CBSE Class 10 Maths Revision Notes Chapter 3 Pair of Linear Equations in Two Variables 2026–27

Pair of Linear Equations in Two Variables explains how two linear equations can be solved together to find common values of x and y.
For CBSE Class 10 Maths, this chapter covers graphical, substitution, elimination and cross multiplication methods.

Pair of Linear Equations in Two Variables deals with two equations that contain the same two variables. These equations help students solve real-life problems related to cost, age, distance, speed and quantity.

Use these CBSE Class 10 Maths Revision Notes Chapter 3 to revise the chapter in a clear order. Start with the meaning of a linear equation, then revise solution types, graphical method, substitution method, elimination method and cross multiplication method.

Key Takeaways

• Linear equation: An equation where the highest power of each variable is 1.
• General form: ax + by + c = 0, where a and b are not both zero.
• Solution: A pair (x, y) that satisfies both equations.
• Nature of lines: Intersecting lines give one solution, parallel lines give no solution and coincident lines give infinitely many solutions.

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CBSE Class 10 Maths Chapter 3 Pair of Linear Equations at a Glance

Chapter 3 connects equations with graphs and algebraic methods. The same pair of equations can be solved through graphs, substitution, elimination or cross multiplication.

Linear Equation in Two Variables Class 10

A linear equation in two variables contains two variables with power 1. In Class 10 Maths Chapter 3 Notes, the variables are usually x and y.

The general form is:

ax + by + c = 0

Here:

• a, b and c are real numbers.
• a and b cannot be zero at the same time.
• x and y are variables.

Copyable format:

ax + by + c = 0

a and b cannot be zero simultaneously.

Example:

2x + y - 4 = 0

This is a linear equation in two variables because the powers of x and y are 1.

Solution of a Linear Equation in Two Variables

A solution of a linear equation in two variables is an ordered pair (x, y) that makes the equation true. One linear equation in two variables can have infinitely many solutions.

Example:

2x + y = 4

Check (0, 4):

2(0) + 4 = 4

4 = 4

So, (0, 4) is a solution.

Check (1, 2):

2(1) + 2 = 4

4 = 4

So, (1, 2) is also a solution.

Copyable format:

If 2x + y = 4, then (0, 4) is a solution.

If 2x + y = 4, then (1, 2) is also a solution.

Pair of Linear Equations in Two Variables Class 10

A pair of linear equations in two variables means two equations in x and y considered together. The common solution satisfies both equations.

General form:

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

Here:

• a1, b1, c1, a2, b2 and c2 are real numbers.
• a1 and b1 are not both zero.
• a2 and b2 are not both zero.

Example:

2x + y - 6 = 0

4x - 2y - 4 = 0

The solution of this pair is the point that satisfies both equations.

Graphical Method of Solution for Pair of Linear Equations

The graphical method of solution shows each linear equation as a straight line. The point where the two lines meet gives the solution.

Steps:

• Write both equations in a form that helps find points.
• Choose values of x and calculate values of y.
• Plot two points for each equation.
• Draw both straight lines.
• Read the point of intersection.

Example:

2x + y - 6 = 0

4x - 2y - 4 = 0

The two lines intersect at (2, 2).

So, the solution is:

x = 2

y = 2

Copyable format:

Solution = point of intersection of the two lines

If the lines intersect at (2, 2), then x = 2 and y = 2.

Nature of Lines and Solution Conditions

The nature of two straight lines decides the number of solutions. This is a key part of CBSE Notes Class 10 Maths Chapter 3.

Copyable format:

If a1/a2 ≠ b1/b2, then the pair has a unique solution.

If a1/a2 = b1/b2 = c1/c2, then the pair has infinitely many solutions.

If a1/a2 = b1/b2 ≠ c1/c2, then the pair has no solution.

Consistent, Inconsistent and Dependent Pair of Linear Equations

A pair of linear equations is consistent when it has at least one solution. Intersecting and coincident lines are consistent.

A pair is inconsistent when it has no solution. Parallel lines are inconsistent.

A pair is dependent when both equations represent the same line. A dependent pair has infinitely many solutions.

Substitution Method for Pair of Linear Equations

The substitution method is useful when one equation can easily give one variable in terms of the other. The value is then substituted in the second equation.

Example:

7x - 15y = 2

x + 2y = 3

From the second equation:

x + 2y = 3

x = 3 - 2y

Substitute this value in the first equation:

7(3 - 2y) - 15y = 2

21 - 14y - 15y = 2

21 - 29y = 2

-29y = -19

y = 19/29

Now substitute y in:

x = 3 - 2y

x = 3 - 2(19/29)

x = 3 - 38/29

x = 87/29 - 38/29

x = 49/29

Answer:

x = 49/29

y = 19/29

Copyable format:

x = 49/29, y = 19/29

Elimination Method for Pair of Linear Equations

The elimination method removes one variable by making its coefficients equal. This method works well when coefficients can be matched easily.

Example:

2x + 3y = 8

4x + 6y = 7

Multiply the first equation by 2:

4x + 6y = 16

Second equation:

4x + 6y = 7

Subtract:

(4x - 4x) + (6y - 6y) = 16 - 7

0 = 9

This is a false statement.

Answer:

The pair has no solution.

So, the equations are inconsistent.

Copyable format:

0 = 9 is false.

Therefore, the pair has no solution.

Cross Multiplication Method for Linear Equations in Two Variables

The cross multiplication method gives direct formulas for x and y. It is useful when the pair is in standard form.

For:

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

The formulas are:

x = (b1c2 - b2c1)/(a1b2 - a2b1)

y = (c1a2 - c2a1)/(a1b2 - a2b1)

Condition:

a1b2 - a2b1 ≠ 0

Copyable format:

x = (b1c2 - b2c1)/(a1b2 - a2b1)

y = (c1a2 - c2a1)/(a1b2 - a2b1)

a1b2 - a2b1 ≠ 0

Another common form:

x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(a1b2 - a2b1)

Equations Reducible to a Pair of Linear Equations

Some equations do not look linear at first. They can become linear after using substitution.

Example:

2/x + 3/y = 4

5/x - 4/y = 9

Let:

1/x = u

1/y = v

Then the equations become:

2u + 3v = 4

5u - 4v = 9

Now this is a pair of linear equations in u and v. After finding u and v, substitute back to get x and y.

Copyable format:

1/x = u

1/y = v

2/x + 3/y = 4 becomes 2u + 3v = 4

5/x - 4/y = 9 becomes 5u - 4v = 9

Formula Table for Class 10 Maths Chapter 3 Notes

Use this table for quick revision before solving questions from Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables.

Solved Examples from Chapter 3 Pair of Linear Equations

Solved examples help students choose the correct method. Use substitution when one variable is easy to isolate, elimination when coefficients can be matched and cross multiplication for direct formula-based solving.

Example 1: Check a Solution of a Linear Equation

Check whether (0, 4) is a solution of:

2x + y = 4

Substitute x = 0 and y = 4:

2(0) + 4 = 4

4 = 4

Answer:

(0, 4) is a solution of 2x + y = 4.

Example 2: Find the Nature of a Pair of Equations

Check whether the pair has one solution, no solution or infinitely many solutions:

2x - 3y = 8

4x - 6y = 9

Write in standard form:

2x - 3y - 8 = 0

4x - 6y - 9 = 0

Compare:

a1 = 2, b1 = -3, c1 = -8

a2 = 4, b2 = -6, c2 = -9

Ratios:

a1/a2 = 2/4 = 1/2

b1/b2 = -3/-6 = 1/2

c1/c2 = -8/-9 = 8/9

So:

a1/a2 = b1/b2 ≠ c1/c2

Answer:

The lines are parallel.

The pair has no solution.

The pair is inconsistent.

Example 3: Solve by Elimination Method

Solve:

x + 2y = 8

2x - 3y = 2

Multiply the first equation by 2:

2x + 4y = 16

Second equation:

2x - 3y = 2

Subtract:

(2x - 2x) + (4y - (-3y)) = 16 - 2

7y = 14

y = 2

Substitute y = 2 in:

x + 2y = 8

x + 2(2) = 8

x + 4 = 8

x = 4

Answer:

x = 4

y = 2

Solution = (4, 2)

Chapter 3 Pair of Linear Equations Summary

This chapter teaches how two linear equations in the same two variables can be solved together. The solution depends on whether the lines intersect, are parallel or coincide.

• A linear equation in two variables has the form ax + by + c = 0.
• A solution is an ordered pair (x, y) that satisfies the equation.
• A pair of linear equations can represent intersecting, parallel or coincident lines.
• Intersecting lines have one solution.
• Parallel lines have no solution.
• Coincident lines have infinitely many solutions.
• Substitution method replaces one variable using the other.
• Elimination method removes one variable by matching coefficients.
• Cross multiplication method gives direct formulas for x and y.
• Some equations can be reduced to linear equations by substitution.

Useful Important Questions Class 10 Maths Links