Many maths problems do not work with one condition alone. They require two conditions to arrive at a correct answer. This idea forms the base of a pair of linear equations. When two equations represent the same situation, their common solution gives the result. When they fail to meet, no valid answer exists. When they overlap, more than one solution appears.

Students apply this concept in questions that compare money, speed, distance, and quantity. The chapter helps you understand how equations behave, how graphs show solutions, and how algebraic methods give exact values. Board questions focus on method selection, formula application, and consistency conditions.

In this blog, you will learn CBSE Class 10 Maths Chapter 3, Pair of Linear Equations in Two Variables. Along with important formulas, you will read all the required definitions and conditions.

Key Definitions and Concepts: Pair of Linear Equations in Two Variables

1. Linear Equation in Two Variables

A linear equation in two variables can be written in the form
ax + by + c = 0,
where a, b, and c are real numbers and a and b cannot both be zero.
The variables x and y represent unknown values.

2. Solution of a Pair of Linear Equations

A solution refers to the values of x and y that satisfy both equations at the same time. Only the common values that make each equation true form the solution of the system.

3. Consistent System of Linear Equations

A system of linear equations is called consistent when it has at least one solution. The equations may have one solution or infinitely many solutions.

4. Inconsistent System of Linear Equations

A system is inconsistent when no pair of values satisfies both equations. In this case, the equations never meet.

5. Graphical Method for Solving a Pair of Linear Equations

Consider the pair:
a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

Each equation represents a straight line on the Cartesian plane. The nature of their intersection decides the solution.

Conditions and Their Graphical Meaning

Graph-Based Interpretation

• Intersecting lines meet at one point. That point gives the solution.
• Coincident lines overlap completely. Every point on the line satisfies both equations.
• Parallel lines never meet. No solution exists.

6. Algebraic Methods to Solve a Pair of Linear Equations

Students solve equations algebraically using these standard methods:

1. Substitution Method – Express one variable in terms of the other and substitute.
2. Elimination Method – Remove one variable by adding or subtracting equations.
3. Cross Multiplication Method – Apply direct formulas when equations are in standard form.

Applications of Pair of Linear Equations in Two Variables

Students use a pair of linear equations to solve problems where two conditions control two unknown values.

Common Applications

• Money problems: Find the cost of items when the total price and quantity are known.
• Speed and distance problems: Calculate speeds when total time and distance remain fixed.
• Number problems: Find two numbers when their sum and difference are given.
• Work and quantity distribution: Divide work, mixtures, or items under fixed conditions.
• Real-life comparisons: Compare plans, prices, or quantities using two equations.

Tips to Memorize Chapter 3 Formulas and Concepts

Tip 1: Remember the Standard Form

Always rewrite equations as:
ax + by + c = 0
This form helps in checking consistency and applying cross multiplication.

Tip 2: Use Ratio Logic for Graph Questions

Link ratios with line behavior:

• Different ratios → lines meet → one solution
• Same ratios → check c values
• Same a and b but different c → no solution

Tip 3: Method Selection Shortcut

• Easy isolation → substitution
• Matching coefficients → elimination
• Direct formulas needed → cross multiplication

Tip 4: Visual Memory Trick

Think of three-line cases only:

• Meet
• Overlap
• Never meet

Every graph question fits one case.

Example Based on a Pair of Linear Equations

Question: The sum of two numbers is 10. Their difference is 2. Find the numbers.

Step 1: Form equations
x + y = 10
x − y = 2

Step 2: Solve using elimination
Add both equations: 2x = 12, x = 6

Step 3: Find y
y = 10 − 6
y = 4

Answer:
The two numbers are 6 and 4.