Linear Equations in One Variable — Class 8

Concise notes with definitions, methods, solved examples, practice questions and step-by-step solutions — optimized for mobile and exams.

1. What is a Linear Equation in One Variable?

A linear equation in one variable is an equation that can be written in the form ax + b = 0, where a and b are constants and a ≠ 0. The variable appears with power 1 only. Example: 3x − 5 = 10.

2. Standard Form & Terminology

  • General form: ax + b = 0
  • Solution / Root: value of x that satisfies the equation.
  • Types of equations: Identity (true for all x), Conditional (true for some x), Inconsistent (no solution).

3. Methods to Solve

  1. Simplify both sides: expand brackets, combine like terms.
  2. Collect variable terms on one side: use addition/subtraction.
  3. Isolate x: divide by coefficient (remember sign).
  4. Check: substitute back to verify.

Special cases

  • If equation reduces to 0 = 0 → infinite solutions (identity).
  • If equation reduces to 0 = c where c ≠ 0 → no solution.

4. Solved Examples (Step-by-step)

Example 1: Solve 3x − 5 = 10.
Add 5: 3x = 15. Divide by 3: x = 5. Check: 3×5−5=10.
Example 2: Solve 2(x − 3) = x + 4.
Expand: 2x − 6 = x + 4. Subtract x: x − 6 = 4. Add 6: x = 10.
Example 3 (No solution): Solve 4(x + 1) = 4x + 5.
Expand: 4x + 4 = 4x + 5. Subtract 4x: 4 = 5 → contradiction → no solution.
Example 4 (Infinite solutions): Solve 2(3x − 1) = 6x − 2.
Expand: 6x − 2 = 6x − 2 → identity → all real numbers are solutions.

5. Practice Questions (with answers)

  1. Solve: 5x + 7 = 2x + 19.
    Answer
    Simplify: 3x = 12 → x = 4
  2. Solve: 3(x + 2) = 2(2x − 1).
    Answer
    Expand: 3x + 6 = 4x − 2 → x = 8
  3. Find x: 7 − 2x = 3x + 2.
    Answer
    7 − 2x = 3x + 2 → 5 = 5x → x = 1
  4. Which of these is an identity? 4(2x − 1) = 8x − 4.
    Answer
    Left = Right after expansion → identity → infinite solutions

Solve on paper first — then confirm using the answers hidden above.

6. Exam Tips & Tricks

  • Always expand brackets before combining like terms.
  • Keep operations balanced: do the same to both sides.
  • Beware of sign mistakes when moving terms across the equals sign.
  • Check your solution by substitution — saves careless errors.

7. References & Further Practice

  • NCERT Class 8 Mathematics — Chapter: Linear Equations in One Variable
  • Practice from sample question papers and worksheets