 → x/5
Example 3: Example 3: Writing Words from Expressions
Problem: Write these expressions in words: (a) y + 4 (b) 7n (c) 2x − 3 (d) m/6
Solution:
- (a) y + 4 → 4 more than a number y
- (b) 7n → 7 times a number n
- (c) 2x − 3 → 3 less than twice a number x
- (d) m/6 → a number m divided by 6
Example 4: Example 4: Substitution
Problem: If x = 5, find the value of: (a) 2x + 3 (b) 4x − 7 (c) x² + 1
Solution:
- (a) 2x + 3 = 2(5) + 3 = 10 + 3 = 13
- (b) 4x − 7 = 4(5) − 7 = 20 − 7 = 13
- (c) x² + 1 = 5² + 1 = 25 + 1 = 26
Example 5: Example 5: Writing a Formula
Problem: Write a formula for the total cost of buying n notebooks at Rs. 40 each.
Solution:
- Cost of 1 notebook = Rs. 40
- Cost of n notebooks = 40 × n
Formula: C = 40n
Where C is the total cost and n is the number of notebooks.
Check: If n = 3, C = 40 × 3 = Rs. 120. This makes sense.
Example 6: Example 6: Age Problem
Problem: Arun is x years old. His father is 28 years older than Arun. Write expressions for: (a) His father's age (b) Their combined age (c) His father's age when Arun was born
Solution:
- (a) Father's age = x + 28
- (b) Combined age = x + (x + 28) = 2x + 28
- (c) When Arun was born (age 0), father's age = 28 (this is a constant, not a variable expression)
Example 7: Example 7: Pattern with Matchsticks
Problem: To make one square with matchsticks, you need 4 matchsticks. For 2 squares in a row (sharing a side), you need 7 matchsticks. For 3 squares in a row, you need 10 matchsticks. Write a rule for n squares.
Solution:
Look at the pattern:
- 1 square → 4 matchsticks
- 2 squares → 7 matchsticks
- 3 squares → 10 matchsticks
The pattern increases by 3 each time. The first square needs 4, and each additional square needs 3 more.
- Number of matchsticks = 4 + 3 × (n − 1) = 4 + 3n − 3 = 3n + 1
Formula: M = 3n + 1
Check: n = 1 → 3(1) + 1 = 4 ✓. n = 2 → 3(2) + 1 = 7 ✓. n = 3 → 3(3) + 1 = 10 ✓.
Example 8: Example 8: Forming an Equation
Problem: Sita has some pencils. She gives 5 to her friend and has 12 left. Write an equation and find how many pencils she had.
Solution:
Let the number of pencils = x.
- She gives away 5: x − 5
- She has 12 left: x − 5 = 12
Equation: x − 5 = 12
Solving: x = 12 + 5 = 17
Answer: Sita had 17 pencils.
Example 9: Example 9: Perimeter Using Variables
Problem: Write the perimeter of a triangle with sides a, b, and c. If a = 5 cm, b = 7 cm, c = 9 cm, find the perimeter.
Solution:
Formula: P = a + b + c
Substituting:
- P = 5 + 7 + 9
- P = 21 cm
Answer: The perimeter is 21 cm.
Example 10: Example 10: Consecutive Numbers
Problem: Write three consecutive numbers using a variable. If the smallest is 15, find all three.
Solution:
Let the smallest number = n.
- The three consecutive numbers are: n, n + 1, n + 2
If n = 15:
- First number: 15
- Second number: 15 + 1 = 16
- Third number: 15 + 2 = 17
Answer: The three numbers are 15, 16, and 17.
Real-World Applications
Algebra is used in many everyday situations, even if you do not notice it:
- Shopping: If one pen costs Rs. 10, the cost of n pens is 10n. This is an algebraic expression you use every time you buy multiple items.
- Formulas in maths: All formulas (perimeter, area, volume) use variables. P = 2(l + b) works for every rectangle, not just one specific one.
- Patterns: If you see a growing pattern (like matchstick designs or number sequences), algebra helps you write a general rule for any position in the pattern.
- Science: Speed = Distance / Time is written as s = d/t. This algebraic formula is used in physics every day.
- Sports: A cricketer's total runs after n matches can be expressed using algebra: Total = average × n.
- Puzzles and riddles: "Think of a number" puzzles are solved using algebra. The unknown number becomes x, and you form an equation to solve it.
Key Points to Remember
- Algebra uses letters to represent unknown or variable numbers.
- A variable is a letter whose value can change (x, y, n, etc.).
- A constant is a fixed number that does not change (5, −3, 100).
- An algebraic expression combines variables, constants, and operations (3x + 7).
- The number in front of a variable is called the coefficient. In 5x, the coefficient is 5.
- We write 2 × x as 2x (no multiplication sign needed).
- Algebra lets you write general rules — P = 4s works for every square, not just one.
- Substitution means replacing the variable with a specific number and calculating.
- An equation uses an equals sign (=) and says two things are equal: x + 3 = 10.
- Algebra is the foundation for solving equations, which you will learn in Grade 7.
Practice Problems
- Identify the variables and constants in: 9m − 4.
- Write algebraic expressions for: (a) 6 more than a number (b) Half of a number (c) 3 times a number, minus 2.
- Write in words: (a) 5x (b) y − 8 (c) 4n + 1.
- If y = 3, find the value of: (a) 7y (b) 2y + 5 (c) y² − 4.
- The cost of one chocolate is Rs. 15. Write a formula for the cost of n chocolates.
- Ravi's age is x years. Write expressions for: (a) his age 5 years from now (b) his age 3 years ago.
- Form an equation: A number increased by 12 equals 30. Solve it.
- Write a rule for the number of matchsticks needed to make n triangles in a row (sharing sides). Check for n = 1, 2, 3.
Frequently Asked Questions
Q1. What is algebra?
Algebra is a branch of maths where letters are used to represent numbers. These letters, called variables, can stand for unknown values or values that change. Algebra helps you write general rules and solve problems.
Q2. What is a variable?
A variable is a letter (like x, y, or n) that represents a number whose value is not yet known or can change. In the expression 2x + 3, the letter x is the variable.
Q3. What is the difference between a variable and a constant?
A variable changes — its value is not fixed. A constant stays the same. In 5x + 8, the variable is x (it can be any number) and the constant is 8 (it is always 8).
Q4. Why do we use letters in maths?
Letters allow us to write general rules that work for any number. For example, P = 4s gives the perimeter of every square, regardless of its size. Without letters, you would need to calculate separately for every case.
Q5. What is an algebraic expression?
An algebraic expression is a mathematical phrase that contains variables, constants, and operations (+, −, ×, ÷) but does NOT have an equals sign. Examples: 3x + 5, 2a − 7, y/4.
Q6. What is the difference between an expression and an equation?
An expression does not have an equals sign: 3x + 5 is an expression. An equation HAS an equals sign and says two things are equal: 3x + 5 = 20 is an equation.
Q7. What does 2x mean?
2x means 2 times x, or 2 multiplied by x. In algebra, we skip the multiplication sign between a number and a variable. So 2 × x is written as 2x.
Q8. Can any letter be used as a variable?
Yes, any letter can be a variable. The most commonly used are x, y, z, a, b, n, p, and m. Some letters have standard uses — like r for radius and t for time — but this is just a convention, not a rule.
