NCERT Solutions Class 10 Maths Chapter 3 teaches you four powerful methods to solve pairs of linear equations: graphical representation, substitution, elimination, and cross-multiplication. You’ll learn when to use each method, how to interpret solutions geometrically (consistent, inconsistent, or dependent systems), and apply these techniques to real-world problems like age, distance, and cost calculations that frequently appear in CBSE exams.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables – Complete Guide
NCERT Class 10 Chapter 3 – Pair of Linear Equations in Two Variables is a fundamental chapter that builds your algebraic problem-solving skills for CBSE board exams. You’ll explore how two linear equations can be represented graphically as straight lines and learn to find their point of intersection, which represents the solution. This chapter carries 5 marks weightage in your board exam and forms the foundation for advanced mathematics in higher classes.
📊 CBSE Class 10 Maths Chapter 3 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 5 Marks |
| Unit Name | Algebra |
| Difficulty Level | Medium |
| Importance | Medium |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 1-2 questions |
You will learn three powerful algebraic methods to solve these equations: the substitution method (replacing one variable with another), the elimination method (adding or subtracting equations to cancel variables), and the cross-multiplication method (a quick formula-based approach). Each method has its advantages, and you’ll develop the judgment to choose the most efficient technique for different problems. The chapter also covers special cases where equations have no solution (parallel lines) or infinitely many solutions (coincident lines).
This topic has excellent real-world applications in business mathematics, economics, and everyday problem-solving. You’ll work with word problems involving ages, numbers, speeds, mixtures, and costs that require forming and solving simultaneous equations. Expect 2-3 questions in your CBSE board exam, typically including one 3-mark word problem and one or two 2-mark problems testing different solving methods.
Quick Facts – Class 10 Chapter 3
| 📖 Chapter Number | Chapter 3 |
| 📚 Chapter Name | Pair of Linear Equations in Two Variables |
| ✏️ Total Exercises | 3 Exercises |
| ❓ Total Questions | 12 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering this chapter will strengthen your analytical thinking and prepare you for coordinate geometry and linear programming in Class 11. Practice diverse problem types, focus on forming equations from word problems correctly, and always verify your solutions by substituting back into the original equations. With consistent practice of NCERT solutions and sample papers, you’ll confidently tackle any question from this chapter.
NCERT Solutions Class 10 Maths Chapter 3 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| Exercise 3.1 | Complete step-by-step solutions for 7 questions | 📥 Download PDF |
| Exercise 3.2 | Complete step-by-step solutions for 3 questions | 📥 Download PDF |
| Exercise 3.3 | Complete step-by-step solutions for 2 questions | 📥 Download PDF |
Pair of Linear Equations in Two Variables – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| General Form of a Linear Equation \(ax + by + c = 0\) | Represents a straight line in a coordinate plane. Note: a, b, and c are real numbers, and a and b cannot both be zero. | When you need to identify coefficients (a, b, c) in a linear equation or to convert an equation to standard form. |
| Condition for Intersecting Lines \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) | Determines if two lines intersect at a unique point. Note: Lines intersect at exactly one point. The system is consistent. | To check if a system of linear equations has a unique solution. |
| Condition for Parallel Lines \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) | Determines if two lines are parallel and never intersect. Note: Lines never intersect. The system is inconsistent. | To check if a system of linear equations has no solution. |
| Condition for Coincident Lines \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) | Determines if two lines are coincident (the same line). Note: Lines overlap completely. The system is consistent and dependent. | To check if a system of linear equations has infinitely many solutions. |
| Substitution Method Solve one equation for one variable and substitute into the other equation. | A method to solve a system of linear equations by expressing one variable in terms of the other. Note: Be careful with signs when substituting. Check your solution by plugging the values back into the original equations. | When one of the equations can be easily solved for one variable (e.g., x = … or y = …). |
| Elimination Method Multiply equations by constants to make the coefficients of one variable equal, then add or subtract the equations to eliminate that variable. | A method to solve a system of linear equations by eliminating one of the variables. Note: Multiply the *entire* equation, not just one term. Watch out for sign errors when adding or subtracting. | When the coefficients of one variable are easy to manipulate to become equal or opposite. |
| Cross-Multiplication Method (Solution for x) \(x = \frac{b_1c_2 – b_2c_1}{a_1b_2 – a_2b_1}\) | Formula to directly find the value of x in a pair of linear equations. Note: Memorize the formula carefully! Ensure equations are in the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0. Avoid using this method if a1b2 – a2b1 = 0, as the denominator will be zero. | When you need to quickly find the value of x, especially in multiple-choice questions or when time is limited. |
| Cross-Multiplication Method (Solution for y) \(y = \frac{c_1a_2 – c_2a_1}{a_1b_2 – a_2b_1}\) | Formula to directly find the value of y in a pair of linear equations. Note: Memorize the formula carefully! Ensure equations are in the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0. Avoid using this method if a1b2 – a2b1 = 0, as the denominator will be zero. | When you need to quickly find the value of y, especially in multiple-choice questions or when time is limited. |
| Equations Reducible to Linear Form Substitute \(\frac{1}{x} = p\) and \(\frac{1}{y} = q\) to transform into a linear equation. | Transforms equations that are not initially linear into linear equations by using substitutions. Note: After solving for p and q, remember to find x and y by taking the reciprocals: \(x = \frac{1}{p}\) and \(y = \frac{1}{q}\). | When you have equations with variables in the denominator (e.g., \(\frac{2}{x} + \frac{3}{y} = 5\)). |
Frequently Asked Questions – NCERT Class 10 Maths Chapter 3
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