NCERT solutions for class 10 maths chapter 3 — Pair of Linear Equations in Two Variables: this page is a directory. It routes you to the right exercise and names the method that exercise tests, so you open the correct page with the correct tool in hand.
The fully worked answers to every question live on the exercise pages one level below this one.
A pair of linear equations in two variables is two straight lines on one graph, and the chapter asks one question: where do those lines meet? Three outcomes are possible, and each exercise below drills one of the tools for finding them.
What chapter 3 does: two lines, three possible answers
A pair of linear equations in two variables is simply two lines drawn on the same graph. To solve the pair is to find where they meet, and exactly three outcomes are possible:
- The lines intersect at one point — one unique solution. The pair is consistent.
- The lines are parallel — no solution. The pair is inconsistent.
- The lines coincide — infinitely many solutions, every point on the line. The pair is dependent and consistent.
The chapter opens with Akhila at a fair: \( x \) rides on the Giant Wheel and \( y \) games of Hoopla. Rides cost ₹3 and Hoopla costs ₹4, and she spent ₹20, which gives \( 3x + 4y = 20 \). The number of Hoopla games is half the rides, so \( y = \frac{1}{2}x \).
This page will not solve that pair — the point is that the chapter then builds the tools to solve any such pair: the graphical method, the substitution method and the elimination method.
Exercise directory: which exercise, which method
Chapter 3 has three exercises, each drilling a different tool. The description under each name tells you what that exercise’s own questions actually test, so you can match your homework in one glance.
Exercise 3.1 (7 questions) is the graphical method from scratch: it forms equations from word problems — the Class X quiz’s boys-and-girls split and the pencils-and-pens costs — solves them on graph paper, and runs the ratio comparison \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), \( \frac{c_1}{c_2} \) to classify pairs as intersecting, parallel or coincident and to test consistency.
Its closing questions ask you to invent a second equation that forces a chosen line type, and to find the triangle a pair cuts out with the x-axis.
Exercise 3.2 (3 questions, 10 subparts) is pure substitution method.
One question works through six systems, including surd coefficients like \( \sqrt{2}x + \sqrt{3}y = 0 \) and decimals like \( 0.2x + 0.3y = 1.3 \); a second solves a system and feeds the result into \( y = mx + 3 \); and six word problems — numbers, supplementary angles, bats and balls, taxi charges, a fraction, ages — are turned into equations and solved by substitution.
Exercise 3.3 (2 questions, 9 subparts) is the elimination method, and it deliberately asks for elimination and substitution together on the same four systems so you can compare the two tools. Its word problems cover a fraction, ages, a two-digit number, bank notes and library charges.
The three methods in chapter 3 and when each wins
Read this section once and you can say what the chapter is for: three tools that answer the same question, each best in different circumstances.
Graphical method
Plot two lines from two solutions each; the intersection point is the solution. Fig 3.1 shows this: lines AB and PQ meet at the point B \((6, 0)\), so \( x = 6 \), \( y = 0 \) solves that pair. Before drawing anything, the ratio test decides the outcome for a pair written in standard form \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \):
| Ratio comparison | Lines | Solutions |
|---|---|---|
| \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) | intersecting | exactly one (consistent) |
| \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) | coincident | infinitely many (dependent) |
| \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) | parallel | none (inconsistent) |
Exercise 3.1 drills this method and its ratio test. The ratio-comparison questions are the quickest marks in the chapter and repeat the exact table the chapter summary restates at the end.
Substitution method
Isolate one variable in one equation, substitute it into the other, solve, then back-substitute for the second variable. The method is called substitution because you substitute the value of one variable written in terms of the other.
When the working collapses to a true statement with no variable (like \( 18 = 18 \)), the pair has infinitely many solutions; a false statement (like \( -4 = 0 \)) means no solution. Exercise 3.2 drills substitution exclusively.
Elimination method
Multiply both equations by suitable non-zero constants so the coefficients of one variable are numerically equal, then add or subtract to eliminate that variable. This is more convenient than substitution when the coefficients align neatly. Exercise 3.3 drills it, and its Question 1 makes you run substitution alongside elimination on the same systems so you can see the two methods agree.
Common mistakes in pair of linear equations
These four errors cost more marks in chapter 3 than any wrong arithmetic:
- Comparing ratios before the equations are in standard form. The ratio test only works when every line is written as \( a_1x + b_1y + c_1 = 0 \), zero on one side. Comparing \( a_1/a_2 \) and \( b_1/b_2 \) on a pair left as it stands gives the wrong verdict. Rearrange first, then compare.
- Confusing dependent with inconsistent. Dependent means coincident lines and infinitely many solutions, and that pair is consistent. Inconsistent means parallel lines and no solution. The two endings sound similar but describe opposite cases.
- Writing a two-digit number as \( x + y \) instead of \( 10x + y \). A number like 56 is \( 10(5) + 6 \), not \( 5 + 6 \). This is the single most-missed step in the two-digit-number word problems of Exercise 3.3.
- Panicking when the variable disappears. Getting \( 18 = 18 \) (true) or \( 0 = 9 \) (false) is a correct ending, not an error. True means infinitely many solutions; false means the pair is inconsistent. Write the conclusion and move on.
How to pick the right exercise for your homework
Run this ten-second diagnostic on your question:
- Says “graphically”, “consistent” or “inconsistent” → Exercise 3.1.
- Says “substitution method” → Exercise 3.2.
- Says “elimination method”, or asks for both → Exercise 3.3.
- A word problem about numbers, ages, coins, charges or a fraction → Exercise 3.2 Q3 or Exercise 3.3 Q2.
One practical warning the chapter itself gives: the graphical method turns unreliable when the solution has non-integral coordinates like \((\sqrt{3}, 2\sqrt{7})\) or \((-1.75, 3.3)\) — reading such points off a graph invites error, which is exactly why the algebraic methods exist.
A selection rule that works most of the time: choose the graphical method for whole-number answers, substitution when one variable is easy to isolate, and elimination when the coefficients line up neatly.
For revision, chapter 3 follows the polynomial work in Class 10 Maths Chapter 2 and leads into quadratic equations in Chapter 4. The Class 10 Maths solutions hub holds every chapter. The official NCERT textbook PDF for Chapter 3 is useful if you want to read the worked examples alongside.
| Exercise | What you get |
|---|---|
| Exercise 3.1 | Solved |
| Exercise 3.2 | Solved |
| Exercise 3.3 | Solved |
Frequently asked questions
Which exercise in class 10 maths chapter 3 uses the substitution method?
Exercise 3.2. Its three questions all use substitution: six pure systems in Question 1, a system feeding into the line \( y = mx + 3 \) in Question 2, and six word problems in Question 3.
How can I tell whether a pair of linear equations is consistent without drawing a graph?
Use the ratio test. Write both equations in standard form, then compare \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \) and \( \frac{c_1}{c_2} \). If the first two ratios differ, the lines intersect and the pair is consistent. If the first two are equal but the third differs, the lines are parallel and the pair is inconsistent.
If all three are equal, the lines coincide and the pair is consistent (dependent).
What does it mean when solving a pair of equations gives 18 = 18 or 0 = 9?
Both are valid endings with no variable left. A true statement like \( 18 = 18 \) means the two equations are equivalent, so there are infinitely many solutions. A false statement like \( 0 = 9 \) (or \( -4 = 0 \)) means the lines are parallel and there is no solution — the pair is inconsistent.
Why is the graphical method not always convenient?
Because the answer must be read off the graph. When the solution is a fraction or surd such as \((\sqrt{3}, 2\sqrt{7})\) or \((-1.75, 3.3)\), you cannot read the coordinates accurately. The algebraic methods — substitution and elimination — return exact values, which is exactly why they exist.
Where do the word problems appear in chapter 3?
They are spread across the exercises. Exercise 3.1 Question 1 sets up quiz and pencils-and-pens problems to solve graphically; Exercise 3.2 Question 3 solves numbers, supplementary angles, bats-and-balls, taxi charges, a fraction and ages by substitution; Exercise 3.3 Question 2 covers a fraction, ages, a two-digit number, bank notes and library charges by elimination.
Reference: NCERT textbooks (CBSE).