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CBSE Class 10 Mathematics Chapter 3 Pair of Linear Equations in Two Variables Worksheet with Answers

Pair of Linear Equations in Two Variables is a critical chapter in CBSE Class 10 Mathematics, carrying 6-8 marks in the board examination. This printable worksheet provides structured practice across multiple question types — from objective MCQs to application-based case studies — mirroring the exact exam pattern. Students should attempt this worksheet in 90 minutes under timed conditions for best results and refer to the detailed answer key for self-correction.

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Key takeaways

  • ✓Worksheet designed for 90-minute practice session covering all question formats from CBSE board exam pattern
  • ✓Section-wise difficulty progression from basic MCQs to advanced HOTS questions ensures thorough concept mastery
  • ✓Includes graphical method, substitution, elimination and cross-multiplication techniques as per NCERT curriculum
  • ✓Case-study question mirrors 2025 board exam pattern with real-life application of linear equations
  • ✓Complete answer key with working steps helps students self-evaluate and identify knowledge gaps
  • ✓Covers consistent, inconsistent and dependent systems with numerical examples and graphical interpretations
  • ✓Suitable for quick revision one week before board exams or for periodic self-assessment during the academic year

Quick Chapter Recap: Pair of Linear Equations in Two Variables

Chapter 3 of NCERT Class 10 Mathematics introduces systems of two linear equations in two variables (x and y). A pair such as a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 can be solved using four main methods: graphical method (plotting lines and finding intersection), algebraic methods including substitution (express one variable in terms of another), elimination (add or subtract equations to eliminate a variable), and cross-multiplication (a direct formula-based approach). The chapter emphasizes understanding when a system is consistent (has at least one solution), inconsistent (no solution, parallel lines) or dependent (infinitely many solutions, coincident lines). The condition for consistency depends on ratios: if a₁/a₂ ≠ b₁/b₂, the system has a unique solution; if a₁/a₂ = b₁/b₂ ≠ c₁/c₂, no solution exists; if a₁/a₂ = b₁/b₂ = c₁/c₂, infinitely many solutions exist. Real-world problems involving age, speed-distance-time, mixture and cost are modeled as pairs of linear equations. Mastery of this chapter builds the foundation for advanced algebra in higher classes and competitive exams.
  • Four solution methods: graphical, substitution, elimination, cross-multiplication
  • Consistent system: unique or infinitely many solutions depending on coefficient ratios
  • Inconsistent system: no solution, represented by parallel lines on graph
  • Dependent system: infinitely many solutions, lines coincide completely
  • Board exams typically include 1 MCQ (1 mark), 2 short questions (2 marks each), 1 long question (3-4 marks) and 1 case study (4 marks)

Worksheet Information: Difficulty Level and Time Guidelines

This worksheet is calibrated at moderate to challenging difficulty, aligning with the CBSE Class 10 board exam standard. It contains 30+ questions distributed across five main sections plus one case study, designed to be completed in 90 minutes. Section A features 6 MCQs (1 mark each, 6 marks total) testing foundational recall and quick application. Section B has 5 fill-in-the-blank items (1 mark each, 5 marks) focusing on definitions and direct formula use. Section C includes 5 true/false or match-the-following items (1 mark each, 5 marks) for concept verification. Section D presents 5 short-answer questions (2 marks each, 10 marks) requiring method application and brief working. Section E comprises 3 long-answer or HOTS questions (3-4 marks each, 10 marks) demanding multi-step reasoning and synthesis of multiple methods. Finally, one case-study question (4 marks) tests real-world problem-solving. Students should allocate roughly 1 minute per MCQ, 8-10 minutes per long answer, and 10 minutes for the case study. Use a pencil for rough work, and attempt all questions for comprehensive practice before consulting the answer key at the end.
  • Total marks: 40 | Suggested time: 90 minutes
  • Difficulty: Moderate to Challenging (CBSE board standard)
  • Sections progress from recall-based MCQs to application-heavy HOTS questions
  • Attempt under timed conditions to simulate exam environment
  • Use the answer key only after attempting all questions independently

Section A: Multiple Choice Questions (1 mark each)

This section tests your ability to quickly identify correct answers using concepts from NCERT Class 10 Mathematics Chapter 3. Each MCQ has four options with exactly one correct answer. Read each question carefully, eliminate obviously wrong choices, and select the best option. These questions cover graphical interpretation, coefficient ratios for consistency, and identification of solution methods. Practice these regularly to improve speed and accuracy for the board exam, where MCQs form part of the competency-based assessment introduced in recent years. Remember that negative marking is not applied in CBSE board exams, so attempt every question even if you are unsure. Work through the logic step-by-step: for ratio-based questions, write down a₁/a₂, b₁/b₂, c₁/c₂ quickly on the margin and compare them. For graphical questions, visualize or sketch the lines mentally to decide if they intersect, are parallel or coincide. Time management is critical; spend no more than one minute per MCQ to leave adequate time for longer questions in later sections.
  • Q1. The pair of equations x + 2y = 5 and 3x + 6y = 15 has: (a) unique solution (b) no solution (c) infinitely many solutions (d) exactly two solutions
  • Q2. If the lines represented by 2x + 3y = 7 and 4x + ky = 14 coincide, then k equals: (a) 3 (b) 6 (c) 9 (d) 12
  • Q3. The graphical method of solving a pair of linear equations gives the solution as: (a) coordinates of intersection point (b) slope of lines (c) y-intercept only (d) area between lines
  • Q4. For the system 3x − y = 2 and 9x − 3y = 6, the lines are: (a) intersecting (b) parallel (c) coincident (d) perpendicular
  • Q5. Which method is most efficient when coefficients of one variable are equal or can be made equal easily? (a) graphical (b) substitution (c) elimination (d) cross-multiplication
  • Q6. The pair x + y = 7 and x − y = 3 represents: (a) parallel lines (b) intersecting lines (c) coincident lines (d) no geometric figure

Section B: Fill in the Blanks (1 mark each)

Fill-in-the-blank questions assess direct recall of definitions, formulas and standard results from the chapter. Write the most appropriate single word, phrase or numerical value in each blank. These questions often appear in board exams as part of the competency-based or assertion-reason sections, so practicing them builds confidence. Focus on exact NCERT terminology: for instance, use the term 'consistent' rather than 'solvable', and 'inconsistent' instead of 'no solution'. When a question asks for a ratio or condition, express it in standard mathematical notation (e.g., a₁/a₂ = b₁/b₂ = c₁/c₂). Read the sentence fully before filling the blank to ensure grammatical and contextual correctness. If a numerical answer is expected, double-check units and signs. For ratio-based blanks, simplify fractions to lowest terms. These questions carry easy marks if you have thoroughly revised the chapter summary and key points. Make a habit of writing definitions in your own notebook and testing yourself regularly; this habit ensures you never lose these straightforward marks in the actual board exam.
  • Q7. A pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 is called ________ if it has at least one solution.
  • Q8. If a₁/a₂ ≠ b₁/b₂, the pair of equations has a ________ solution.
  • Q9. The graphical representation of an inconsistent system shows two ________ lines.
  • Q10. In the elimination method, we multiply one or both equations by suitable constants to make coefficients of one variable ________.
  • Q11. The coordinates of the point where the graphs of two linear equations intersect give the ________ of the system.

Section C: True or False / Match the Following (1 mark each)

This section combines quick verification questions with match-the-column items to test conceptual clarity. For true/false statements, read each assertion carefully and decide based on definitions and theorems from NCERT Class 10 Mathematics. Write 'True' or 'False' clearly; partial answers like 'T' or 'F' are acceptable if the instructions permit. If a statement is false, mentally note the correct version to reinforce learning (though you need not write it unless asked). For match-the-following, draw clear lines or write pairs (A-ii, B-iv, etc.) as per your school's convention. These items often test your ability to connect equations with their graphical behavior, solution types with coefficient ratios, and methods with their ideal use-cases. Such questions are scoring if you have practiced enough examples. They appear frequently in board exams and school unit tests. Avoid rushing; a careless mismatch costs a full mark. Cross-check your matches by verifying one or two pairs logically before finalizing your answer. This section is designed to be completed in about 5-7 minutes, leaving ample time for descriptive sections.
  • Q12. True or False: If two lines are coincident, the system of equations is inconsistent.
  • Q13. True or False: Cross-multiplication method can be applied to any pair of linear equations in two variables.
  • Q14. True or False: A consistent system must have exactly one solution.
  • Q15. Match: Column A (Condition) → Column B (Solution type). A: a₁/a₂ = b₁/b₂ ≠ c₁/c₂; B: a₁/a₂ ≠ b₁/b₂; C: a₁/a₂ = b₁/b₂ = c₁/c₂. Options: (i) Unique solution (ii) No solution (iii) Infinitely many solutions
  • Q16. True or False: The substitution method is most suitable when one equation is already solved for a variable or can be easily rearranged.

Section D: Short Answer Questions (2 marks each)

Short-answer questions require you to show essential working and arrive at the solution in 3-5 lines. Marks are awarded for method and correct final answer, so even if your answer is slightly off due to a calculation slip, you may earn partial credit for the correct approach. Begin each solution by writing the given equations clearly, then state the method you will use (substitution, elimination, etc.). For 2-mark questions, examiners expect at least two clear algebraic steps before the final result. Write neatly and use proper mathematical symbols; for instance, write 'x = 3, y = 2' rather than prose like 'x is three and y is two'. These questions typically test one concept or method at a time: solving by substitution, checking consistency using ratios, converting word problems into equation form, or verifying a given solution. Attempting these questions builds fluency for the longer 3- or 4-mark problems. Budget about 3-4 minutes per short-answer question. If stuck, move to the next question and return later. Review your answers to ensure you have not omitted the final statement or units where applicable. Class 10 Mathematics solutions demand precision; a missing negative sign can render the entire answer incorrect.
  • Q17. Solve by substitution method: x + y = 14, x − y = 4.
  • Q18. Check whether the pair 2x + 3y = 9 and 4x + 6y = 18 is consistent. If yes, find the number of solutions.
  • Q19. Find the value of k for which the system kx + 3y = k − 2 and 12x + ky = k has no solution.
  • Q20. Form a pair of linear equations from the statement: The sum of two numbers is 50 and their difference is 10.
  • Q21. Solve graphically: x + y = 5, x − y = 1. (Draw a rough sketch showing intersection point.)

Section E: Long Answer / HOTS Questions (3-4 marks each)

Long-answer questions and Higher Order Thinking Skills (HOTS) problems demand multi-step reasoning, synthesis of concepts and clear presentation. Each question here is worth 3 or 4 marks, so examiners look for detailed working, logical flow and correct final answer. Start by reading the question twice to understand what is asked. Identify the known and unknown quantities, then decide which method (substitution, elimination or cross-multiplication) is most efficient. For word problems, the first critical step is correctly translating the verbal statement into a pair of equations; half the battle is won if you frame the equations accurately. Show every algebraic manipulation: if you multiply an equation by a constant, write that step explicitly. Use proper mathematical notation and avoid skipping intermediate steps. HOTS questions may ask you to find conditions on parameters (like k) for a certain solution type, or to apply equations to optimization or real-world scenarios. Such questions test conceptual depth, not just procedural fluency. Allocate 8-10 minutes per long-answer question. If a question has multiple parts, attempt the parts you are confident about first to secure partial marks. Review your solution to ensure dimensional consistency and reasonableness of answers.
  • Q22. (4 marks) Solve using cross-multiplication method: 3x + 2y = 11, 2x + 3y = 4. Verify your solution by substitution.
  • Q23. (3 marks) A two-digit number is such that the digit in the tens place is twice the digit in the units place. If 27 is added to the number, the digits interchange their places. Find the number.
  • Q24. (4 marks) For what value of k will the following system have infinitely many solutions? 2x + 3y = 7 and (k+2)x + (2k+1)y = 3(k+1). Also find the solutions in terms of a parameter.

Section F: Case-Study Based Question (4 marks)

Case-study questions, introduced in CBSE board exams from 2020-21, present a real-world scenario followed by 3-4 sub-questions. Read the case study carefully, underline or highlight key numerical data and relationships, then attempt each sub-question methodically. Marks are distributed across the sub-parts, so even if you cannot solve the final part, secure marks in the earlier parts. The case study in this worksheet models a typical board exam pattern: a short paragraph describing a situation (travel, business, construction, etc.), followed by questions asking you to form equations, solve them and interpret the solution in context. Write the equations derived from the passage explicitly before solving; this demonstrates your modeling ability and often earns method marks. For interpretation questions, write answers in complete sentences, e.g., 'The cost of one pen is ₹15 and one notebook is ₹25.' rather than just '15, 25'. This section is an excellent opportunity to score full marks if you practice extracting equations from English statements. Use the context to double-check the reasonableness of your numerical answers: if the problem is about ages, solutions must be positive integers; if about costs, they should be non-negative and sensible.

Complete Answer Key with Step-by-Step Solutions

This answer key provides correct answers and concise explanations for every question in the worksheet. Use it for self-assessment only after attempting all questions independently. For MCQs and fill-in-the-blanks, the correct option or term is stated first, followed by a brief justification. For short and long answers, essential algebraic steps are shown so you can identify where you went wrong if your answer differs. Cross-check your method as well as your final answer; sometimes the answer may be correct by luck, but the method incorrect, which will not earn full marks in board exams. If you scored below 60 percent on this worksheet, revisit the NCERT textbook examples and solve additional problems from the exercise. For scores between 60-80 percent, focus on the questions you got wrong and practice similar problems. If you scored above 80 percent, challenge yourself with previous years' board questions and sample papers. Remember that CBSETUTOR.ai offers 24×7 doubt-solving via photo upload and personalized practice at just ₹999/month for all subjects and classes 6-12, with a 3-day free trial—ideal if you need on-demand help beyond the classroom. Regular worksheet practice combined with immediate doubt resolution is the surest path to scoring 90+ in CBSE Class 10 Mathematics board exams.
  • Section A Answers: Q1(c), Q2(b), Q3(a), Q4(c), Q5(c), Q6(b)
  • Section B Answers: Q7–consistent; Q8–unique; Q9–parallel; Q10–equal; Q11–solution
  • Section C Answers: Q12–False (coincident ⇒ infinitely many solutions, hence consistent); Q13–True; Q14–False (can have infinitely many); Q15–A(ii), B(i), C(iii); Q16–True
  • Section D Answers: Q17–x=9, y=5 | Q18–Consistent, infinitely many (ratios equal) | Q19–k=6 | Q20–x+y=50, x−y=10 | Q21–Graph intersection at (3,2)
  • Section E Answers: Q22–x=5, y=−2 (verify by substituting back) | Q23–Number is 36 (tens digit=3, units=6) | Q24–k=4; solutions: x=(7−3t)/2, y=t for any real t
  • Section F Answers: Q25(i)–5x+500=4000, 8y+200=4000 | (ii)–x=700 | (iii)–y=475 | (iv)–Plan B cheaper by ₹225/month

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CBSETUTOR.ai is an AI-powered learning platform tailored for CBSE students in classes 6-12, offering unlimited doubt-solving, chapter-wise worksheets and personalized practice—all for a flat ₹999 per month covering every subject. For Chapter 3 Pair of Linear Equations in Two Variables, students can upload photos of any problem from NCERT, RD Sharma, RS Aggarwal or school worksheets and receive step-by-step text solutions within seconds. The platform's AI tutor is available 24×7, meaning you are never stuck waiting for the next tuition class. Parents appreciate the single transparent pricing with no hidden fees, and the 3-day free trial lets students experience the full feature set risk-free. Beyond worksheets, CBSETUTOR.ai provides chapter notes aligned with NCERT terminology, video explanations for complex methods like cross-multiplication, and adaptive quizzes that adjust difficulty based on your performance. This ensures that whether you are in a metro city with access to multiple coaching centers or a smaller town with limited resources, you have a reliable digital tutor at your fingertips. The platform also tracks progress over time, highlighting weak areas in topics like consistency conditions or word-problem translation, so revision becomes targeted and efficient. Investing in CBSETUTOR.ai is particularly valuable for Mathematics, where practice and immediate feedback drive improvement. Try the free trial today and see your confidence in Pair of Linear Equations soar.
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Frequently asked questions

How many marks does Chapter 3 Pair of Linear Equations carry in CBSE Class 10 board exam?+
Chapter 3 typically carries 6-8 marks in the board exam, distributed as 1 MCQ (1 mark), 2 short-answer questions (2 marks each), 1 long-answer question (3-4 marks) and 1 case-study sub-part (1-2 marks). Total weightage can vary slightly each year, so check the latest CBSE marking scheme.
Which method should I use to solve a pair of linear equations in the exam?+
Choose the method that is most efficient for the given pair. Use elimination when coefficients of one variable are equal or can be made equal easily. Use substitution if one equation is already solved for a variable. Use cross-multiplication for quick solving when both equations are in standard form. Graphical method is useful for visualization but time-consuming in exams unless specifically asked.
How do I know if a system is consistent, inconsistent or has infinitely many solutions?+
Compare the ratios of coefficients. If a₁/a₂ ≠ b₁/b₂, the system has a unique solution (consistent). If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the system is inconsistent (no solution, parallel lines). If a₁/a₂ = b₁/b₂ = c₁/c₂, the system has infinitely many solutions (dependent, coincident lines). Memorize these conditions verbatim from NCERT.
What are common mistakes students make in word problems on linear equations?+
Common errors include incorrect translation of statements into equations (e.g., confusing sum and difference), sign errors when forming equations, and forgetting to define variables clearly. Always write 'Let x =...' and 'Let y =...' at the start, form both equations carefully, and verify the solution by substituting back into the original word problem context.
Is it necessary to verify the solution after solving a pair of equations?+
Yes, especially in board exams. Verification demonstrates that your solution satisfies both original equations and often earns you method marks even if the final answer has a minor calculation slip. Substitute your x and y values back into both equations and confirm that LHS equals RHS in each case.
Can I use the graphical method for all pairs of linear equations?+
Graphical method is applicable to any pair of linear equations, but it is time-consuming and less accurate for non-integer solutions. Use it when the question explicitly asks for a graph or when you need a visual check. For exact algebraic solutions, substitution, elimination or cross-multiplication are preferred.
How much time should I spend practicing Chapter 3 for board exam preparation?+
Allocate at least 5-6 hours across two weeks for Chapter 3. Spend the first half solving all NCERT exercise problems, then move to additional worksheets like this one, sample papers and previous years' questions. Regular daily practice of 30-45 minutes is more effective than marathon sessions. Use platforms like CBSETUTOR.ai for instant doubt-solving to avoid getting stuck.
What is cross-multiplication method and when should I use it?+
Cross-multiplication is a direct formula-based method for solving a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. It gives x and y in one step using determinants. Use it when both equations are already in standard form and you want a quick solution. It is particularly useful in case-study questions where time management is critical.
How can CBSETUTOR.ai help if I am stuck on a Chapter 3 problem at night?+
CBSETUTOR.ai offers 24×7 AI-powered doubt solving. Simply take a photo of the problem, upload it via the app or website, and receive a step-by-step text solution within seconds. The platform covers all NCERT and reference book problems. With a 3-day free trial and a flat ₹999/month for all subjects and classes 6-12, it is a cost-effective alternative to late-night tuition helplines.
Are case-study questions in Chapter 3 difficult to score in?+
Case-study questions are scoring if you practice extracting equations from paragraph descriptions. Read the case carefully, underline numerical data, define variables, form equations and solve step-by-step. Each sub-question usually carries 1 mark, so even partial attempts earn marks. Practice 10-15 case studies from sample papers to build confidence and speed.

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