Chapter 3 of 14

21%

Rajasthan (RBSE)Class 10 Mathematics★ 5–6 marks · RBSE Class 10 Maths board (Unit — Algebra)

Pair of Linear Equations in Two Variables

Two straight lines, one shared solution — RBSE Class 10 Maths on solving a pair of linear equations by graph, substitution, elimination and cross-multiplication. Learn to read consistency from the coefficient ratios (one solution, none, or infinitely many) and to set up word problems, with fully solved board-pattern questions.

⏱ 32 min readMedium difficulty✓ Free · NCERT-aligned

Worksheet

Podcast

Listen as a conversation

A teacher & a student talk through the chapter — in your language.

Mind Map

The chapter at a glance

Every concept and exam-favourite branch — revisable in 5 minutes.

…

Study Guide

Night-before notes

Key points, glossary and the questions most likely to show up.

…

VocabularyBuild your vocabularyLexiGenome mines the key words and schedules them with spaced repetition.Mine words →

By the end of this chapter you'll be able to…

1Represent a pair of linear equations graphically and interpret the intersection
2Classify a pair as consistent (unique / infinite) or inconsistent using coefficient ratios
3Solve a pair by substitution, elimination and cross-multiplication
4Reduce equations of the form a/x + b/y = c to linear form and solve
5Translate word problems (ages, digits, speed, fractions) into equations and solve them

💡

Why this chapter matters

A high-yield algebra chapter: it almost always carries a short 'method' question plus a long word problem worth 4–5 marks. The coefficient-ratio consistency test is a standing one-mark question, and the solving methods reappear throughout higher maths.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

🔗

Linear Equations in Two Variables (Class 9)

Plotting a single linear equation as a straight line.

Pair of Linear Equations in Two Variables — RBSE Class 10 (Mathematics)

One equation in two unknowns has endless answers — a whole line of them. Pair it with a second equation and, usually, exactly one point satisfies both: the place where the two lines cross. This chapter is about finding that point cleanly, and about spotting the special cases where the lines never meet or lie exactly on top of each other.

1. What a pair looks like

A linear equation in two variables has the form $a x + b y + c = 0$ (with a, b not both zero). A pair is two such equations: $a_{1} x + b_{1} y + c_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0$

A solution is a pair $(x, y)$ that satisfies both. Graphically each equation is a straight line, so a solution is a point of intersection.

2. Three geometric outcomes

Compare the ratios of the coefficients:

Condition	Lines	Solutions	Called
$\frac{a _{1}}{a _{2}} \neq = \frac{b _{1}}{b _{2}}$	intersecting	exactly one	consistent (independent)
$\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}}$	parallel	none	inconsistent
$\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}$	coincident	infinitely many	consistent (dependent)

This ratio test is a favourite one-mark question — memorise it.

3. Algebraic methods

Substitution

Make one variable the subject in one equation, substitute into the other. Best when a coefficient is already $\pm 1$ .

Elimination

Multiply the equations to match the coefficient of one variable, then add or subtract to eliminate it. The most reliable all-purpose method.

Example — solve $2 x + 3 y = 11$ and $2 x - 4 y = - 24$ . Subtract: $(3 y - (- 4 y)) = 11 - (- 24) \Rightarrow 7 y = 35 \Rightarrow y = 5$ . Then $2 x + 15 = 11 \Rightarrow x = - 2$ . $x = - 2, y = 5$

Cross-multiplication

For $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ : $\frac{x}{b _{1} c _{2} - b _{2} c _{1}} = \frac{y}{c _{1} a _{2} - c _{2} a _{1}} = \frac{1}{a _{1} b _{2} - a _{2} b _{1}}$ A direct formula — useful when the numbers are awkward, provided $a_{1} b_{2} - a_{2} b_{1} \neq = 0$ .

4. Equations reducible to linear form

Some equations look non-linear but become linear after a substitution. For example, with $\frac{2}{x} + \frac{3}{y} = 13$ , put $u = \frac{1}{x}, v = \frac{1}{y}$ to get $2 u + 3 v = 13$ — an ordinary linear pair. Solve for u, v, then invert to get x, y.

5. Word problems — the real exam prize

The five-mark questions are almost always word problems: ages, two-digit numbers, speed of boat/stream, fractions, and geometry. The skill is translation:

Name the two unknowns clearly (let the number be $10 x + y$ , etc.).
Turn each sentence into one equation.
Solve by elimination (safest), then check against the original words.

Example — the sum of a two-digit number and the number formed by reversing its digits is 66; the digits differ by 2. Find the number. Let digits be x (tens) and y (units): $(10 x + y) + (10 y + x) = 66 \Rightarrow x + y = 6$ ; and $x - y = \pm 2$ . Solving gives $x = 4, y = 2$ or $x = 2, y = 4$ → the number is 42 or 24.

6. Closing thought

Every method here lands at the same place — the crossing point of two lines. Pick elimination as your default, keep the ratio test ready for "how many solutions" questions, and treat word problems as translation exercises. In the RBSE board this chapter reliably carries a short algebra question and a long word problem, so it is worth a large share of your practice time.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

General pair

a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0

Two lines in the plane.

Unique solution

a₁/a₂ ≠ b₁/b₂

Intersecting lines — consistent, independent.

No solution

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Parallel lines — inconsistent.

Infinite solutions

a₁/a₂ = b₁/b₂ = c₁/c₂

Coincident lines — consistent, dependent.

Cross-multiplication

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

Valid when a₁b₂−a₂b₁ ≠ 0.

⚠️

Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Confusing the parallel and coincident conditions

✓ Both need a₁/a₂ = b₁/b₂. Parallel (no solution) ALSO needs ≠ c₁/c₂; coincident (infinite) needs = c₁/c₂ as well.

WATCH OUT

✗ Sign errors in cross-multiplication

✓ Write all equations as = 0 first and use the formula exactly; a single sign slip flips the answer. Cross-check by substituting back.

WATCH OUT

✗ Not simplifying reducible equations before solving

✓ For 2/x + 3/y = 13, substitute u = 1/x, v = 1/y FIRST, solve for u, v, then take reciprocals — don't clear fractions blindly.

WATCH OUT

✗ Forgetting to answer the actual question in word problems

✓ After finding x and y, re-read the question — it may ask for the number (10x+y), the sum of ages, etc., not x and y themselves.

WATCH OUT

✗ Reading consistency off the graph carelessly

✓ Plot at least two accurate points per line; a rough sketch can make intersecting lines look parallel.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 3.1

Exercise 3.1

Forming equations and graphical representation

Questions

Ex 3.2

Exercise 3.2

Graphical method and the coefficient-ratio consistency test

Questions

Ex 3.3

Exercise 3.3

Substitution and elimination methods; word problems

Questions

Examples (in-text)

Worked solutions by every method, including reducible forms

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Consistency

For what value of k do the equations 2x + 3y = 4 and (k+2)x + 6y = 3k+2 represent intersecting lines? State the condition.

▶ Show solution

Step 1 — Intersecting lines need a₁/a₂ ≠ b₁/b₂.

Step 2 — 2/(k+2) ≠ 3/6 = 1/2 ⇒ (k+2) ≠ 4 ⇒ k ≠ 2.

✦ Answer: any k ≠ 2.

Q2EASY· Method

Solve x + y = 14 and x − y = 4.

▶ Show solution

Step 1 — Add: 2x = 18 ⇒ x = 9.

Step 2 — y = 14 − 9 = 5.

✦ Answer: x = 9, y = 5.

Q3EASY· Classify

Are the lines 3x + 2y = 5 and 6x + 4y = 9 consistent?

▶ Show solution

Step 1 — 3/6 = 2/4 = 1/2 but 5/9 ≠ 1/2, so a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Step 2 — Parallel lines ⇒ no solution.

✦ Answer: Inconsistent (no solution).

Q4MEDIUM· Elimination

Solve 2x + 3y = 11 and 2x − 4y = −24 by elimination.

▶ Show solution

Step 1 — Subtract: (3y − (−4y)) = 11 − (−24) ⇒ 7y = 35 ⇒ y = 5.

Step 2 — 2x + 15 = 11 ⇒ x = −2.

✦ Answer: x = −2, y = 5.

Q5MEDIUM· Substitution

Solve s − t = 3 and s/3 + t/2 = 6 by substitution.

▶ Show solution

Step 1 — s = t + 3.

Step 2 — (t+3)/3 + t/2 = 6 ⇒ multiply by 6: 2(t+3) + 3t = 36 ⇒ 5t + 6 = 36 ⇒ t = 6.

Step 3 — s = 9.

✦ Answer: s = 9, t = 6.

Q6MEDIUM· Reducible

Solve 2/x + 3/y = 13 and 5/x − 4/y = −2.

▶ Show solution

Step 1 — Let u = 1/x, v = 1/y: 2u + 3v = 13 and 5u − 4v = −2.

Step 2 — Multiply: (×4) 8u + 12v = 52; (×3) 15u − 12v = −6. Add: 23u = 46 ⇒ u = 2.

Step 3 — 2(2) + 3v = 13 ⇒ v = 3.

Step 4 — x = 1/u = 1/2, y = 1/v = 1/3.

✦ Answer: x = 1/2, y = 1/3.

Q7MEDIUM· Infinite-solution

Find k so that x + 2y = 3 and 5x + ky + 15 = 0 have infinitely many solutions.

▶ Show solution

Step 1 — Write both as = 0: x + 2y − 3 = 0 and 5x + ky + 15 = 0.

Step 2 — Need 1/5 = 2/k = −3/15 = −1/5.

Step 3 — But 1/5 ≠ −1/5, so there is NO value of k giving infinite solutions.

✦ Answer: no such k exists (the constant ratio contradicts the coefficient ratio).

Q8HARD· Word problem (digits)

The sum of a two-digit number and the number formed by reversing its digits is 66. The digits differ by 2. Find the number(s).

▶ Show solution

Step 1 — Let tens digit x, units y: (10x+y)+(10y+x)=66 ⇒ 11(x+y)=66 ⇒ x+y=6.

Step 2 — Digits differ by 2: x − y = ±2.

Step 3 — x+y=6, x−y=2 ⇒ x=4, y=2 (number 42); x−y=−2 ⇒ x=2, y=4 (number 24).

✦ Answer: 42 or 24.

Q9HARD· Word problem (speed)

A boat goes 30 km upstream and 44 km downstream in 10 hours. It goes 40 km upstream and 55 km downstream in 13 hours. Find the speed of the boat in still water and of the stream.

▶ Show solution

Step 1 — Let boat speed x, stream y: upstream (x−y), downstream (x+y). Put u = 1/(x−y), v = 1/(x+y).

Step 2 — 30u + 44v = 10 and 40u + 55v = 13.

Step 3 — Solve: (×4) 120u+176v=40; (×3) 120u+165v=39 ⇒ 11v = 1 ⇒ v = 1/11 ⇒ x+y = 11.

Step 4 — 30u + 44/11 = 10 ⇒ 30u = 6 ⇒ u = 1/5 ⇒ x−y = 5.

Step 5 — x+y=11, x−y=5 ⇒ x=8, y=3.

✦ Answer: boat 8 km/h, stream 3 km/h.

Q10HARD· Word problem (ages)

Five years ago Nuri was thrice as old as Sonu. Ten years later Nuri will be twice as old as Sonu. Find their present ages.

▶ Show solution

Step 1 — Let present ages Nuri x, Sonu y: (x−5) = 3(y−5) ⇒ x − 3y = −10.

Step 2 — (x+10) = 2(y+10) ⇒ x − 2y = 10.

Step 3 — Subtract: −y = −20 ⇒ y = 20; then x = 10 + 2(20) = 50.

✦ Answer: Nuri 50 years, Sonu 20 years.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•A solution of a pair is the point where the two lines meet.
•Unique solution: a₁/a₂ ≠ b₁/b₂ (intersecting).
•No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel).
•Infinite solutions: a₁/a₂ = b₁/b₂ = c₁/c₂ (coincident).
•Methods: substitution, elimination, cross-multiplication — all give the same point.
•Reducible equations: substitute u = 1/x, v = 1/y, solve, then invert.
•Word problems: name unknowns, one equation per condition, solve, verify against the words.

Rajasthan (RBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 5–6 marks

Question type	Marks each	Typical count	What it tests
MCQ / very short	1	1	Consistency from coefficient ratios
Short answer	2–3	1	Solve by elimination/substitution; reducible forms
Long answer	4–5	1	Word problem (age/digit/speed/fraction)

Prep strategy

Make elimination your default method — it rarely fails
Memorise the three coefficient-ratio cases for the consistency question
Drill one word problem of each type: ages, digits, boat-and-stream, fractions
Always substitute the answer back into the original equations to catch sign errors

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Cost and break-even

Two cost/revenue lines meet at the break-even quantity — a linear-pair solution.

Mixtures and pricing

Finding quantities of two items given a total and a total cost is a standard pair of equations.

Speed, time and current

Boat-and-stream and aeroplane-and-wind problems model with upstream/downstream speeds.

Supply and demand

Economics finds equilibrium where the supply line crosses the demand line.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Show every step of elimination — method marks are awarded even if arithmetic slips.
For consistency questions, quote the ratio condition explicitly before concluding.
In word problems, clearly define the variables in words before forming equations.
End word problems with the quantity actually asked, in the correct units.
Verify the final (x, y) in BOTH original equations before writing the answer.

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Systems of three equations in three unknowns and when they are consistent.
Determinants (Cramer's rule) as the reason cross-multiplication works.
Diophantine (integer-only) solutions of ax + by = c.
Geometric meaning: rank of a system and dimension of its solution set.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

RBSE Class 10 Board (BSER Ajmer)High — a method question plus a long word problem almost every year

NTSE / state scholarshipMedium — algebra MCQs

JEE FoundationMedium — linear systems feed into matrices later

Maths Olympiad (IMO/NMTC)Medium — word problems and integer solutions

AI helpers for this chapter

Generate more practice on the fly, or have the chapter explained at a different level.

⚡

Generate 5 more questions

Fresh MCQs + short-answer items based on this chapter, plus answer keys.

🧠

Explain at a different level

Re-explain the chapter at the level you actually want.

📝 My notes

Saved on this device — sign in to sync · how this works

0/600

Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Yes. RBSE (BSER, Ajmer) prescribes the NCERT Mathematics textbook, so chapters and exercises match the national syllabus; RBSE sets its own exam pattern.

Elimination is the safest default. Use substitution when a coefficient is ±1, and cross-multiplication when the numbers are awkward or the question specifically asks for it.

Write both equations as = 0 and compare a₁/a₂, b₁/b₂, c₁/c₂. Ratios all equal → infinite; first two equal but not the third → none; first two unequal → exactly one.

Equations with 1/x or 1/y terms. Substituting u = 1/x and v = 1/y turns them into an ordinary linear pair; solve, then take reciprocals to get x and y.

Practice more, ask doubts, find a tutor

Rajasthan (RBSE) Class 10 mock tests

Rajasthan (RBSE) Class 10 PYQs

Ask in Q&A

Mathematics flashcards

Related chapters

Students who read Pair of Linear Equations in Two Variables usually read these next.

Real Numbers

The opening chapter of RBSE Class 10 Maths — the Fundamental Theorem of Arithmetic and prime factorisation, finding HCF and LCM (and the handy HCF × LCM = product rule), and proving numbers like √2 irrational. Includes the decimal-expansion test for terminating vs non-terminating, with fully solved board-pattern problems.

Polynomials

RBSE Class 10 Maths Chapter 2 — degree and types of polynomials, the geometric meaning of a zero (where the graph cuts the x-axis), and the all-important relationship between the zeroes and coefficients of a quadratic (sum = −b/a, product = c/a). Includes forming a quadratic from its zeroes, with fully solved board-pattern problems.

Quadratic Equations

RBSE Class 10 Maths Chapter 4 — the standard form ax² + bx + c = 0 and three ways to solve it (factorisation, completing the square, the quadratic formula), the discriminant test for the nature of the roots, and translating word problems into quadratics. Includes fully solved board-pattern problems with the discriminant reasoning shown.

Arithmetic Progressions

RBSE Class 10 Maths on sequences that grow by a fixed step — the nth term aₙ = a + (n−1)d and the sum Sₙ = n/2[2a + (n−1)d]. Learn to spot the common difference, find any term, sum a series, and solve real-life AP word problems, with fully solved board-pattern questions.

🧑‍🏫