NCERT Solutions for Class 10 Maths Chapter 3: Pair of Linear Equations in Two Variables
For students looking for step-by-step NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables, Myclass24 offers the most comprehensive and student-friendly solutions available online. This chapter is one of the highest-weightage topics in the CBSE Class 10 Maths paper, covering multiple methods of solving linear equations and their real-life applications.
Chapter 3 is unique in the sense that it combines algebraic methods with graphical representation, making it both visual and analytical. At Myclass24, we have solved each problem using all relevant methods — graphical, substitution, elimination, and cross-multiplication — so that students understand which approach is fastest for a given problem type. The chapter also includes important real-life word problems that test a student's ability to frame equations from given conditions, which is a skill tested heavily in board examinations. With the right practice using Myclass24's solutions, students not only crack Chapter 3 questions but also improve their overall problem-solving approach in mathematics.
Download PDF – NCERT Solutions for Class 10 Maths Chapter 3 Linear Equations
Myclass24 provides a free, downloadable PDF of NCERT Solutions for Class 10 Maths Chapter 3 covering Exercise 3.1 through Exercise 3.7. The PDF is optimised for mobile devices and can be easily read on any screen size, making it ideal for students who prefer digital study materials over printed books.
Chapter 3 – Key Concepts, Solution Methods & Exercise Overview
Chapter 3 starts with the graphical method of solving a pair of linear equations. Two lines on a graph can either intersect at one point (unique solution), be parallel (no solution), or coincide (infinitely many solutions). These three scenarios correspond to consistent, inconsistent, and dependent systems, respectively. The algebraic methods taught in this chapter — substitution, elimination, and cross-multiplication — are powerful tools. The cross-multiplication method is particularly efficient when coefficients are complex. The chapter also introduces equations reducible to linear form, where substitution transforms non-linear equations into simple linear ones — a skill that appears in higher-level competitive exams as well.
| Exercise | Topics Covered | No. of Questions |
| Exercise 3.1 | Graphical Method | 3 |
| Exercise 3.2 | Substitution Method | 7 |
| Exercise 3.3 | Elimination Method | 3 |
| Exercise 3.4 | Cross-Multiplication Method | 2 |
| Exercise 3.5 | Equations Reducible to Linear Form | 4 |
| Exercise 3.6 | Word Problems – Applications | 2 |
| Exercise 3.7 | Additional Problems (Optional) | 8 |
| Type of System | Graphical Appearance | Number of Solutions | Algebraic Condition |
| Consistent (Unique) | Lines intersect | Exactly 1 | a₁/a₂ ≠ b₁/b₂ |
| Inconsistent | Lines are parallel | No solution | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ |
| Dependent (Consistent) | Lines coincide | Infinitely many | a₁/a₂ = b₁/b₂ = c₁/c₂ |
A key fact students must remember: the cross-multiplication method gives the solution as x/(b₁c₂ - b₂c₁) = y/(c₁a₂ - c₂a₁) = 1/(a₁b₂ - a₂b₁). This formula looks complicated but is very systematic once practised. Word problems in Exercise 3.6 and Exercise 3.7 are gold for board exams — they involve scenarios like age problems, number problems, boat-and-stream, work problems, and geometry-based linear equation problems. CBSE frequently picks 4–5 mark word problems directly from these exercises.
| Method | Best Used When | Speed | CBSE Preference |
| Substitution | One variable is easily expressed | Medium | Common |
| Elimination | Coefficients are equal or easily matched | Fast | Most Common |
| Cross-Multiplication | Direct formula, coefficients complex | Fastest | High |
| Graphical | Need to show nature of system visually | Slow | MCQ/VSA |
NCERT Solutions for Class 10 Maths Chapter 3: Linear Equations in Two Variables
A linear equation in two variables is an equation that can be written in the form ax + by + c = 0, where a, b, and c are constants and x and y are variables. Such equations represent straight lines when plotted on a graph. Students learn how to identify solutions and understand the relationship between the variables. Linear equations are important because they are used to model real-life situations involving two quantities. This chapter lays the groundwork for coordinate geometry and advanced algebraic concepts studied in higher classes.
A solution of a linear equation in two variables is an ordered pair of values that satisfies the equation. Since a linear equation represents a straight line, it has infinitely many solutions. Every point lying on the line is a valid solution. Students learn how to find solutions by substituting values and checking whether the equation holds true. Understanding solutions is important because it helps connect algebraic equations with graphical representations. Questions related to finding and verifying solutions are common in examinations.
To represent a linear equation graphically, first find at least two solutions of the equation. Plot these points on a coordinate plane and join them using a straight line. The resulting line represents all possible solutions of the equation. Graphical representation helps students visualize the relationship between variables and understand how changes in one variable affect the other. This method also makes mathematical concepts easier to interpret and apply. Graph-based questions are frequently included in school examinations and board assessments.
A linear equation in two variables represents a straight line on a graph. Every point on this line satisfies the equation, meaning each point corresponds to a valid solution pair. Since a straight line contains infinitely many points, the equation also has infinitely many solutions. This concept highlights the difference between equations with one variable and equations with two variables. Understanding this idea helps students interpret graphs correctly and develop a deeper understanding of algebraic relationships between variables.
Coordinate geometry provides the graphical framework needed to represent linear equations. It allows students to plot points, draw lines, and interpret mathematical relationships visually. Through coordinate geometry, abstract algebraic equations become easier to understand because students can see how solutions form a straight line. This approach improves conceptual clarity and strengthens analytical skills. Coordinate geometry also serves as a foundation for advanced topics involving graphs, distances, and geometric reasoning. Mastering these concepts is important for success in future mathematics chapters.
Linear equations in two variables are used to represent relationships between two changing quantities. They can model situations involving cost and quantity, distance and time, production and profit, or other practical scenarios. By analyzing these equations, people can make predictions and solve everyday problems. Students learn that mathematics is not limited to textbooks but can be applied to real-world decision-making. Understanding these applications increases interest in the subject and demonstrates the usefulness of algebra in daily life.
Students often make mistakes while plotting points, selecting incorrect coordinates, using unequal scales on axes, or drawing inaccurate lines through plotted points. Another common error is substituting values incorrectly while finding solutions. Such mistakes can lead to incorrect graphs and wrong conclusions. To avoid these errors, students should verify calculations, label axes clearly, and plot points carefully. Regular practice improves accuracy and confidence. Understanding these common challenges helps students perform better in graph-based questions and examinations.