✖️ Class 10 · Algebra

Pair of Linear Equations in Two Variables for Class 10: notes, worked examples and practice

Two equations with two unknowns pin down a single answer. Substitution and elimination are the two reliable ways to find it.

Updated 21 October 2026. Every worked example and answer on this page is checked by automated tests.

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Notes

Pair of Linear Equations in Two Variables: what you need to know

Chapter names and order differ by board and textbook edition, so follow your own textbook for the order.

What a pair is

Two linear equations in x and y, such as x + y = 10 and x − y = 4, form a pair. A solution must satisfy both equations at once.

Substitution

Write one variable in terms of the other from one equation, then put that into the second equation. This leaves one equation in one variable.

Elimination

Multiply the equations so that one variable has equal coefficients, then add or subtract to remove it. Solve for the other variable and substitute back.

How many solutions?

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂: if a₁/a₂ ≠ b₁/b₂ there is one solution; if a₁/a₂ = b₁/b₂ ≠ c₁/c₂ there is no solution; if all three ratios are equal there are infinitely many.

Check both equations

Substitute your answer into both original equations. Checking only one is a common reason for lost marks.

Formulas and rules to remember

Unique solution: a₁/a₂ ≠ b₁/b₂
No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Infinitely many: a₁/a₂ = b₁/b₂ = c₁/c₂

Solved examples

Example 1

Solve x + y = 10 and x − y = 4.

Add the equations: 2x = 14, so x = 7.
Then y = 10 − 7 = 3.

Answer: x = 7, y = 3

Example 2

Solve 2x + 3y = 12 and x − y = 1 by substitution.

From the second equation x = y + 1.
Put it in the first: 2(y + 1) + 3y = 12, so 5y = 10 and y = 2.
Then x = 3.

Answer: x = 3, y = 2

Example 3

Solve 3x + 2y = 16 and 2x − 2y = 4 by elimination.

The y coefficients are +2 and −2, so add: 5x = 20 and x = 4.
Put x = 4 in the second equation: 8 − 2y = 4, so y = 2.

Answer: x = 4, y = 2

Example 4

Does the pair 2x + 3y = 5 and 4x + 6y = 12 have a solution?

a₁/a₂ = 2/4 = 1/2 and b₁/b₂ = 3/6 = 1/2.
c₁/c₂ = 5/12, which is not 1/2.
The ratios of the coefficients are equal but not the constants, so there is no solution.

Answer: No solution

Avoid these

Common mistakes

Sign errors when subtracting equations

Subtracting a whole equation changes the sign of every term in it.

Checking only one equation

A true solution satisfies both. Substitute into both.

Not matching coefficients before eliminating

Multiply one or both equations first so that one variable cancels.

Mixing up x and y in the answer

Write x and y clearly and check which you solved for.

10 practice questions with answers

Try each question on paper first, then tap it to check your answer. The questions are generated by code and each answer is checked.

Solve the pair of equations: 2x − 2y = 12 and 4x + 3y = 38
Answer: x = 8, y = 2
Solve the pair of equations: 2x + 4y = 12 and 2x + y = 15
Answer: x = 8, y = −1
Solve the pair of equations: x + y = 6 and 4x + 3y = 22
Answer: x = 4, y = 2
Solve the pair of equations: 2x + 2y = −2 and 2x − 2y = −2
Answer: x = −1, y = 0
Solve the pair of equations: 2x − y = −1 and 3x − 2y = 0
Answer: x = −2, y = −3
Solve the pair of equations: 3x + 4y = −11 and 3x − y = −1
Answer: x = −1, y = −2
Solve the pair of equations: 2x + y = 6 and 3x + y = 10
Answer: x = 4, y = −2
Solve the pair of equations: 2x − 2y = 12 and 3x − y = 22
Answer: x = 8, y = 2
Solve the pair of equations: x − y = −1 and 2x + y = −2
Answer: x = −1, y = 0
Solve the pair of equations: 3x − y = −3 and x + 2y = 6
Answer: x = 0, y = 3

More practice

Games and lessons

Maths dictionary

FAQ

Pair of Linear Equations in Two Variables Class 10: questions students ask

What are the methods to solve a pair of linear equations?

Substitution, elimination and the graphical method.

When does a pair of linear equations have no solution?

When a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel and never meet.

How do I check my solution?

Substitute the values of x and y into both original equations.

Which is better, substitution or elimination?

Use substitution when one variable has coefficient 1, and elimination when the coefficients line up or can be matched easily.

Which board is this pair of linear equations in two variables page for?

It covers the ideas taught in CBSE, ICSE and state-board maths. Chapter names and order differ by board and textbook edition, so follow your own textbook for the exact order.

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