Class 10 Math Pair of Linear Equation Notes 2025 & Study Material PDF

Chapter 3 in Class 10 Mathematics, Pair of Linear Equations in Two Variables, is an important algebraic concept that involves solving linear equations with two unknowns in several different ways. It involves three such methods: graphical, substitution, and elimination methods—all of them relevant to where two lines may cross and determine their solution. In other words, it must be grasped not just in the academic context but for practical purposes of solving problems in business, physics, and engineering. Class 10 Maths Chapter 3 Notes covers the solution of linear equations, interpretation of graphical solutions, and the use of such techniques for solving word problems.

Practising questions from class 10 ch 3 math can help in scoring at least 3 marks in the final examination. On this platform, students may get pertinent study materials like class 10 maths chapter 3 notes PDF, experiential exercises, DoE worksheets, and assistance materials from Educart.

CBSE Class X Pair of Linear Equations in Two Variables Notes

Class 10 math Chapter 3 notes cover all the main concepts like the substitution method and solving linear equations. Pair of Linear Equations in Two Variables include every number other than the complex numbers, and the downloadable notes PDFs provided below are detailed and in easy-to-understand language.

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What is a Linear Equation?

In algebra, a linear equation is an equation where the highest power of the variable is one. In the case of two variables, such as x and y, the general form of a linear equation is:

ax+by=c

Where:

• a and b are the coefficients of x and y, respectively.
• c is a constant.

An example of a linear equation in two variables is:

2x+3y=5

Pair of Linear Equations in Two Variables

A pair of linear equations in two variables represents a system of two equations with two unknowns. The general form is:

a₁x+b₁y=c₁

a_2​x+b₂​y=c₂

Where:

• a₁,b₁​, and c₁​ are constants in the first equation.
• a₂,b₂​, and c₂​ are constants in the second equation.

The goal is to determine the values of x and y that satisfy both equations simultaneously.

Methods to Solve a Pair of Linear Equations

There are three primary methods to solve a system of linear equations:

1. Graphical Method
2. Substitution Method
3. Elimination Method

Graphical Method

This method involves plotting both equations on a Cartesian plane and finding the point where they intersect, which gives the solution (x,y).

Steps to solve using the Graphical Method:

• Rewrite each equation in the form y=mx+c
• Plot both equations on a graph.
• The point where the two lines intersect is the solution (x,y).

Example:

Solve the system of equations graphically:

x+y=4

2x+y=5

After plotting these equations, the intersection point is (1, 3), meaning x=1 and y=3 is the solution.

Substitution Method

In this method, one equation is solved for one variable in terms of the other, and then substituted into the second equation.

Steps to solve using the Substitution Method:

• Rearrange one equation to express either xxx or yyy in terms of the other.
• Substitute this value into the second equation.
• Solve for the remaining variable.
• Substitute back to find the first variable.

Example:

Solve the system:

3x+4y=10

x−y=1

• From the second equation: x=y+1
• Substituting into the first equation: 3(y+1)+4y=10

3y+3+4y=10

7y=7 

y=1

Substituting y=1into x=y+1

x=2

Thus, the solution is x=2,y=1.

Elimination Method

In this method, equations are added or subtracted to eliminate one variable, making it easier to solve for the other.

Steps to solve using the Elimination Method:

• Multiply one or both equations to make the coefficients of one variable equal.
• Add or subtract the equations to eliminate one variable.
• Solve for the remaining variable.
• Substitute the found value into one of the original equations to find the other variable.

Example:

Solve the system:

2x+3y=12

4x−3y=6

Adding both equations:

(2x+3y)+(4x−3y)=12+6

6x=18 

x=3

Substituting x=3 into the first equation:

2(3)+3y=12 

6+3y=12

3y = 6

y=2

Thus, the solution is x=3,y=2

Types of Solutions for a Pair of Linear Equations

A system of two linear equations can have three types of solutions:

Unique Solution:

• The lines intersect at one point.
• The equations are consistent and independent.

No Solution:

• The lines are parallel and never intersect.
• The equations are inconsistent.

Infinite Solutions:

• The lines coincide, meaning they are the same equation.
• The system is dependent.

How to Identify the Type of Solution:

• Parallel lines → No solution.
• Coinciding lines → Infinite solutions.
• Intersecting lines → Unique solution.

Applications of a Pair of Linear Equations in Real Life

Linear equations are not just theoretical; they have practical applications in various fields:

Business & Economics:

• Helps in calculating profit, loss, expenses, and break-even points.

Travel Problems:

• Used to determine time, speed, and distance relationships, such as calculating when two vehicles will meet.

Mixture Problems:

• Helps in determining the right proportions of different substances in chemistry and manufacturing.

Geometry & Physics:

• In geometry, they determine where two lines or planes intersect.
• In physics, they express relationships between different quantities.

Important Questions from Chapter 3: Pair of Linear Equations in Two Variables

Practising these questions will help you understand key concepts and score well in exams.

Basic Concept Questions

• Define a linear equation in two variables with an example.
• What are the possible types of solutions for a pair of linear equations?
• How can you determine whether a system of equations has one solution, no solution, or infinitely many solutions?

Graphical Method Questions

• Solve the following pair of equations graphically: x+y=5x + y = 5 x+y=5 2x−y=42x - y = 4 2x−y=4 Find the point of intersection.
• What does it mean if two equations result in parallel lines when plotted on a graph?

Substitution & Elimination Method Questions

• Solve using the substitution method

3x+4y=10

x−y=2

• Solve using the elimination method

2x+3y=12

4x−3y=6

• Find the value of ‘k’ if the system has infinitely many solutions:

2x+3y=7

(k+2)x+6y=14

Word Problems

• The sum of two numbers is 25, and their difference is 5. Find the numbers using a system of linear equations.
• The cost of 2 pencils and 3 pens is ₹18, while the cost of 4 pencils and 6 pens is ₹36. Form a pair of linear equations and solve to find the price of a pencil and a pen.
• A boat travels 30 km downstream in 3 hours and the same distance upstream in 5 hours. Find the speed of the boat in still water and the speed of the current.

Higher-Order Thinking Questions

• Prove that the equations x + 2y = 4 and 2x + 4y = 8 represent the same line graphically.
• The ages of a father and his son add up to 50 years. Five years ago, the father’s age was three times the son’s age. Find their present ages.
• The perimeter of a rectangle is 34 cm. The length is 5 cm more than the breadth. Find its length and breadth using a pair of linear equations.

These questions cover all important concepts from Chapter 3, helping you prepare effectively for exams. Let me know if you need solutions or additional practice questions!

Important Formulas from Chapter 3: Pair of Linear Equations in Two Variables

Understanding and memorising these formulas will help you solve problems efficiently.

General Form of a Linear Equation in Two Variables

A linear equation in two variables is written as:

ax+by=c

Where:

• a and b are coefficients of x and y, respectively.
• c is a constant.

General Form of a Pair of Linear Equations

A system of two linear equations in two variables is given by:

a₁​x+b₁​y=c₁

a_2​x+b₂​y=c₂​

Where a₁,b₁,c₁,a₂,b₂,c₂​ are constants.

Methods to Solve a Pair of Linear Equations

Graphical Method

• Convert each equation into the form y=mx+c (slope-intercept form).
• Plot both equations on a graph.
• The point of intersection gives the solution (x,y).

Substitution Method

• Express one variable in terms of the other using one equation.
• Substitute this value into the second equation.
• Solve for the remaining variable.

Elimination Method

• Multiply one or both equations to make the coefficient of one variable the same.
• Add or subtract the equations to eliminate one variable.
• Solve for the remaining variable.

Cross Multiplication Method

For the equations:

a​x+b​y=c​

a​x+b

​y=c

x= (b_1​c₂​−b₂​c₁​​)/(a_1​b₂​−a₂​b₁​)

y= (c₁​a_2​−c₂​a₁​)/(a_1​b₂​−a₂​b₁​)

This method is useful when elimination and substitution are complex or time-consuming.

Types of Solutions & Their Conditions

The solution is found using the formula:

‍

Important Formulas for Word Problems

Time, Speed, and Distance:

Distance=Speed×Time

Cost and Revenue Problems:

Total Cost=Cost per Unit×Number of Units

Age Problems:

If the present age is x, then:

• Age n years ago = x−n
• Age n years later = x+n

Mixture Problems:

The total quantity of one component in a mixture = Percentage×Total Quantity

Common Mistakes Students Make in Linear Equations

Many students struggle with linear equations in two variables due to common mistakes that can be avoided with proper understanding and practice. Here are some frequent errors and how to prevent them:

Misinterpreting the Question

• Not fully understanding what the problem is asking.
• Ignoring key terms such as "find the solution," "graph the equation," or "identify the type of solution."

Tip: Read the question carefully and underline important information.

Incorrectly Plotting Graphs

• Using an inconsistent scale.
• Plotting points inaccurately.
• Forgetting to label the axes and intersection points.

Tip: Always check your calculations before plotting, and use a ruler for accuracy.

Arithmetic Errors

• Making mistakes in addition, subtraction, multiplication, or division.
• Misplacing negative signs.

Tip: Double-check your calculations, especially when simplifying equations.

Choosing the Wrong Method

• Using substitution when elimination would be quicker (or vice versa).
• Not simplifying equations before solving.

Tip: Identify the most efficient method based on the given equations.

Forgetting to Check the Solution

• Substituting incorrect values.
• Solving for only one variable and forgetting the other.

Tip: Always substitute your solution back into the original equations to verify its correctness.

How to Make Effective Notes for Linear Equations

Taking well-organised notes is essential for revision and better retention. Follow these strategies to create clear and effective study notes:

Use a Structured Format

• Write the chapter name and topic at the top.
• Use headings and subheadings to organise different concepts.

Include Definitions & Key Formulas

• Write the general form of equations.
• List important formulas for solving equations using different methods.

Step-by-step Worked Examples

• Include detailed examples for each method.
• Highlight key steps and common pitfalls.

Summarise Key Concepts

• Create a quick revision box with essential points.
• Use a comparison table to summarise graphical, substitution, and elimination methods.

Highlight Common Mistakes & Useful Tips

• Note down common errors and ways to avoid them.
• Add shortcuts or simple tricks to solve equations faster.

Use Visual Aids & Diagrams

• Draw graphs to help visualise solutions.
• Use colour-coding for different types of equations.

Keep It Concise & Neat

• Avoid unnecessary information.
• Leave space for additional notes or queries.

By following these tips, you can create clear, structured, and easy-to-revise notes for your exams. 

Important Points to Remember

• The graphical method provides a visual way to solve equations by plotting them on a graph.
• The substitution and elimination methods are algebraic techniques that are often faster for finding exact solutions.
• If a system of equations has no solution, the lines will be parallel.
• If a system has infinitely many solutions, the lines will coincide (overlap completely).
• Always verify your solution by substituting the values of x and y back into the original equations.

In this chapter, you have learned how to solve pairs of linear equations in two variables using different methods. Each technique—graphical, substitution, and elimination—plays a crucial role in tackling algebraic challenges and real-life applications. Mastering these methods through consistent practice will help in confidently solving linear equations in exams and practical situations