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📘 Linear Equations in Two Variables Class 10 Notes (Chapter 3)

🔷 Introduction

Linear equations in two variables are an important part of algebra. In simple terms, these equations contain two variables, usually x and y.

For example:$$ax + by + c = 0$$ax+by+c=0

Here, a, b, and c are real numbers, and both a and b are not zero.

🔹 Standard Form of Linear Equation

The standard form of a linear equation in two variables is:$$ax + by + c = 0$$ax+by+c=0

Therefore, every linear equation can be written in this form.

🔹 What Does It Represent?

A linear equation in two variables represents a straight line on a graph.

Hence, solving such equations means finding the point where lines intersect.

🔷 Conditions for Solutions

When we have two linear equations:$$a_1x + b_1y + c_1 = 0$$a1​x+b1​y+c1​=0 $$a_2x + b_2y + c_2 = 0$$a2​x+b2​y+c2​=0

We compare ratios:

🔹 Case 1: Unique Solution (One Solution)

$$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$$a2​a1​​=b2​b1​​

In this case, the lines intersect at one point.
Therefore, the system has a unique solution.

👉 Example:
2x + 2y + 5 = 0
4x + 3y + 6 = 0

🔹 Case 2: Infinite Solutions

$$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$a2​a1​​=b2​b1​​=c2​c1​​

Here, both equations represent the same line.
As a result, there are infinitely many solutions.

👉 Example:
4x + 5y + 10 = 0
8x + 10y + 20 = 0

🔹 Case 3: No Solution

$$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$$a2​a1​​=b2​b1​​=c2​c1​​

In this situation, the lines are parallel.
Therefore, they never meet and have no solution.

👉 Example:
3x + 5y + 6 = 0
6x + 10y + 5 = 0

🔷 Graphical Representation

• Intersecting lines → One solution
• Coincident lines → Infinite solutions
• Parallel lines → No solution

Thus, graphs help us understand solutions visually.