Pair of Linear Equations in Two Variables is one of the most important chapters in Class 10 Maths because it combines algebra, graphs, concept-based reasoning, and application questions in one place. This chapter is highly scoring for students who understand the basic form of a linear equation, the meaning of a solution, the graphical representation of two lines, and the standard methods used to solve the equations.
Many students feel that this chapter is difficult only because it includes more than one method. In reality, the chapter becomes much easier when students first understand the concept and then learn when to use the graphical method, substitution method, and elimination method. Once that clarity is built, the chapter becomes straightforward and board-friendly.
At Deeksha Vedantu, we always encourage students to treat this chapter as a pattern-based topic. If the definition, solution conditions, and standard methods are clear, students can solve both direct and competency-based questions with much more confidence.
Chapter Overview at a Glance
This quick table helps students revise the whole chapter faster.
| Topic | Key idea |
| Linear equation in two variables | Equation of the form ax + by + c = 0 |
| Graph of a linear equation | Always a straight line |
| Pair of linear equations | Two such equations taken together |
| Solution of the pair | Common values of x and y satisfying both equations |
| Unique solution | Two lines intersect |
| Infinitely many solutions | Two lines coincide |
| No solution | Two lines are parallel |
| Main solving methods | Graphical, substitution, elimination |
What Is a Linear Equation in Two Variables
A linear equation in two variables is an equation of the form:
ax + by + c = 0
Here:
- x and y are variables
- a, b, and c are real numbers
- a and b cannot both be zero
Why a and b Cannot Both Be Zero
If both a and b become zero, then the x and y terms disappear, and the equation no longer remains a linear equation in two variables.
Examples of Linear Equations in Two Variables
- 2x + 3y = 7
- x – 4y + 5 = 0
- 3x – 2y = 9
All these equations are linear because they involve x and y only in the first degree.
Graph of a Linear Equation in Two Variables
Whenever a linear equation in two variables is represented on a graph, it gives a straight line. This is a very important idea because the whole chapter depends on understanding how two straight lines behave when taken together.
What Is a Pair of Linear Equations in Two Variables
When two linear equations in two variables are taken together, they are called a pair of linear equations in two variables.
The general form is:
- a₁x + b₁y + c₁ = 0
- a₂x + b₂y + c₂ = 0
Here, the coefficients are written with subscripts 1 and 2 to distinguish the first equation from the second equation.
What Is the Solution of a Pair of Linear Equations
The solution of a pair of linear equations means the common values of x and y that satisfy both equations together.
Simple Meaning of Solution
A solution is the point where both equations agree. If one value of x and one value of y satisfy both equations at the same time, then that ordered pair is called the solution.
Graphical Meaning of the Solution
Since each linear equation gives a straight line, a pair of linear equations gives two straight lines on a graph. There are three possibilities.
Case 1: Intersecting Lines
If the two straight lines intersect each other at one point, then there is exactly one solution. The common intersection point satisfies both equations, so the pair is called a consistent pair with a unique solution.
Case 2: Coincident Lines
If both lines overlap each other completely, then every point on one line lies on the other line as well. There are infinitely many common points, so the pair is called a consistent pair with infinitely many solutions.
Case 3: Parallel Lines
If the two straight lines are parallel, they never meet. There is no common point, so the pair is called an inconsistent pair with no solution.
Nature of Solutions Summary Table
| Graphical condition | Number of solutions | Nature of pair |
| Intersecting lines | One | Consistent |
| Coincident lines | Infinitely many | Consistent |
| Parallel lines | None | Inconsistent |
Quick Chart for Nature of Solutions
This is one of the most important parts of the chapter for board exams.
For the equations:
- a₁x + b₁y + c₁ = 0
- a₂x + b₂y + c₂ = 0
Nature of Solutions Condition Table
| Condition | Result |
| a₁/a₂ ≠ b₁/b₂ | Unique solution, intersecting lines, consistent pair |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Infinitely many solutions, coincident lines, consistent pair |
| a₁/a₂ = b₁/b₂ but not equal to c₁/c₂ | No solution, parallel lines, inconsistent pair |
Why This Solution Chart Is Very Important
This chart helps students answer many questions without solving the equations fully. In competency-based and board-style questions, students are often asked to identify the nature of the solution directly from coefficients.
Example: Find the Value of k for Infinite Solutions
For what value of k will the equations have infinitely many solutions?
- 2x + ky = 6
- 4x + 6y = 12
Given
- First equation: 2x + ky = 6
- Second equation: 4x + 6y = 12
- Condition needed: infinitely many solutions
Step 1: Use the Condition for Infinite Solutions
For infinitely many solutions:
a₁/a₂ = b₁/b₂ = c₁/c₂
Step 2: Compare the Ratios
- a₁/a₂ = 2/4 = 1/2
- b₁/b₂ = k/6
- c₁/c₂ = 6/12 = 1/2
Step 3: Equate the Ratios
Since k/6 must be equal to 1/2:
2k = 6
k = 3
Answer
The value of k is 3.
Methods of Solving Pair of Linear Equations in Two Variables
There are four standard ways students usually learn this chapter, but the main board-relevant focus here is on graphical method, substitution method, and elimination method.
Method Summary Table
| Method | Best use |
| Graphical method | Useful for visual understanding and graph-based questions |
| Substitution method | Useful when one variable can be isolated easily |
| Elimination method | Very useful for board exams when coefficients can be matched quickly |
Graphical Method
In the graphical method, students find a few solution points for each equation, plot them on graph paper, and draw the two lines. The point where the lines intersect gives the solution.
How to Use Graphical Method
| Step | What to do |
| Step 1 | Take one equation |
| Step 2 | Choose values of one variable |
| Step 3 | Find corresponding values of the other variable |
| Step 4 | Plot at least two or three points |
| Step 5 | Repeat the process for the second equation |
| Step 6 | Draw both lines |
| Step 7 | Identify the intersection point |
Example of Graphical Method Setup
Solve graphically:
- 2x – y = 9
- 5x + 2y = 27
Given
- Equation 1: 2x – y = 9
- Equation 2: 5x + 2y = 27
Step 1: Rearrange the First Equation
2x – y = 9
2x = 9 + y
x = (9 + y)/2
Step 2: Choose Suitable Values for the First Equation
A convenient point table is:
| y | x |
| -1 | 4 |
| 1 | 5 |
| 3 | 6 |
So the points are:
- (4, -1)
- (5, 1)
- (6, 3)
Step 3: Rearrange the Second Equation
5x + 2y = 27
5x = 27 – 2y
x = (27 – 2y)/5
Step 4: Choose Suitable Values for the Second Equation
A convenient point table is:
| y | x |
| 1 | 5 |
| 6 | 3 |
| -4 | 7 |
So the points are:
- (5, 1)
- (3, 6)
- (7, -4)
Solution
After plotting these points and drawing the two lines, the lines intersect at:
(5, 1)
Answer
The graphical solution is:
x = 5, y = 1
Substitution Method
In this method, students first express one variable in terms of the other from one equation. Then they substitute that expression into the second equation.
Steps of Substitution Method
| Step | What to do |
| Step 1 | Take one equation |
| Step 2 | Express x in terms of y or y in terms of x |
| Step 3 | Substitute that expression into the second equation |
| Step 4 | Solve for one variable |
| Step 5 | Put the value back to get the second variable |
Example of Substitution Method
Solve:
- 2x – y = 9
- 5x + 2y = 27
Given
- First equation: 2x – y = 9
- Second equation: 5x + 2y = 27
Step 1: Express y from the First Equation
2x – y = 9
y = 2x – 9
Step 2: Substitute in the Second Equation
5x + 2(2x – 9) = 27
5x + 4x – 18 = 27
9x = 45
x = 5
Step 3: Find y
Put x = 5 in y = 2x – 9
y = 10 – 9
y = 1
Answer
The solution is:
x = 5, y = 1
Elimination Method
In the elimination method, students make the coefficient of one variable equal in both equations and then add or subtract the equations so that one variable gets eliminated. This is one of the most important methods for board exams.
Steps of Elimination Method
| Step | What to do |
| Step 1 | Write both equations clearly |
| Step 2 | Multiply one or both equations if needed to make coefficients equal |
| Step 3 | Add or subtract the equations |
| Step 4 | Eliminate one variable |
| Step 5 | Solve for the remaining variable |
| Step 6 | Substitute back to get the second variable |
Example of Elimination Method
Solve:
- 2x – y = 9
- 5x + 2y = 27
Given
- First equation: 2x – y = 9
- Second equation: 5x + 2y = 27
Step 1: Make the Coefficients of y Equal
Multiply the first equation by 2:
4x – 2y = 18
Second equation remains:
5x + 2y = 27
Step 2: Add the Equations
4x – 2y = 18
5x + 2y = 27
Add:
9x = 45
x = 5
Step 3: Find y
Put x = 5 in the first equation:
2(5) – y = 9
10 – y = 9
y = 1
Answer
The solution is:
x = 5, y = 1
Which Method Is Best for Boards
Students often ask which method should be preferred. The answer depends on the form of the equations.
Method Choice Table
| Method | Best situation |
| Graphical method | When the question specifically asks for graph-based solution or interpretation |
| Substitution method | When one variable can be isolated easily |
| Elimination method | When coefficients can be matched quickly and board-style written solution is needed |
According to the chapter focus, elimination method is especially important from the board exam point of view.
Important Board Exam Areas from This Chapter
The most repeated and important board areas are listed below.
Important Board Areas Table
| Area | Why it matters |
| Nature of solutions | Common in direct coefficient comparison questions |
| Elimination method | Highly important for written board exam solutions |
| Word problems | Frequently asked in application-based form |
| Graph-based conceptual questions | Useful in competency-based questions |
| Case study questions | Can be built from graphs, coefficients, or practical situations |
Common Word Problem Themes
Board exams frequently ask application-based questions from areas such as:
- age problems
- speed and distance
- money problems
- cost and quantity problems
Important Concepts Students Must Revise Quickly
Here is a quick revision list for last-minute preparation.
Key Definitions
- linear equation in two variables
- pair of linear equations
- solution of the pair
- consistent pair
- inconsistent pair
Key Conditions
- a₁/a₂ ≠ b₁/b₂ gives one solution
- a₁/a₂ = b₁/b₂ = c₁/c₂ gives infinitely many solutions
- a₁/a₂ = b₁/b₂ but not equal to c₁/c₂ gives no solution
Key Methods
- graphical method
- substitution method
- elimination method
Quick Revision Table
| Revision point | What to remember |
| Standard form | ax + by + c = 0 |
| General pair | a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 |
| Unique solution | a₁/a₂ ≠ b₁/b₂ |
| Infinite solutions | a₁/a₂ = b₁/b₂ = c₁/c₂ |
| No solution | a₁/a₂ = b₁/b₂ but not equal to c₁/c₂ |
Common Mistakes Students Make in This Chapter
Common Mistakes Table
| Mistake | What students should do instead |
| Forgetting the standard form | Arrange equations properly before comparing coefficients |
| Mixing up the nature-of-solutions conditions | Memorise the full ratio chart clearly |
| Sign errors in substitution and elimination | Simplify each step carefully |
| Plotting inaccurate points in graphical method | Choose clean values and mark points accurately |
| Using the wrong operation in elimination method | Decide carefully whether to add or subtract after matching coefficients |
Study Strategy for Quick Revision
This chapter becomes easy when revised in the right order.
Step-by-Step Revision Table
| Step | What to do |
| Step 1 | Revise the basic form of linear equation |
| Step 2 | Memorise the nature-of-solutions chart |
| Step 3 | Practise one question in each method |
| Step 4 | Focus on word problems |
| Step 5 | Revise signs and transposition carefully |
Practice Questions for Students
Important Practice Questions
- Check the nature of solutions of two given equations without solving them.
- Solve a pair of equations by substitution method.
- Solve a pair of equations by elimination method.
- Draw the graph of two equations and identify the solution.
- Solve a word problem based on age or money using pair of linear equations.
FAQs
Q1. What is a pair of linear equations in two variables?
A pair of linear equations in two variables means two linear equations involving the same two variables, usually x and y, taken together.
Q2. What is the standard form of a linear equation in two variables?
The standard form is ax + by + c = 0, where a, b, and c are real numbers and a and b do not both become zero.
Q3. What is the solution of a pair of linear equations?
The solution is the common value of x and y that satisfies both equations at the same time.
Q4. How do I know whether a pair of equations has one solution, no solution, or infinitely many solutions?
Compare the ratios a₁/a₂, b₁/b₂, and c₁/c₂. Their relationship tells you the nature of the solution.
Q5. Which method is most important for board exams in this chapter?
Elimination method is very important for board exams, though graphical and substitution methods are also important.
Q6. What happens when two lines intersect on a graph?
When two lines intersect, there is exactly one common point, so the pair has a unique solution.
Q7. What happens when two lines are parallel?
When two lines are parallel, they do not meet, so there is no solution.
Q8. What kind of word problems come from this chapter?
Common word problems are based on age, speed and distance, money, cost, and quantity.
Conclusion
Pair of Linear Equations in Two Variables is a highly important Class 10 Maths chapter because it combines algebraic solving, graphical interpretation, and real-life application. Students who understand the standard form, solution meaning, nature of solutions chart, and main solving methods can perform very well in both school exams and board exams.
The best way to prepare this chapter is to revise the solution conditions thoroughly, practise elimination method regularly, and solve a few word problems along with direct equations. At Deeksha Vedantu, we always believe that quick revision works best when the basics are clear, the methods are organised, and practice is consistent.
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