📗 Chapter 3: Pair of Linear Equations in Two Variables – Class 10 Maths Notes

 

🧠 Introduction

In this chapter, you will learn how to solve two-variable linear equations, represent them graphically, and apply algebraic methods like substitution and elimination.

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📌 What is a Linear Equation?

A linear equation in two variables is of the form:

$$ax + by + c = 0$$

Where:

• $a, b, c$ are real numbers

• $x$ and $y$ are variables

• Degree = 1

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📉 Graphical Method of Solution

Each linear equation represents a straight line on the graph.

• If lines intersect: One solution (consistent & independent)

• If lines are parallel: No solution (inconsistent)

• If lines coincide: Infinite solutions (consistent & dependent)

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✍️ Algebraic Methods to Solve Pair of Linear Equations

1. Substitution Method:

   • Solve one equation for one variable

   • Substitute in the second

2. Elimination Method:

   • Multiply to make coefficients equal

   • Add/Subtract equations to eliminate a variable

3. Cross Multiplication Method:

For equations:

$$a_1x + b_1y + c_1 = 0$$ $$a_2x + b_2y + c_2 = 0$$

The solution is:

$$\frac{x}{(b_1c_2 - b_2c_1)} = \frac{y}{(c_1a_2 - c_2a_1)} = \frac{1}{(a_1b_2 - a_2b_1)}$$

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📊 Types of Solutions (Consistency Table)

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✅ Key Points to Remember

• Linear equations form straight lines on graphs

• Use algebraic methods for accurate solutions

• Understand types of solutions based on coefficients

• Graphical method helps in visualizing solutions

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📘 Previous Chapter: Chapter 2 – Polynomials »
📚 Next Chapter: Chapter 4 – Quadratic Equations »

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