Class 10 NCERT Maths Chapter 3: Pair of Linear Equations in Two Variables

1. Introduction

In Class 9, you learned about single linear equations. In Class 10 Chapter 3, we study pairs of linear equations. We learn to solve them using graphical and algebraic methods. This is an important step toward advanced algebra topics in higher classes.

2. Topics in the 2025–26 Syllabus

• Pair of linear equations and their representation
• Graphical method of solution and the concept of consistency/inconsistency
• Algebraic methods: Substitution, Elimination
• Simple word problems

Note: Cross-multiplication method has been removed from the syllabus.

3. Graphical Solution & Consistency

The solution type depends on the ratios of coefficients:

\[ \frac{a_1}{a_2},\ \frac{b_1}{b_2},\ \frac{c_1}{c_2} \quad\Rightarrow\quad \begin{cases} \text{Unique solution: } \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \\ \text{Infinite solutions: } \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \\ \text{No solution: } \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \end{cases} \]

4. Algebraic Methods

Substitution Method:

Example: \[ \begin{cases} x + y = 10, \\ x - y = 4 \end{cases} \] From the second equation: \(x = y + 4\). Substituting into the first: \[ (y+4) + y = 10 \quad\Rightarrow\quad 2y = 6 \quad\Rightarrow\quad y = 3 \] Then \(x = 3 + 4 = 7\).

Elimination Method:

Example: \[ \begin{cases} 2x + 3y = 11, \\ 2x - 4y = -24 \end{cases} \] Subtract the second from the first: \[ (2x + 3y) - (2x - 4y) = 11 - (-24) \\ 7y = 35 \quad\Rightarrow\quad y = 5 \] Substitute into \(2x + 3(5) = 11\): \[ 2x + 15 = 11 \quad\Rightarrow\quad 2x = -4 \quad\Rightarrow\quad x = -2 \]

5. Summary

• Graphical method gives a visual understanding.
• Substitution and elimination are the only algebraic methods in the latest syllabus.
• Cross-multiplication method is no longer included.
• Practice with real-life word problems to master the topic.