Class 8 Mathematics Chapter 2 Linear Equations in One Variable — Formulas & Key Points

Core Definitions and Fundamental Concepts

• Linear equation: An equation of the form ax + b = c where a, b, c are constants and a ≠ 0
• Variable: The unknown quantity represented by a letter (usually x, y, or z)
• Coefficient: The numerical factor multiplying the variable (in 5x, the coefficient is 5)
• Constant term: A number without any variable attached
• Solution/Root: The value of the variable that satisfies the equation
• LHS and RHS: Left-Hand Side and Right-Hand Side of the equation separated by the equals sign

Standard Forms and Equation Types

• Simple form: ax = b → solution x = b/a
• Standard form: ax + b = 0 → solution x = -b/a
• Two-term form: ax + b = c → solution x = (c - b)/a
• Variables on both sides: ax + b = cx + d → collect terms then solve
• Fractional equations: Clear denominators by multiplying through by LCM
• Multi-step equations: Simplify brackets and combine like terms first

Core Formula Table — Equation Solving Rules

• Addition Property: If a = b, then a + c = b + c for any number c
• Subtraction Property: If a = b, then a - c = b - c for any number c
• Multiplication Property: If a = b, then a × c = b × c for any non-zero number c
• Division Property: If a = b, then a ÷ c = b ÷ c for any non-zero number c
• Symmetric Property: If a = b, then b = a
• Transitive Property: If a = b and b = c, then a = c

Transposition Rules and Sign Changes

• Transposing an added term: x + 5 = 12 becomes x = 12 - 5
• Transposing a subtracted term: x - 8 = 10 becomes x = 10 + 8
• Transposing a coefficient: 4x = 20 becomes x = 20 ÷ 4 or x = 20/4
• Transposing a divisor: x/3 = 7 becomes x = 7 × 3
• Multiple transpositions: Handle one term at a time, constants first, then coefficients
• Sign rule: When transposing, + changes to −, − changes to +, × changes to ÷, ÷ changes to ×

Step-by-Step Solving Methods

• Step 1: Simplify both sides — remove brackets, combine like terms
• Step 2: Move all variable terms to one side using transposition
• Step 3: Move all constant terms to the other side
• Step 4: Combine like terms on each side
• Step 5: Isolate the variable by dividing or multiplying
• Step 6: Verify solution by substituting back into the original equation

Equations with Variables on Both Sides — Formula Approach

• General form: ax + b = cx + d
• Collect variables: ax - cx = d - b
• Factor out x: (a - c)x = d - b
• Solve: x = (d - b)/(a - c) where a ≠ c
• Strategy tip: Move the smaller coefficient to avoid negative coefficients when possible
• Always check: the value of x must satisfy the original equation on both sides

Word Problems — Forming Equations from Statements

• Identify the unknown: 'Let the number be x' or 'Let the present age be x years'
• Translate relationships: 'five more than twice a number' → 2x + 5
• Common phrases: 'sum of' → +, 'difference between' → −, 'times/product' → ×, 'divided by' → ÷
• Formulate equation: Use the main condition stated in the problem
• Solve systematically: Apply equation-solving rules
• Interpret the answer: Check if it makes sense in the original problem context

Memory Tricks and Mnemonics for Quick Recall

• CSCS: Change Side, Change Sign (for transposition)
• BODMAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction (simplification order)
• READ: Represent, Express, Apply, Determine (word problem strategy)
• FLIP: For Like terms, Isolate, Perform inverse operation (solving strategy)
• Golden Rule of Equations: 'Balance the scale — whatever you do to one side, do to the other'
• Verification Rhyme: 'Substitute back, check your track, LHS and RHS must match exact'

Common Mistakes, Sign Errors and Notation Pitfalls

• Sign error: Transposing x - 3 = 10 as x = 10 - 3 is wrong; correct is x = 10 + 3
• Partial operation: From 2x + 5 = 15, writing 2x = 15 × 2 is wrong; correct is 2x = 15 - 5
• Notation: Never write 3x = 12 = x = 4 in one line; separate each equation on a new line
• Zero division: If solution makes denominator zero, it is NOT valid
• Like terms: Only combine terms with identical variable parts (3x and 5x can combine, but 3x and 5x² cannot)
• Verification: Skipping the check step means you will not catch calculation errors

Three Solved Mini-Examples Applying Chapter 2 Formulas

One-Glance Last-Minute Revision Box

• Definition: Linear equation in one variable has form ax + b = 0, where a ≠ 0, degree = 1
• Equality rule: Same operation on both sides preserves equality
• Transposition: Change side → change sign (+↔−, ×↔÷)
• Solving steps: Simplify → collect variables one side → constants other side → isolate variable → verify
• Variables both sides: Move all x-terms to LHS, constants to RHS, then solve
• Word problems: Let unknown = x, translate words to symbols, form equation, solve, check context
• Common errors: Sign mistakes in transposition, forgetting to verify, partial operations
• Standard form solution: ax + b = c → x = (c - b)/a
• Verification is mandatory: Substitute solution back into original equation, LHS must equal RHS
• Notation: Write each step on a new line, never chain equations with multiple = signs

Important Constants, Values and Standard Notations

• Variable notation: x, y, z (lowercase letters)
• Coefficient: Any real number multiplying the variable
• Constant: A term without a variable
• Equals sign: = (denotes equality between LHS and RHS)
• Not equals: ≠ (used in stating conditions like a ≠ 0)
• Solution set: For linear equations, typically contains one unique value
• Identity: An equation true for all values (e.g., 2x = 2x)
• Contradiction: An equation with no solution (e.g., x + 1 = x + 2)