CBSE • Class 10 • Mathematics • Chapter 3 (Pair of Linear Equations in Two Variables) • Exercise 3.1

Exercise 3.1: Pair of Linear Equations in Two Variables — NCERT Solutions

Algebraic methods — solving simultaneous linear equations by substitution.

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What this exercise covers

Substitution methodWord problems

Step-by-step solutions — Exercise 3.1

3 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 3.1 Q1 • 3 marks

Solve the following pair of linear equations by the substitution method: x+y=14x + y = 14 and x−y=4x - y = 4.
Hint (Socratic — try this first)▾
Can you express one variable in terms of the other from the simpler equation before substituting?
Step-by-step solution▾

From the first equation, x+y=14x + y = 14, express xx in terms of yy:

x=14−yx = 14 - y

Substitute this into the second equation x−y=4x - y = 4:

(14−y)−y=4(14 - y) - y = 4 14−2y=414 - 2y = 4 −2y=−10-2y = -10 y=5y = 5

Now substitute y=5y = 5 back into x=14−yx = 14 - y:

x=14−5=9x = 14 - 5 = 9

Solution: x=9, y=5x = 9,\ y = 5.

Common mistake:
Making a sign error when substituting, e.g. writing 14−y−y=414 - y - y = 4 as 14−0=414 - 0 = 4 by cancelling the two yy terms incorrectly.
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Exercise 3.1 Q2 • 3 marks

Solve by substitution: 3x−y=33x - y = 3 and 9x−3y=99x - 3y = 9. Comment on the number of solutions.
Hint (Socratic — try this first)▾
After substituting, what happens if the variable terms cancel out entirely?
Step-by-step solution▾

From the first equation, 3x−y=33x - y = 3, express yy in terms of xx:

y=3x−3y = 3x - 3

Substitute into 9x−3y=99x - 3y = 9:

9x−3(3x−3)=99x - 3(3x - 3) = 9 9x−9x+9=99x - 9x + 9 = 9 9=99 = 9

This is a true statement with no variables left, meaning the equations are dependent — they represent the same line.

Conclusion: The pair has infinitely many solutions. Every point on y=3x−3y = 3x - 3 satisfies both equations.

Common mistake:
Concluding 'no solution' when the statement reduces to a true identity like 9=99=9; a true identity means infinitely many solutions, whereas a false statement means no solution.
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Exercise 3.1 Q3 • 2 marks

The sum of two numbers is 30 and one number is twice the other. Form a pair of linear equations and solve by substitution.
Hint (Socratic — try this first)▾
If one number is xx, how would you write 'twice the other' using the same variable?
Step-by-step solution▾

Let the two numbers be xx and yy, with xx being the larger.

Given: x+y=30(1)x + y = 30 \quad (1) x=2y(2)x = 2y \quad (2)

Substitute equation (2) into equation (1):

2y+y=302y + y = 30 3y=303y = 30 y=10y = 10

Then x=2y=2(10)=20x = 2y = 2(10) = 20.

Solution: The numbers are 2020 and 1010.

Common mistake:
Writing the relation as x+2y=30x + 2y = 30 instead of forming two separate equations, mixing the two conditions into one.
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How to approach Exercise 3.1

  1. Re-read the chapter summary first. Open Pair of Linear Equations in Two Variables and refresh the key concepts: Substitution method, Elimination method, Cross-multiplication, Consistent and inconsistent systems.
  2. Try each problem yourself before opening the solution. Ten focused minutes per problem usually beats reading two finished solutions.
  3. Use Guru AI for the questions you get stuck on. The Socratic AI tutor walks you through it with hints instead of dictating the answer.
  4. Mark the questions you got wrong and revisit them after 24 hours — the spacing is what locks the method into long-term memory.

All exercises in Pair of Linear Equations in Two Variables

  1. Exercise 3.1 — Algebraic methods — solving simultaneous linear equations by substitution.
  2. Exercise 3.2 — Elimination method for simultaneous linear equations and consistency by ratios of coefficients.
  3. Exercise 3.3 — Word problems — translating real-life situations into pairs of linear equations.

Exercise 3.2 →

Elimination method for simultaneous linear equations and consistency by ratios of coefficients.

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