NCERT Class 10 Linear Equations in Two Variables

Imagine you buy notebooks and pens from a shop, and the total cost depends on how many of each you purchase. To find the exact price of each item, you need to work with two unknown values. Situations like these can be solved using a pair of linear equations in two variables. In this chapter, students learn how to represent such problems mathematically and solve them using graphical and algebraic methods. In this article, we will explain the concepts, methods, solved examples, and practice questions to help Class 10 students understand the topic easily.

What is a Pair of Linear Equations in Two Variables?

Definition

A linear equation in two variables is an equation that contains two variables and each variable has the highest power of 1. Such equations represent straight lines when plotted on a graph.

The general form of a linear equation in two variables is:

ax + by + c = 0

Here:

• a and b are called the coefficients of the variables. These are real numbers that multiply the variables.
• c is the constant term.
• x and y are the variables whose values we need to determine. 

For example:

2x + 3y = 6

This equation has two variables (x and y) and represents a straight line when graphed.

A linear equation in two variables can have infinitely many solutions because many pairs of values of x and y can satisfy the equation.

What is a Pair of Linear Equations?

A pair of linear equations in two variables consists of two linear equations involving the same variables.

Example:

Equation 1
x + 2y = 6

Equation 2
2x – y = 3

In this case:

• There are two equations.
• Both equations contain the same variables (x and y).
• The goal is to find the values of x and y that satisfy both equations at the same time. 

The pair of equations is usually written in the form:

a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

Solving these equations helps us find a common solution, which means the values of x and y that satisfy both equations simultaneously.

Real-Life Situations Represented by Pair of Linear Equations

Many real-life problems involve two unknown quantities. To find these unknown values, we can represent the situation using a pair of linear equations.

Let us understand this with examples.

Example 1: Fair Ride Example

Suppose a girl goes to a fair and spends money on two activities: rides and hoopla games.

Let:

x = number of rides
y = number of hoopla games

We are given the following information:

• The number of hoopla games is half the number of rides.
• Each ride costs ₹3 and each hoopla game costs ₹4.
• The total amount spent is ₹20. 

From this information we can form two equations.

Equation 1

y = x / 2

Equation 2

3x + 4y = 20

Here, the real-life situation has been converted into two linear equations, which can be solved to find the values of x and y.

Example 2: Shopping Example

Suppose a student buys notebooks and pens from a shop.

Let:

x = cost of one notebook
y = cost of one pen

If:

5 notebooks and 2 pens cost ₹70
3 notebooks and 4 pens cost ₹60

Then we can form the equations:

5x + 2y = 70
3x + 4y = 60

By solving these equations, we can find the cost of each item.

Example 3: Age Problem

Age-related questions are also common examples.

Let:

x = age of a father
y = age of his son

Suppose:

Five years ago, the father's age was three times the son's age.

Equation:

x – 5 = 3(y – 5)

Another condition may give us a second equation. By solving both equations, we can determine the actual ages.

These examples show how real-life situations can be translated into mathematical equations and solved systematically.

Graphical Method of Solving a Pair of Linear Equations

One of the simplest ways to solve a pair of linear equations is the graphical method.

What is the Graphical Method?

In the graphical method:

• Each linear equation represents a straight line on a graph.
• When both equations are drawn on the same coordinate plane, the point where the lines intersect represents the solution. 

That intersection point gives the values of x and y that satisfy both equations.

Steps to Solve Using the Graphical Method

Step 1
Rewrite the equation in a convenient form if required.

Step 2
Find two or more solution points for each equation.

Step 3
Plot these points on a coordinate graph.

Step 4
Join the points to draw a straight line for each equation.

Step 5
Identify the point where the two lines intersect. That point represents the solution of the pair of equations.

If the lines intersect at a point such as (3,2), it means:

x = 3
y = 2

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Types of Solutions in Graphical Method

When two linear equations are represented on a graph, three different situations can occur depending on how the lines are positioned.

Unique Solution (Intersecting Lines)

A unique solution occurs when the two lines intersect at exactly one point.

Condition:

a₁ / a₂ ≠ b₁ / b₂

Example:

x + 3y = 6
2x – 3y = 12

When these equations are plotted on a graph, the two lines intersect at a single point.

That point represents the unique solution, meaning there is only one pair of values (x, y) that satisfies both equations.

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Infinitely Many Solutions (Coincident Lines)

Sometimes both equations represent the same line on the graph.

Condition:

a₁ / a₂ = b₁ / b₂ = c₁ / c₂

Example:

2x + 3y – 9 = 0
4x + 6y – 18 = 0

If we observe carefully, the second equation is just twice the first equation. Therefore, both equations represent the same straight line.

Since every point on the line satisfies both equations, the pair of equations has infinitely many solutions.

No Solution (Parallel Lines)

In some cases, the two lines are parallel and never intersect.

Condition:

a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂

Example:

x + 2y – 4 = 0
2x + 4y – 12 = 0

When these equations are plotted on a graph, the lines remain parallel and never meet.

Since there is no common point, the pair of equations has no solution.

These three situations help students understand the relationship between equations and their graphical representation, which is an important concept in the chapter Pair of Linear Equations in Two Variables.

Example of Graphical Method (Step-by-Step)

Let us solve the following pair of linear equations using the graphical method:

Step 1: Convert Each Equation into a Table of Values

Equation 1:

Equation 2:

Step 2: Plot the Points on a Graph

Plot the points of both equations on a graph paper and draw straight lines passing through them.

Step 3: Find the Point of Intersection

The point where both lines intersect represents the solution of the pair of linear equations.

Explanation:
If the two lines intersect at a single point, that point satisfies both equations simultaneously.

Algebraic Methods to Solve Pair of Linear Equations

While the graphical method helps us understand the solution visually, it is not always practical.

Sometimes:

• The coordinates may involve fractions
• The values may be large numbers
• The intersection may lie between decimal points, making it difficult to read accurately. 

Because of these limitations, mathematicians use algebraic methods to solve pairs of linear equations.

The two most common algebraic methods are:

1. Substitution Method
2. Elimination Method 

These methods give exact solutions without drawing graphs.

Substitution Method

The substitution method involves expressing one variable in terms of the other and substituting it into the second equation.

Steps of the Substitution Method

Step 1:
Find the value of one variable from one equation.

Step 2:
Substitute that value into the second equation.

Step 3:
Solve the resulting equation to find the value of one variable.

Step 4:
Substitute this value back into the first equation to find the second variable.

Example (Solved)

Solve:

Step 1: Express from the second equation

Step 2: Substitute in the first equation

Step 3: Solve

Step 4: Substitute the value of

Final Solution

Word Problem Example (Age Problem)

Suppose a father and daughter have the following relationship between their ages.

Let:

• = father’s age
• = daughter’s age 

Equation 1

Equation 2

These equations can be solved using the substitution method by expressing one variable in terms of the other and substituting it into the second equation.

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Elimination Method

The elimination method works by removing one variable from the equations.

This is done by:

• Multiplying equations
• Making coefficients of one variable equal
• Adding or subtracting the equations 

Once one variable is eliminated, the remaining variable can be easily solved.

Steps of Elimination Method

1. Multiply the equations if necessary.
2. Make the coefficients of one variable the same.
3. Add or subtract the equations to eliminate that variable.
4. Solve the remaining variable.
5. Substitute the value back to find the other variable. 

Example (Solved)

Solve:

Step 1: Make coefficients of equal

Multiply:

First equation by 3

Second equation by 4

Step 2: Subtract the equations

Step 3: Substitute the value of

Substitute in any original equation to find .

Example: Digit Problem

Let:

• = tens digit
• = units digit 

Given:

Add the equations:

Substitute in the first equation:

Number Formed

Important Conditions for Pair of Linear Equations

The general form of two linear equations is:

The number of solutions depends on the ratio of coefficients.

Case 1: One Solution (Intersecting Lines)

The two lines intersect at one point, giving one unique solution.

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Case 2: No Solution (Parallel Lines)

The lines are parallel and never intersect, so there is no solution.

Case 3: Infinite Solutions (Coincident Lines)

Both equations represent the same line, so there are infinitely many solutions.

Category-Wise Practice Questions

Practicing different types of questions helps students build a strong understanding of pair of linear equations in two variables. The following questions are divided into categories so that students can practice concepts, graphical methods, algebraic methods, and word problems effectively.

Concept-Based Questions

1. What is a pair of linear equations in two variables?
2. Write the general form of a pair of linear equations.
3. Define a consistent pair of linear equations.
4. Define an inconsistent pair of linear equations.
5. What is a dependent pair of linear equations?
6. What does the point of intersection of two lines represent in a graphical solution?
7. What condition represents parallel lines in a pair of linear equations?
8. What condition represents coincident lines?
9. What do you mean by unique solution?
10. How can we determine the number of solutions using the ratios of coefficients?  

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Graphical Method Practice

Solve the following pairs of linear equations using the graphical method.

x + y = 4
x – y = 2

2x + y = 6
x + 3y = 6

x + 2y = 6
x – y = 3

2x + y = 5
x + y = 4

x + y = 8
x – y = 0

2x + 3y = 12
x + y = 5

x – y = 1
x + y = 7

x + 2y = 10
2x – y = 1

x + y = 5
2x + y = 7

3x + y = 9
x + y = 5

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Substitution Method Practice

Solve the following equations using the substitution method.

x + y = 10
x – y = 4

3x – y = 5
x + y = 7

2x + y = 9
x – y = 1

x + 2y = 8
x – y = 2

3x + y = 11
x + y = 5

x + y = 6
2x – y = 3

2x + y = 7
x + y = 5

x + 3y = 13
x – y = 1

2x – y = 5
x + y = 9

x + y = 12
2x – y = 3

Elimination Method Practice

Solve the following equations using the elimination method.

2x + 3y = 11
2x – 4y = –24

3x + 4y = 10
2x – 2y = 2

4x + 3y = 19
2x + 3y = 11

3x + 2y = 13
2x + 2y = 10

5x + 2y = 14
3x + 2y = 10

4x + y = 13
2x + y = 7

3x + 5y = 21
3x – 2y = 6

2x + 3y = 13
4x + 3y = 19

5x + 3y = 16
2x + 3y = 7

4x + 2y = 12
3x + 2y = 10

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Word Problems

Solve the following real-life problems by forming pairs of linear equations.

1. Cost of Pencil and Pen Problem
   The cost of a pen and a pencil together is ₹30. The pen costs ₹10 more than the pencil. Find the cost of each.
2. Taxi Fare Problem
   A taxi charges ₹50 as base fare and ₹10 per kilometre. Another taxi charges ₹30 as base fare and ₹15 per kilometre. For how many kilometres will both fares be the same?
3. Age Problem
   The sum of the ages of a father and son is 50 years. The father is 20 years older than the son. Find their ages.
4. Number Digit Problem
   The sum of the digits of a two-digit number is 9. The number obtained by reversing the digits is 27 more than the original number. Find the number.  
5. Library Charges Problem
   A library charges ₹5 for the first day and ₹3 for each additional day. Another library charges ₹2 per day but requires a membership fee of ₹12. After how many days will both charges be equal?
6. The sum of two numbers is 15 and their difference is 3. Find the numbers.
7. Two numbers differ by 8 and their sum is 24. Find the numbers.
8. The sum of two numbers is 18 and one number is twice the other. Find the numbers.
9. The sum of the digits of a number is 10 and the difference is 2. Find the digits.
10. A shopkeeper sells two types of notebooks whose total cost is ₹120. One type costs ₹10 more than the other. Find the cost of each notebook. 

Common Mistakes Students Make

While solving pairs of linear equations, students often make the following mistakes:

• Incorrect substitution when replacing variables in equations
• Sign mistakes while adding or subtracting equations
• Plotting wrong points while drawing graphs
• Misunderstanding coefficient ratios in solution conditions
• Forgetting to substitute back to find the second variable 

Avoiding these mistakes can significantly improve accuracy in exams.

Tips to Master Pair of Linear Equations

Here are some simple tips that can help students solve these problems more effectively:

• Understand the concept first before attempting questions
• Practice different types of word problems regularly
• Use elimination method when coefficients are easy to match
• Always verify your solution in both equations
• Practice plotting graphs to understand solutions visually 

Consistent practice will help students build both speed and accuracy.

How PlanetSpark Helps Students Learn Maths Better

Learning mathematics becomes easier when students receive the right guidance and structured practice. Platforms like PlanetSpark help students strengthen their concepts through engaging learning methods.

PlanetSpark supports students by providing:

• Interactive math learning sessions that make concepts easy to understand
• Personal mentorship from experienced educators
• Concept clarity through real-life examples and simplified explanations
• Strong problem-solving practice with guided exercises
• Confidence-building for exams through regular assessments
• A fun and engaging learning environment that keeps students motivated 

This approach helps students develop strong analytical thinking and problem-solving skills.

Building Strong Concepts in Pair of Linear Equations

Pairs of linear equations in two variables are an important topic in Class 10 mathematics. Students learn different ways to solve them, including the graphical method, substitution method, and elimination method. While graphical solutions help visualize the intersection of lines, algebraic methods provide more accurate results. 

Understanding when to use substitution or elimination can make problem-solving easier. With regular practice and clear conceptual understanding, students can confidently solve these equations and perform well in exams.

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