Class 10 · Chapter 3 · Algebra unit (20 of 80 marks)

Pair of Linear Equations in Two Variables Class 10: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 51 board-style questions.

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51 questions
14 multiple choice, 3 assertion–reason, 12 very short answer, 13 short answer, 6 long answer, 3 case study
About 12 hours to master

Algebra unit: 20 of 80 theory marks (Polynomials, Pair of Linear Equations, Quadratic Equations, Arithmetic Progressions).

NCERT textbook: Chapter 3 on ncert.nic.in (PDF)

Revision notes

Pair of Linear Equations in Two Variables — revision notes

1. General form and graphs

$a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ . Each equation is a straight line; a solution is a point lying on both lines.

2. Number of solutions (consistency)

Ratios	Lines	Solutions	Pair is
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$	intersecting	exactly one	consistent
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$	coincident	infinitely many	consistent (dependent)
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$	parallel	none	inconsistent

3. Graphical method

Make a table of at least 2–3 points for each line (intercepts are easiest), plot, and read the intersection point.
Triangles formed with an axis: find where each line meets that axis, plus the intersection point; area $= \frac{1}{2} \times base \times height$ .

4. Algebraic methods

Substitution: make one variable the subject of one equation and substitute into the other.
Elimination: multiply to make the coefficients of one variable equal, then add/subtract.
Symmetric coefficients (e.g. $47 x + 31 y, 31 x + 47 y$ ): add and subtract the equations to get $x + y$ and $x - y$ .

Worked example 1

Solve $4 x + 3 y = 18, 2 x - y = 4$ . From the second, $y = 2 x - 4$ ; then $4 x + 6 x - 12 = 18 \Rightarrow x = 3, y = 2$ .

Worked example 2

For which $k$ is $3 x + k y = 9, 6 x + 4 y = 18$ dependent? $\frac{3}{6} = \frac{k}{4} = \frac{9}{18} \Rightarrow k = 2$ .

Worked example 3

A number of ₹10 and ₹20 notes total 30 notes worth ₹440: $x + y = 30, 10 x + 20 y = 440 \Rightarrow y = 14, x = 16$ .

Common errors

Comparing only $a$ and $b$ ratios and forgetting $c$ when deciding between parallel and coincident.
Sign errors when subtracting equations; check the answer in both original equations.
“Consistent” means at least one solution — not necessarily a unique one.
In graphs: unlabelled axes/lines, or too few points plotted.

Board-exam tips

Word problems: define your variables in words first (“let the cost of one pen be ₹ $x$ ”) — this earns the method mark.
If a method is specified (graphical/elimination/substitution), you must use it.
Write the final answer in context with units.

Topics in this chapter: Consistency and number of solutions · Elimination method · Graphical method of solution · Substitution method · Situational problems.

Formulas for this chapter: Chapter 3 on the Class 10 formula sheet (free PDF and A3 poster).

Pair of Linear Equations in Two Variables Class 10: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 4 pages · 177 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceConsistency and number of solutions

The pair $2 x + 3 y = 7$ and $6 x + k y = 21$ has infinitely many solutions when $k$ equals

Show worked answer

Need $\frac{2}{6} = \frac{3}{k} = \frac{7}{21}$ , i.e. $\frac{3}{k} = \frac{1}{3}$ , so $k = 9$ A1
Correct option: (c) $9$

Q2·1 mark·Multiple choiceElimination method

The solution of $x + y = 14$ and $x - y = 4$ is

Q3·1 mark·Assertion–reasonConsistency and number of solutions

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The lines $3 x - y = 5$ and $6 x - 2 y = 7$ are parallel.

Reason (R): Lines $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ are parallel when $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ .

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerSubstitution method

Solve by substitution: $2 x + y = 11$ and $x - 3 y = - 5$ .

Show worked answer

Answer:

x = 4

y = 3

From the second equation $x = 3 y - 5$ ; then $2 (3 y - 5) + y = 11 \Rightarrow 7 y = 21$ M1
$y = 3$ and $x = 3 (3) - 5 = 4$ A1

Q5·2 marks·Short answerConsistency and number of solutions

Find all values of $k$ for which the lines $k x + 4 y = 8$ and $3 x + 2 y = 5$ meet in exactly one point.

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerElimination method

Solve by elimination: $3 x + 4 y = 10$ and $2 x - 2 y = 2$ .

Show worked answer

Answer:

x = 2

y = 1

Multiply the second equation by $2$ : $4 x - 4 y = 4$ ; add to the first: $7 x = 14$ M1
$x = 2$ A1
$2 (2) - 2 y = 2 \Rightarrow y = 1$ A1

Q7·3 marks·Short answerSituational problems

The digits of a two-digit number add up to $11$ . Reversing the digits gives a number $27$ more than the original. Find the number.

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerSituational problems

A gym charges a one-time joining fee plus a fixed fee each month. Ananya paid ₹ $4200$ in all for her first $5$ months and Kabir paid ₹ $6600$ in all for his first $9$ months. Find the joining fee and the monthly fee. How much does a member pay in all for the first $12$ months?

Show worked answer

Answer: Joining fee ₹

1200

, monthly fee ₹

600

; ₹

8400

Joining fee ₹ $x$ , monthly fee ₹ $y$ : $x + 5 y = 4200$ and $x + 9 y = 6600$ M1
Subtracting: $4 y = 2400$ M1
$y = 600$ A1
$x = 4200 - 3000 = 1200$ A1
First $12$ months: $1200 + 12 \times 600 =$ ₹ $8400$ A1

Q9·5 marks·Long answerGraphical method of solution

Solve graphically: $x + y = 5$ and $2 x - y = 4$ . Find the vertices of the triangle formed by the two lines and the $y$ -axis, and the area of this triangle.

Section E style · Case study (4 marks)

Q10·4 marks·Case studySituational problems

School concert tickets. For a school concert, adult tickets cost ₹ $x$ each and child tickets ₹ $y$ each. One family paid ₹ $1100$ for $2$ adult and $3$ child tickets; another paid ₹ $1400$ for $3$ adult and $2$ child tickets.

Write the two equations described. [1 mark]
Without finding $x$ and $y$ separately, find the cost of one adult and one child ticket together. [1 mark]
Find the price of each kind of ticket. [2 marks]

Show worked answer

$2 x + 3 y = 1100$ and $3 x + 2 y = 1400$ A1
Adding: $5 x + 5 y = 2500 \Rightarrow x + y = 500$ , so ₹ $500$ A1
Subtracting the first equation from the second: $x - y = 300$ M1
With $x + y = 500$ : $x = 400$ , $y = 100$ : adult ₹ $400$ , child ₹ $100$ A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 10 assertion–reason questions · the Class 10 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can solve a pair of equations by substitution or elimination and tell from the ratios whether lines meet, are parallel or coincide.

Read first: 1. General form and graphs; 2. Number of solutions (consistency); 4. Algebraic methods 12 practice questions · checkpoint: 4 questions, 8 marks, pass 80%

Practise step 1

Step 2

Board standard

You can find k for a given number of solutions, solve by the graph, and turn a board-style word problem into two equations.

Read first: 2. Number of solutions (consistency); 3. Graphical method; 4. Algebraic methods; Worked examples 1-3 19 practice questions · checkpoint: 4 questions, 10 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can produce neat graphs with labelled lines and complete word-problem solutions that define variables and state the final answer.

Read first: 3. Graphical method; Worked example 3; Board-exam tips 7 practice questions · checkpoint: 3 questions, 15 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle consistency questions with two unknowns, clever combinations of the solution and the slightly unusual word problems that separate 95+ scripts.

Read first: 2. Number of solutions (consistency); Common errors 5 practice questions · checkpoint: 3 questions, 10 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 12 of the 51 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceGraphical method of solution

The graph of $3 x + 2 y = 12$ meets the $y$ -axis at

(a) $(4, 0)$
(b) $(0, 4)$
(c) $(6, 0)$
(d) $(0, 6)$

Q2·1 mark·Multiple choiceGraphical method of solution

The lines $x = 3$ and $y = - 2$ intersect at

(a) $(3, - 2)$
(b) $(- 2, 3)$
(c) $(3, 0)$
(d) $(0, - 2)$

Q3·1 mark·Multiple choiceConsistency and number of solutions

The pair of equations $2 x + 3 y = 7$ and $4 x + 6 y = 14$ has

(a)a unique solution
(b)no solution
(c)exactly two solutions
(d)infinitely many solutions

Worked solution·2 marks·Very short answerSituational problems

The sum of the digits of a two-digit number is $12$ . The number obtained by reversing its digits exceeds the original number by $18$ . Find the number.

Full worked solution Free, no sign-in

Tens digit $x$ , units digit $y$ : $x + y = 12$ , $(10 y + x) - (10 x + y) = 18 \Rightarrow y - x = 2$ M1
$x = 5, y = 7$ : the number is $57$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original algebra problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I117 · Problem I124 · Problem I27 · Problem I109. All 29 →

Themed lesson: Children’s Day (India) maths lesson · Holi maths lesson · Prior-knowledge relay and maths bingo maths lesson (starter, activity and worked answers). All themed maths

Knowledge organiser: Pair of Linear Equations in Two Variables on one page (key facts, formulas and mistakes to avoid; printable PDF)

Where marks are lost in Pair of Linear Equations in Two Variables

Deciding 'parallel' vs 'coincident' from a1/a2 and b1/b2 only. Fix: always compare c1/c2 as well and write all three ratios.
Graphs with too few points, unlabelled axes or unnamed lines. Fix: make a table of at least 3 points per line, label each line with its equation and mark the intersection with its coordinates.
Using elimination when the question says 'by substitution' (or graphically). Fix: if a method is named, use it; otherwise the answer marks can be lost.
Word problems without 'Let the cost of one pen be ₹x…'. Fix: define both variables in words; this line earns the first M1.
Sign errors when subtracting equations. Fix: check the solution in both original equations before writing the final answer.
Triangle-with-axis questions: using the wrong base or forgetting the intercept points. Fix: list the three vertices, then area = ½ × base × height.

Exam strategy: how answers are step-marked · competency-based questions

Examiner Insights: common mistakes in CBSE Class 10 Maths, with fixes (our analysis of public sources) →

Pair of Linear Equations in Two Variables in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 3, 27
Sample paper 2 (Board standard): questions 3, 22
Sample paper 3 (Above board): questions 4, 27
Sample paper 4 (Above board): questions 4, 27
Sample paper 5 (95+ challenge): questions 3, 32, 38

Want it against the clock? Take a timed 30-mark chapter test on Pair of Linear Equations in Two Variables, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Pair of Linear Equations in Two Variables: common questions

What is in CBSE Class 10 Maths Chapter 3, Pair of Linear Equations in Two Variables?

The topics are: Consistency and number of solutions, Elimination method, Graphical method of solution, Substitution method, Situational problems. This page has revision notes for each, and 51 board-style practice questions with step-marked solutions.

How many marks is Pair of Linear Equations in Two Variables worth?

It is in the Algebra unit, which carries 20 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Pair of Linear Equations in Two Variables?

Deciding 'parallel' vs 'coincident' from a1/a2 and b1/b2 only. Fix: always compare c1/c2 as well and write all three ratios. Graphs with too few points, unlabelled axes or unnamed lines. Fix: make a table of at least 3 points per line, label each line with its equation and mark the intersection with its coordinates.

Are the Pair of Linear Equations in Two Variables important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Pair of Linear Equations in Two Variables — revision notes

1. General form and graphs

2. Number of solutions (consistency)

3. Graphical method

4. Algebraic methods

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Pair of Linear Equations in Two Variables Class 10: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Pair of Linear Equations in Two Variables

Pair of Linear Equations in Two Variables in our sample papers

Pair of Linear Equations in Two Variables: common questions

Work through the Route to 95 with your progress saved