Linear Equations in One Variable Notes
A linear equation in one variable has the variable to the first power only and an equals sign. This chapter covers solving equations with the variable on one side and on both sides, equations involving brackets and fractions, and forming equations from word problems.
- 3 min read
- 10 practice questions
- Aligned to CBSE 2025–26 syllabus
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- Updated Sept 2026
Use this map for a first read, then return to the checks and formulas during revision.
- 1.What Is a Linear Equation?
- 2.Variable on Both Sides; Brackets and Fractions
- 3.Forming Equations from Word Problems
- 1 concept checks
What Is a Linear Equation?
An equation says two expressions are equal. It is linear in one variable if the variable (say x) appears only to the power 1 and there is just one variable. Examples: 2x + 5 = 11, and (x − 3)/4 = 2.
Solving means finding the value of the variable that makes the two sides equal. That value is the solution or root.
The main tool is 'do the same thing to both sides': add, subtract, multiply, or divide both sides by the same non-zero number, so the equality is preserved while the equation gets simpler. Transposing a term to the other side simply changes its sign — it is a shortcut for adding or subtracting the same amount on both sides.
Variable on Both Sides; Brackets and Fractions
When the variable is on both sides, bring all variable terms to one side and all constants to the other by transposing, then simplify.
For equations with brackets, expand using the distributive law first. For equations with fractions, multiply every term on both sides by the LCM of the denominators to clear the fractions, then solve.
Always check the answer by substituting it back into the original equation.
Forming Equations from Word Problems
Read the problem, choose a letter for the unknown, and express the other quantities in terms of it. Translate the given relationship into an equation, solve it, and interpret the answer in the words of the problem.
Common set-ups: 'a number and its parts', consecutive integers, ages 'now and later', perimeter of a rectangle, and 'sum of digits' problems for two-digit numbers.
Example: The sum of three consecutive multiples of 5 is 555. Let them be x, x + 5, x + 10. Then 3x + 15 = 555, so x = 180 and the multiples are 180, 185, 190.
Linear Equations in One Variable: revision sheet
Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.
Recall checklist
- What Is a Linear Equation?
- Variable on Both Sides; Brackets and Fractions
- Forming Equations from Word Problems
For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.
Explain your method
Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.
Check worked solution 1
Check worked solution 2
Turn mistakes into revision
Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.
Practice and Revision
Test your understanding with quick chapter-level practice.
Chapter Q&A
What does 'transposing' a term mean?
Moving a term from one side of the equation to the other. When it crosses the equals sign, its sign changes (a '+' becomes '−' and vice versa). It is a shortcut for adding or subtracting the same quantity on both sides.
Why do we multiply by the LCM to solve equations with fractions?
Multiplying every term by the LCM of the denominators clears all fractions in one step, leaving a simpler equation with only whole-number coefficients.
How do I check whether my solution is correct?
Substitute the value back into the original equation. If the left-hand side equals the right-hand side, the solution is correct.