Overview
Chapter: Algebra (Mathematics – VI) Introduction: This chapter introduces algebra as a compact way to express patterns and relationships using symbols (usually letters). Students move from concrete arithmetic to simple abstraction by using letters to stand for numbers, building and evaluating algebraic expressions, and solving basic equations. The chapter uses everyday examples and number patterns to show why algebra is useful and how it makes generalization and problem solving easier. Importance: Algebra is the bridge between arithmetic and higher mathematics. Early exposure develops logical thinking, modelling skills and the ability to generalize rules. It helps students translate word problems into mathematical language and prepares them for topics like equations, functions and geometry in higher classes. Key themes and learning focus: The chapter covers variables and constants, terms, coefficients, writing and reading algebraic expressions, simplifying expressions, evaluating expressions by substitution, forming and solving simple linear equations in one variable, and using algebra to describe patterns and solve real-life problems. Emphasis is on understanding notation,…
Learning Objectives
- Define key algebraic terms: variable, constant, coefficient and term.
- Identify variables, constants, coefficients and like/unlike terms in algebraic expressions.
- Classify expressions as monomials, binomials or polynomials based on the number of terms.
- Translate common verbal phrases and simple word problems into algebraic expressions.
- Evaluate algebraic expressions by substituting given numerical values for variables.
- Simplify algebraic expressions by collecting like terms and performing arithmetic operations.
- Apply the order of operations (BODMAS) correctly when simplifying algebraic expressions.
- Form simple linear equations from word problems involving one unknown.
Topics in this chapter
12 topics · tap a topic title to jump straight to it.
Introduction to Algebra
Algebra is the branch of mathematics that uses letters or symbols (called variables) to represent numbers. In Class 6, algebra helps us write rules, describe patterns, and solve problems where some numbers are unknown. Using letters (like x, y, a, b) makes general statements about numbers and makes problems easier to handle.
Basic terms:
- Variable: A letter that stands for an unknown number (e.g., x).
- Constant: A fixed number (e.g., 5, 12).
- Term: A single part of an expression separated by + or − (e.g., 3x is one term; 7 is another).
- Coefficient: The number multiplied by a variable (in 3x, 3 is the coefficient).
- Expression: A combination of numbers, variables and operations (e.g., 2x + 5).
- Equation: A statement that two expressions are equal (e.g., 2x + 5 = 17).
What you do with algebra:
- Form: Convert words or situations into algebraic expressions or equations (e.g., "three more than twice a number" → 2x + 3).
- Evaluate: Substitute a value for the variable and calculate (if x = 4, 2x + 3 = 2×4 + 3 = 11).
- Simplify: Combine like terms (e.g., 3x + 2x = 5x).
- Solve equations: Find the value of the variable (e.g., 3x + 5 = 20 → 3x = 15 → x = 5).
Steps to solve a simple linear equation (one-step or two-step):
- Use inverse operations to isolate the variable (undo addition with subtraction, undo multiplication with division).
- Keep the equality balance—what you do to one side, do to the other side.
- Check the result by substituting it back into the original equation.
Why algebra is useful: It gives a compact way to write rules, describe relationships (like distance = speed × time), solve puzzles, and model real-life situations where some values are unknown.
- Shopping: If one apple costs ₹5, and you buy x apples, total cost = 5x. If you pay ₹20 and the cost is 5x, the equation 5x = 20 gives x = 4.
- Age problem: If Ayesha is 3 years older than Rahul and Rahul's age is r, then Ayesha's age = r + 3.
- Perimeter of a rectangle: For length l and breadth b, perimeter = 2(l + b). If b = 4 and l is unknown x, perimeter = 2(x + 4).
- Missing number in a pattern: If a pattern shows 2, 5, 8, … the nth term can be written as 3n − 1 (for n = 1, 2, 3 → 2, 5, 8).
- Area model: If the area of a rectangle is 24 and breadth is 3, using A = l × b gives 24 = l × 3 so l = 8.
- Commutative laws: a + b = b + a ; a × b = b × a
- Associative laws: (a + b) + c = a + (b + c) ; (a × b) × c = a × (b × c)
- Distributive law: a(b + c) = ab + ac
- Additive identity: a + 0 = a ; Multiplicative identity: a × 1 = a
- Zero multiplication: a × 0 = 0
- Like terms combine: px + qx = (p + q)x
Algebraic Expressions
What is an algebraic expression? An algebraic expression is a combination of numbers and letters (called variables) joined by operations such as addition, subtraction, multiplication or division. Example: 3x + 5, 2a - b, 4xy.
Parts of an algebraic expression
- Variable: a letter that stands for a number (e.g., x, y, a).
- Coefficient: the numerical factor of a term (in 5x, 5 is the coefficient).
- Constant: a number without a variable (e.g., 7 in 3x + 7).
- Term: each part separated by + or − signs (in 3x + 5y − 2, there are three terms).
Types of expressions: monomial (one term, e.g., 7x), binomial (two terms, e.g., x + 3), trinomial (three terms, e.g., x^2 + 2x + 1), polynomial (one or more terms).
What you do with expressions
- Evaluate: Substitute a value for the variable and compute. Example: evaluate 3x + 5 for x = 2 gives 3(2) + 5 = 11.
- Simplify: Combine like terms (terms with the same variable part). Example: 4x + 3x = 7x.
- Use distributive property to remove brackets: a(b + c) = ab + ac.
How to simplify step-by-step: (1) Remove brackets using distributive law; (2) Group like terms; (3) Add/subtract coefficients of like terms; (4) Write the final simplified expression.
Why it matters: Algebraic expressions let us write general rules and formulas, model real-life situations (cost, distance, area), and solve many problems without knowing exact numbers at first.
- Evaluate 3x + 5 for x = 4: 3(4) + 5 = 12 + 5 = 17.
- Simplify 4x + 3x − 2: Combine like terms 4x + 3x = 7x, so result is 7x − 2.
- Use distributive property: 2(x + 3) = 2x + 6.
- Identify parts in 5ab + 2a − 7: terms are 5ab, 2a, −7; constants = −7; coefficients of a in 2a is 2.
- Form a real-life expression: If one pencil costs Rs. p and you buy x pencils, total cost = px. If there is a fixed bag charge of Rs. 10, total cost = px + 10.
- Perimeter of a rectangle with length l and breadth b: P = 2(l + b) = 2l + 2b (an algebraic expression showing perimeter in terms of l and b).
- Term form: coefficient × variable (e.g., 7x means coefficient 7 and variable x).
- Combine like terms: ax + bx = (a + b)x. Example: 3x + 5x = 8x.
- Distributive property: a(b + c) = ab + ac and a(b − c) = ab − ac.
- Constant term: terms without variables remain unchanged when combining (e.g., in 3x + 4 + 2x, combine x-terms to get 5x + 4).
- General linear form (one variable): ax + b (useful for modeling many straight-line relationships).
Terms, Factors and Coefficients
What is an algebraic expression? An algebraic expression is made of numbers, variables (like x, y) and operations (+, -, ×, ÷). Example: 5x + 3y - 7.
Terms
A term is each part of an expression separated by plus or minus signs. In 5x + 3y - 7 the terms are 5x, 3y and -7. A term can be a number (called a constant term), a variable (x), or a product of numbers and variables (5x, 2xy).
Coefficients
The coefficient of a term is the numerical factor multiplying the variable part. In 5x the coefficient is 5. In -7 (a constant) the coefficient of any variable is 0. If a term is just x, its coefficient is 1 (because x = 1x).
Factors
A factor is one of the quantities being multiplied. For a product 3 × x × y, the factors are 3, x and y. For the term 6xy the factors are 6, x and y. Factoring an expression means writing it as a product of its factors: e.g. 4x + 8 = 4(x + 2) because 4 is a common factor.
Key differences
- Terms are parts added or subtracted.
- Factors are parts multiplied together.
- Coefficient is the number part of a term (the multiplier of the variables).
How to identify quickly
1. Split the expression at + and - to get terms.
2. For each term, separate the number (coefficient) and the variable part. If no number is written, the coefficient is 1 or -1 depending on the sign.
3. To find factors, look for multiplication inside each term or factor common terms from the whole expression.
Simple tips for Class 6
- Write signs with terms: 5x + 3y - 7 → +5x, +3y, -7.
- Constant term means no variable (here -7).
- Use factorization to simplify expressions by taking out common factors.
- Expression: 5x + 3y - 7. Terms: 5x, 3y, -7. Coefficients: 5 (for x), 3 (for y). Constant term: -7.
- Term-only example: x has coefficient 1. -x has coefficient -1.
- Product as factors: 6xy. Factors: 6, x, y. Coefficient is 6.
- Factoring common factor: 4x + 8 = 4(x + 2). Here 4 is a common factor and x+2 is the remaining factor.
- Real-life: If one pencil costs x rupees, buying 3 pencils costs 3x rupees — term is 3x, coefficient 3.
- Area model: Rectangle with length = x m and width = 5 m has area 5x m^2 — factors are 5 and x, coefficient of x is 5.
- General form of a single term: coefficient × variables. Example: a x^n y^m (coefficient = a).
- Coefficient of term ax = a. If term is x, coefficient = 1. If term is -x, coefficient = -1.
- Constant term: a term without variables (example: 7 or -7).
- Factoring common factor: bx + by = b(x + y).
- Product of factors: if A and B are factors then expression = A × B. Example: 6xy = 2 × 3 × x × y.
Like Terms and Unlike Terms
What is a term? A term is a number, a variable, or a product of numbers and variables (for example, 5, x, 3x, 2xy, 4x2y). The number in front of the variable(s) is the coefficient. The part with variable(s) and their powers is the variable part.
Like terms: Terms that have exactly the same variable part (same variables with the same powers) are called like terms. Only the coefficients may differ. Examples: 3x and 7x; 4xy and −5xy; 2x2y and 9x2y; 5 and −2 (constants are like terms with no variables).
Unlike terms: Terms with different variable parts (different variables or different powers) are unlike terms. Examples: x and x2; 3x and 3xy; 2x2 and 2xy.
How to identify like terms: Compare the variable part. If every variable and its power match exactly, the terms are like. Coefficients do not matter for this decision.
Combining like terms (rule): Add or subtract only the coefficients of like terms and keep the common variable part unchanged. In algebraic form:
- axn + bxn = (a + b)xn
- axy + bxy = (a + b)xy
- Constants: p + q = (p + q)
Steps to combine like terms: (1) Identify and group like terms, (2) add/subtract their coefficients, (3) write the result with the common variable part, (4) remove any term with coefficient 0.
Note: Unlike terms cannot be combined algebraically (you keep them separated). Combining like terms simplifies expressions and is a key step before solving equations.
- 3x + 5x → like terms (both are x). Combine coefficients: 3x + 5x = 8x.
- 4xy − 7xy → like terms (both are xy). 4xy − 7xy = −3xy.
- 2x^2 + 5x^2 = 7x^2 (both are x^2).
- x + x^2 → unlike terms (variables' powers differ). Cannot combine; keep as x + x^2.
- 3a + 4b → unlike terms (different variables). Keep as 3a + 4b.
- 5 + (−2) → constants are like terms. 5 − 2 = 3.
- ax^n + bx^n = (a + b)x^n (combine coefficients of like terms)
- axy + bxy = (a + b)xy
- Constants: p + q = p + q (constants are like terms: no variables)
- If variable parts differ → terms are unlike and cannot be combined algebraically
- If (a + b) = 0 then ax + bx = 0 (term cancels out)
Monomials, Binomials and Polynomials
What are terms, coefficients and degree?
A term is a number, a variable or a product of numbers and variables (for example 5, x, 3x, 4x2y). The coefficient is the numerical factor of a term (in 4x2, coefficient = 4). The degree of a term (for single-variable expressions) is the exponent of the variable (degree of 4x2 is 2). For multivariable monomials the degree is the sum of exponents (degree of 3x2y is 3).
Definitions
Monomial: An algebraic expression with one term only (examples: 7, 5x, 3x2).
Binomial: An algebraic expression with exactly two terms (examples: x + 5, 2x2 + 3x).
Polynomial: A sum of one or more terms with non-negative integer exponents (examples: x2 + 3x + 2, 4x3 − x + 6). A polynomial with three terms is often called a trinomial.
Key points
- Terms are separated by + or − signs. Combine like terms (terms with the same variable part).
- The degree of a polynomial is the highest degree among its terms (e.g., degree of 3x4 + 2x2 + 7 is 4).
- The zero polynomial (0) has all coefficients zero; its degree is not defined at this level.
Operations
- Addition/Subtraction: combine like terms, e.g., (3x + 5) + (2x − 1) = 5x + 4.
- Multiplication (monomial × monomial): multiply coefficients and add exponents, e.g., (3x2)(4x3) = 12x5.
- Distributive law (monomial × polynomial): multiply each term, e.g., 2x(x + 3) = 2x2 + 6x.
Why this matters
Polynomials form the basis of algebra. Recognising monomials, binomials and polynomials and knowing how to combine and multiply them helps solve equations, model real situations (like cost, area, speed × time) and prepares you for graphs of algebraic expressions.
- Monomial: 5x (cost if one apple costs 5 rupees and you buy x apples, total cost = 5x).
- Monomial: x^2 (area of a square with side x units is x^2 square units).
- Binomial: x + 5 (a number x increased by 5).
- Binomial: 2l + 2b (perimeter of a rectangle with length l and breadth b; written as 2(l + b)).
- Trinomial (polynomial with three terms): x^2 + 5x + 6 (a typical quadratic expression).
- Polynomial: 3x^3 − x + 4 (a cubic polynomial; degree is 3).
- \[General form of a polynomial: a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\]\[where a_i are coefficients and n is a non-negative integer.\]
- Degree of a polynomial: highest power n for which a_n ≠ 0.
- Degree of a monomial (single variable): exponent of the variable. For multivariable monomial: sum of exponents.
- Addition (like terms): a x^k + b x^k = (a + b) x^k.
- \[Multiplication (monomials): (a x^m)(b x^n) = (a b) x^{m+n}.\]
- Distributive law: a(b + c) = ab + ac. So a(x^2 + 3x + 2) = ax^2 + 3ax + 2a.
Evaluation of Algebraic Expressions (Substitution)
What is an algebraic expression? An algebraic expression is a combination of numbers, variables (letters), and arithmetic operations. Example: 5x + 3, 2a + 3b - 4.
What does "evaluation by substitution" mean? To evaluate an algebraic expression by substitution means to replace each variable with a given number and then perform the arithmetic operations to find a numerical value.
- Step 1: Identify the variable(s) and the given value(s).
- Step 2: Substitute the given number(s) in place of the variable(s). Use parentheses if the value is negative or a fraction.
- Step 3: Follow the order of operations (BODMAS/BIDMAS): Brackets, Orders (powers), Division and Multiplication (left to right), Addition and Subtraction (left to right).
- Step 4: Simplify step by step until you get the final numerical result.
Important tips:
- Always use parentheses when substituting negative numbers, e.g. replace x by (−2) not −2 without brackets.
- If the expression contains products like 3(x + 2), substitute first then use distributive law or compute inside the bracket first.
- When there are two or more variables, substitute each with its given value before simplifying.
Why this matters (real life): Many formulae in daily life are algebraic expressions. Substitution lets you compute actual values: cost calculations, speed × time, temperature conversions, areas, and more.
- 1) Evaluate 5x + 3 for x = 2. Substitute x → 2: 5(2) + 3 = 10 + 3 = 13.
- 2) Evaluate 2a + 3b when a = 2 and b = 4. Substitute: 2(2) + 3(4) = 4 + 12 = 16.
- 3) Evaluate 4x - 7 for x = -3. Use parentheses: 4(−3) - 7 = −12 - 7 = −19.
- 4) Evaluate (3x + 2)/5 for x = 5. Substitute: (3(5) + 2)/5 = (15 + 2)/5 = 17/5 = 3.4.
- 5) Real-life: Cost = 50n + 20 (₹). For n = 3 items, Cost = 50(3) + 20 = 150 + 20 = ₹170.
- 6) Physics example: Distance = speed × time. If speed = 60 km/h and time = 2 h, Distance = 60 × 2 = 120 km.
- Substitution rule: To evaluate expression f(x), replace x by the given number a to get f(a).
- Order of operations (BODMAS): Brackets, Orders (powers), Division/Multiplication (left to right), Addition/Subtraction (left to right).
- Distributive property (useful when substituting in products): a(b + c) = ab + ac.
- Multiple variables: For expression in a and b, e.g. g(a,b) = 2a + 3b, evaluate g(2,4) = 2(2) + 3(4).
- Use parentheses for negative/fractional substitutions: replace x by (−2) or (1/2) before multiplying or adding.
Simplification of Algebraic Expressions
What is an algebraic expression? An algebraic expression is a mathematical phrase that can contain numbers (constants), letters (variables) and operations (+, -, ×, ÷). Examples: 3x + 5, 2a - b, 7.
Parts of an expression
- Term: each addend or part separated by + or - (for example, in 4x + 3y - 5 the terms are 4x, 3y and -5).
- Coefficient: the number multiplied by the variable (in 4x, 4 is the coefficient).
- Variable: a letter that stands for a number (x, y, a etc.).
- Constant: a number without a variable (for example, 5 or -7).
Like terms and unlike terms
- Like terms have the same variable parts (same variables raised to the same powers). Example: 3x and -5x are like terms; 2xy and 4xy are like terms.
- Unlike terms have different variable parts. Example: 3x and 4y are unlike.
What does simplification mean? Simplification means making an expression shorter or easier by combining like terms and removing grouping symbols (brackets) using basic algebra rules.
Common steps to simplify
- Remove brackets using the distributive property: a(b + c) = ab + ac. Also handle negative signs: -(a + b) = -a - b.
- Combine like terms by adding or subtracting their coefficients: ax + bx = (a + b)x.
- Simplify numerical operations among constants.
Useful tips
- Only like terms can be combined.
- Arrange terms so like terms are next to each other (e.g., group x-terms, y-terms, constants).
- Keep signs carefully when removing brackets, especially with a negative sign in front.
Simple examples in words: If you have 3 apples and someone gives 2 more apples, total apples = 3a + 2a = 5a. Here a represents one apple.
- Simplify 3x + 5x: Both are like terms. Add coefficients: (3 + 5)x = 8x.
- Simplify 4a - 2a + 7: Combine like terms 4a and -2a: (4 - 2)a + 7 = 2a + 7.
- Remove brackets: Simplify 2(x + 3): Use distributive law: 2x + 6.
- Handle negative sign: Simplify -(3x - 4) + 2x: = -3x + 4 + 2x = (-3 + 2)x + 4 = -x + 4.
- Combine different variables: Simplify 3x + 4y - 2x + y: Combine x-terms and y-terms: (3 - 2)x + (4 + 1)y = x + 5y.
- Real-life: If one pencil costs p rupees and you buy 3 pencils and then 2 more, cost = 3p + 2p = 5p rupees.
- a(b + c) = ab + ac (Distributive property)
- ax + bx = (a + b)x (Combine like terms)
- ax - bx = (a - b)x
- a + b = b + a and (a + b) + c = a + (b + c) (Commutative and associative laws of addition)
- Constants add normally: 5 + 3 = 8
- If coefficient becomes 0: 2x - 2x = 0
Addition and Subtraction of Algebraic Expressions
What is an algebraic expression? An algebraic expression is a combination of numbers, variables (like x, y) and operation signs (+, −, ×, ÷). Examples: 3x, 5 + y, 2x + 3y − 7.
Parts of an expression: A term is a single part of an expression separated by + or − signs (e.g. 3x, 7). A coefficient is the numerical factor of a term (in 4x the coefficient is 4). A constant is a term without a variable (e.g. 5). Like terms have the same variable part (e.g. 2x and 5x). Unlike terms have different variable parts (e.g. 3x and 4y).
Goal of addition and subtraction: When we add or subtract algebraic expressions we combine like terms only. Unlike terms cannot be combined directly.
- Step 1: Remove parentheses if any (watch signs).
- Step 2: Arrange terms so like terms are together (you may reorder terms).
- Step 3: Add or subtract the coefficients of like terms and keep the common variable part.
- Step 4: Write the simplified expression; include remaining unlike terms unchanged.
Examples of rules: ax + bx = (a + b)x, ax − bx = (a − b)x. Also, adding 0 leaves an expression unchanged: E + 0 = E. Subtracting an expression is same as adding its negative: E − F = E + (−F).
Worked simple example: Simplify 3x + 5 + 2x − 3.
- Group like terms: (3x + 2x) + (5 − 3).
- Add coefficients: 5x + 2.
Result: 5x + 2.
- Example 1 (simple addition): Simplify 4x + 3x. Solution: Both are like terms → (4 + 3)x = 7x.
- Example 2 (addition and constant): Simplify 2x + 5 + 3x − 2. Solution: Group like terms: (2x + 3x) + (5 − 2) = 5x + 3.
- Example 3 (different variables): Simplify 3x + 4y − 2x + y. Solution: Combine x-terms and y-terms: (3x − 2x) + (4y + y) = x + 5y.
- Real-life Example 1 (money): Riya has 3x rupees (x rupee packets) and gets 5x more. Total = 3x + 5x = 8x rupees. If x = 10, total = 80 rupees.
- Real-life Example 2 (fruits): A basket has 2a apples and 4b bananas. If you add 3a apples and remove 1b banana, new count = (2a + 3a) + (4b − 1b) = 5a + 3b.
- Real-life Example 3 (lengths): A rod of length 4x cm joined with another of length 6x cm gives a total length of 10x cm. If x = 2 cm, total = 20 cm.
- ax + bx = (a + b)x
- ax − bx = (a − b)x
- a + b = b + a (commutative property for addition of coefficients)
- (a + b) + c = a + (b + c) (associative property for addition)
- E − F = E + (−1)·F (subtraction as addition of negative)
- E + 0 = E (additive identity)
Formation of Expressions from Word Problems
What is an algebraic expression? An algebraic expression is a mathematical phrase that uses numbers, variables (letters that represent unknown numbers) and operation signs (+, −, ×, ÷). Example: 3x + 5.
Steps to form an expression from a word problem
- Read carefully: Understand what is given and what is unknown.
- Identify the unknown: Choose a letter (variable) to represent the unknown quantity, e.g., x or n.
- Find known quantities: Note constants and quantities given in the problem.
- Translate words to symbols: Use keyword mappings (see list below) to convert phrases into operations.
- Build and simplify: Write the expression using the variable, combine like terms if possible, and include units where needed.
- Check: Read the expression back in words to ensure it matches the problem.
Common keyword → operation translations
- sum/total/added to → +
- difference/less/after subtracting → −
- product/times/multiplied by → ×
- quotient/divided by/every → ÷
- more than → + (but watch word order: "5 more than x" → x + 5)
- less than → − ("5 less than x" → x − 5)
Tips: Always pay attention to word order ("5 more than x" is x + 5, not 5 + x in meaning it is the same numerically but order matters with phrases), include units when needed (rupees, apples), and use parentheses when a whole group is operated on (e.g., 2 times the sum of x and 3 → 2(x + 3)).
- Example 1 — Simple addition (real life): Rama had x rupees and her mother gave her 15 more rupees. Expression: x + 15. (Step: unknown → x, extra rupees → +15.)
- Example 2 — Repeated items (shopping): One notebook costs 30 rupees. If n notebooks are bought, total cost expression: 30n. (Step: cost per item × number of items.)
- Example 3 — Multiplication with boxes (real life): Each box has 12 chocolates. For m boxes, total chocolates = 12m.
- Example 4 — Increase then multiply (word order & parentheses): Three times a number increased by 2 → 3(x + 2). (Means: first increase the number x by 2, then multiply by 3.)
- Example 5 — Subtraction (real life): A bus had y passengers; 7 got down. Remaining passengers: y − 7.
- Example 6 — Perimeter of rectangle (combining terms): If length = l and width = 4, perimeter expression: 2(l + 4) or 2l + 8. (Shows expansion/simplification.)
- Definition: Algebraic expression = terms joined by + or −. Each term = coefficient × variable^power or a constant. Example term: 4x, 7, 3ab (Class 6 uses single-variable terms mainly).
- Variable: a letter representing an unknown number, e.g., x, n, m.
- Coefficient: the number multiplied by the variable. In 5x, 5 is the coefficient.
- Like terms: terms with the same variable part; they can be added/subtracted. Example: 3x + 2x = 5x.
- Distributive idea (useful for forming/simplifying): a(b + c) = ab + ac. Example: 2(x + 4) = 2x + 8.
- Order of operations (BODMAS): Brackets, Orders (powers), Division and Multiplication (left to right), Addition and Subtraction (left to right).
Simple Equations and Number Sentences (Introductory)
What is a number sentence? A number sentence is a mathematical statement made using numbers, operations (like +, −, ×, ÷) and symbols that can be true or false. Examples: 7 + 5 = 12 (true), 9 − 4 = 2 (false).
What is a simple equation? A simple equation (introductory) is a number sentence that contains an unknown (usually written as a letter such as x) and an equal sign. It tells us that two expressions are equal. Example: x + 5 = 12.
How to form equations from words: Read the sentence, choose a variable for the unknown, write expressions for known parts, and use '=' between left and right parts. Example: "I have 5 apples and need 12 in total; how many more?" → x + 5 = 12.
How to solve (balance method):
- Step 1: Identify the unknown (x).
- Step 2: Use inverse operations to isolate x. Whatever you do to one side, do to the other side (keep the equation balanced).
- Step 3: Simplify to find x.
- Step 4: Check by substituting x back into the original equation.
Common patterns (one- and two-step equations): For equations like x + b = c, subtract b from both sides. For x − b = c, add b to both sides. For ax = c, divide both sides by a (a ≠ 0). For ax + b = c, first subtract b, then divide by a.
Checking solutions: Substitute the found value into the original number sentence. If both sides are equal, the solution is correct.
Useful visual tools: balance-scale diagrams, bar models and number lines help students understand why inverse operations work and what the solution represents.
- 1) x + 5 = 12 → subtract 5 from both sides: x = 12 − 5 → x = 7. Check: 7 + 5 = 12 (true).
- 2) x − 7 = 8 → add 7 to both sides: x = 8 + 7 → x = 15. Check: 15 − 7 = 8.
- 3) 3x = 15 → divide both sides by 3: x = 15 ÷ 3 → x = 5. Check: 3 × 5 = 15.
- 4) 2x + 3 = 11 → subtract 3: 2x = 8 → divide by 2: x = 4. Check: 2×4 + 3 = 8 + 3 = 11.
- 5) Word problem: "Riya has some stamps. She buys 4 more and now has 13." Let x be original stamps: x + 4 = 13 → x = 9.
- General one-variable linear form: ax + b = c
- Solve ax + b = c: x = (c − b) / a (provided a ≠ 0)
- Special cases: x + b = c → x = c − b; x − b = c → x = c + b; ax = c → x = c / a
- Always check: Substitute x back: left side = right side?
Patterns, Generalisation and Rules
What is a pattern? A pattern is a regular and repeating arrangement or sequence in numbers, shapes, or objects. Patterns can be numeric (2, 4, 6, 8…), pictorial (red, blue, red, blue…), geometric (tiles, bricks) or positional (chairs in rows).
What is generalisation? Generalisation means finding a concise statement or rule that describes the pattern for any position. Instead of writing many terms, we give a rule that works for the nth term.
Types of rules
- Verbal rule: described in words, e.g., "add 3 each time".
- Tabular rule (input-output): use a table that maps position n to the term a_n.
- Algebraic rule: use a formula with a variable (usually n), e.g., a_n = 2n or a_n = n^2.
How to find a rule (step-by-step)
- Observe the pattern and write the first several terms.
- Look for how terms change: constant increase/decrease, multiplication, or repeating blocks.
- Make a table with position n and term a_n (or input and output).
- Try a verbal description and then convert to an algebraic form using n as the position.
- Test the rule on known terms and predict further terms.
Common simple patterns
- Arithmetic increase by constant d: terms change by adding the same number each time (e.g., 5, 8, 11, 14…).
- Multiplicative pattern: each term is multiplied by a constant (e.g., 2, 4, 8, 16…).
- Repeating (periodic) pattern: a fixed block repeats (e.g., red, blue, blue, red, blue, blue…).
Using variables for generalisation — use n to denote the position. For example, the even numbers 2, 4, 6, 8… can be written as a_n = 2n. Odd numbers 1, 3, 5, 7… are a_n = 2n - 1. Square numbers 1, 4, 9, 16… are a_n = n^2.
Checking your generalisation — always plug in several values of n to see if the formula gives the terms you observe. If it fails, revise the rule.
Why this matters — patterns and generalisation build the foundation for algebra: writing rules with variables, making predictions, and solving problems using formulas.
- Numeric pattern: 2, 4, 6, 8, ... → Verbal rule: add 2 each time → Algebraic rule: a_n = 2n. (Check: n=1 → 2, n=2 → 4.)
- Odd numbers: 1, 3, 5, 7, ... → Algebraic rule: a_n = 2n - 1.
- Squares: 1, 4, 9, 16, ... → Algebraic rule: a_n = n^2.
- Repeating colour pattern: red, blue, blue, red, blue, blue, ... → Block of 3 repeats; position tells colour using n mod 3 (positions 1,4,7 are red).
- Object arrangement: 3 chairs per table. For n tables total chairs = 3n (input-output: n → 3n).
- Input-output table example: n: 1,2,3,4 → y: 4,7,10,13. Difference constant = +3 → rule y = 3n + 1.
- nth term of an arithmetic-type pattern: a_n = a_1 + (n - 1)d (where d is the common difference)
- Even numbers: a_n = 2n
- Odd numbers: a_n = 2n - 1
- Square numbers: a_n = n^2
- Linear rule (input-output): y = m n + c (used for patterns with constant rate m and starting value c)
Basic Properties of Operations Applied to Algebraic Terms
Algebraic terms are expressions like 3x, −2y, 5, x^2. Each term has a coefficient (number part), variable part (x, y) and may have a power (x^2). When we operate on algebraic terms, a few basic arithmetic properties help us simplify and manipulate them. The most important properties for Class 6 are:
- Like and unlike terms: Like terms have exactly the same variable part (same variables raised to the same powers), e.g. 3x and 5x. Unlike terms, e.g. 4x and 3y, cannot be combined by addition or subtraction.
- Commutative property: Order does not matter for addition or multiplication: a + b = b + a and a·b = b·a. For terms: 2x + 3 = 3 + 2x, and 4x·2y = 2y·4x.
- Associative property: Grouping does not matter for addition or multiplication: (a + b) + c = a + (b + c) and (ab)c = a(bc). For terms: (2x + 3x) + x = 2x + (3x + x).
- Distributive property: Multiplication distributes over addition: a(b + c) = ab + ac. This is used to expand expressions and to factor common factors: 3(x + 2) = 3x + 6, and 5x + 10 = 5(x + 2).
- Identity elements: Adding 0 leaves a term unchanged: a + 0 = a. Multiplying by 1 leaves a term unchanged: a·1 = a.
- Inverse (additive): a + (−a) = 0, so + and − allow cancellation of identical terms.
Using these properties we simplify expressions by combining like terms and using distribution to expand or factor. Example methods: convert subtraction to adding negatives, group like terms using commutativity/associativity, and apply distributive law to expand brackets.
- Combining like terms: 3x + 5x = (3 + 5)x = 8x.
- Do not combine unlike terms: 4x + 3y (remains 4x + 3y).
- Distributive property: 2(x + 4) = 2x + 8.
- Factoring a common factor: 6x + 9 = 3(2x + 3).
- Using identity and inverse: x + 0 = x, and x + (−x) = 0.
- Associative and commutative use: (2x + 3x) + x = 2x + (3x + x) = 6x.
- Commutative (addition): a + b = b + a
- Commutative (multiplication): a·b = b·a
- Associative (addition): (a + b) + c = a + (b + c)
- Associative (multiplication): (a·b)·c = a·(b·c)
- Distributive: a(b + c) = ab + ac
- Additive identity: a + 0 = a
Key Concepts
- Variable
- A symbol (usually a letter) that represents an unknown or changing number.
- Constant
- A fixed number that does not change its value.
- Algebraic expression
- A combination of numbers, variables and operations (addition, subtraction, multiplication, division, powers).
- Term
- A part of an expression separated by plus or minus signs.
- Coefficient
- The numerical factor multiplied by a variable in a term.
- Monomial
- An algebraic expression with only one term.
- Binomial
- An algebraic expression with exactly two terms.
- Trinomial
- An algebraic expression with exactly three terms.
- Polynomial
- An algebraic expression made up of one or more terms with nonnegative integer exponents.
- Degree of a term
- The sum of the exponents of the variables in that term.
- Degree of a polynomial
- The highest degree among all its terms.
- Like terms
- Terms that have the same variable factors raised to the same powers.
- Unlike terms
- Terms that are not like terms; they have different variable parts or powers.
- Exponent (Power)
- A small number written above and to the right of a base that shows how many times the base is multiplied by itself.
- Evaluate
- To find the value of an expression by substituting numbers for variables and performing operations.
- Simplify
- To rearrange and combine terms in an expression to make it as simple as possible.
- Substitution
- Replacing a variable with a given number or expression to evaluate or simplify.
- Factor
- A number or expression that divides another exactly; to express an expression as a product of factors.
- Equation
- A mathematical statement that two expressions are equal, shown with an equals sign.
- Identity
- An equation that is true for all values of the variable(s) involved.
End-of-Chapter Trial Paper & Test Questions
Topic-wise questions to test your understanding of every concept in this chapter.
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In the expression 7x + 3, the coefficient of x is: / व्यंजक 7x + 3 में x का गुणांक है: (a) 3 / 3 (b) 7 / 7 (c) 10 / 10 (d) x / x
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(b) 7 / 7 — The coefficient is the numerical factor multiplying the variable. In 7x, the number 7 is multiplied by x, so 7 is the coefficient of x. / गुणांक वह संख्यात्मक गुणक है जो चर को गुणा करता है। 7x में, 7 को x से गुणा किया गया है, अतः 7, x का गुणांक है।
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Simplify: 5x + 3y − 2x + y. / सरल करें: 5x + 3y − 2x + y। (a) 3x + 4y / 3x + 4y (b) 7x + 2y / 7x + 2y (c) 3x + 2y / 3x + 2y (d) 5x + 4y / 5x + 4y
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(a) 3x + 4y / 3x + 4y — Combine like terms: x-terms: 5x − 2x = 3x; y-terms: 3y + y = 4y. Result: 3x + 4y. / सजातीय पदों को मिलाएँ: x-पद: 5x − 2x = 3x; y-पद: 3y + y = 4y। परिणाम: 3x + 4y।
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If x = 3, what is the value of 4x − 5? / यदि x = 3, तो 4x − 5 का मान क्या है? (a) 7 / 7 (b) 17 / 17 (c) 12 / 12 (d) -2 / -2
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(a) 7 / 7 — Substitute x = 3: 4(3) − 5 = 12 − 5 = 7. / x = 3 रखें: 4(3) − 5 = 12 − 5 = 7।
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The algebraic expression for 'five more than three times a number n' is ______. / 'संख्या n का तीन गुना और पाँच अधिक' का बीजीय व्यंजक ______ है।
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3n + 5 / 3n + 5 — 'Three times a number n' = 3n; 'five more than' means add 5. So the expression is 3n + 5. / 'संख्या n का तीन गुना' = 3n; 'पाँच अधिक' का अर्थ है 5 जोड़ना। अतः व्यंजक 3n + 5 है।
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If 2x + 3 = 11, then x = ______. / यदि 2x + 3 = 11, तो x = ______।
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4 / 4 — Subtract 3 from both sides: 2x = 8. Divide both sides by 2: x = 4. Check: 2(4) + 3 = 11. / दोनों पक्षों से 3 घटाएँ: 2x = 8। दोनों पक्षों को 2 से भाग दें: x = 4। जाँच: 2(4) + 3 = 11।
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True or False: 3x and 3y are like terms. / सत्य या असत्य: 3x और 3y सजातीय पद हैं।
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False / असत्य — Like terms must have exactly the same variable part. 3x has variable x and 3y has variable y; these are different, so they are unlike terms and cannot be combined. / सजातीय पदों का चर भाग बिल्कुल समान होना चाहिए। 3x का चर x है और 3y का y; ये भिन्न हैं, अतः ये असजातीय पद हैं और मिलाए नहीं जा सकते।
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A shopkeeper has x pens priced at ₹12 each. He also has a fixed discount coupon of ₹5. Write an algebraic expression for the cost after using the coupon and find the cost for x = 8. / एक दुकानदार के पास x पेन हैं, प्रत्येक ₹12 में। उसके पास ₹5 की छूट का कूपन भी है। कूपन के बाद लागत का बीजीय व्यंजक लिखें और x = 8 के लिए लागत निकालें।
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Expression: 12x − 5; for x = 8, cost = 12(8) − 5 = 96 − 5 = ₹91. / व्यंजक: 12x − 5; x = 8 के लिए, लागत = 12(8) − 5 = 96 − 5 = ₹91। — 'x pens at ₹12 each' is 12x; subtracting the ₹5 discount gives 12x − 5. / 'x पेन ₹12 प्रत्येक' = 12x; ₹5 छूट घटाने पर 12x − 5।
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The number pattern 3, 7, 11, 15, ... has the rule: nth term = ______. / संख्या पैटर्न 3, 7, 11, 15, ... का नियम: n वाँ पद = ______।
Show answer
4n − 1 / 4n − 1 — The pattern increases by 4 each time (common difference d = 4). Using nth term = a₁ + (n−1)d = 3 + (n−1)×4 = 3 + 4n − 4 = 4n − 1. Check: n=1 → 3, n=2 → 7. / पैटर्न हर बार 4 बढ़ता है (सार्व अंतर d = 4)। n वाँ पद = a₁ + (n−1)d = 3 + (n−1)×4 = 4n − 1। जाँच: n=1 → 3, n=2 → 7।
Related Laws & Principles
Explore allFoundational laws & principles behind this chapter. Each one opens a full page — what it says, why it matters, five practice questions and the mistakes to avoid.