Linear Equations in Two Variables is a foundational algebra chapter in Class 9 Mathematics that introduces equations involving two variables and their graphical representation. This chapter helps students understand how such equations represent straight lines on the Cartesian plane, preparing them for systems of linear equations in higher classes.
NCERT Solutions for Class 9 Maths Chapter 4 – Linear Equations in Two Variables are prepared strictly according to the latest CBSE syllabus and exam pattern. The solutions are explained in simple, step-by-step language with worked examples and graphs, helping students grasp concepts clearly and score well in Class 9 examinations.
NCERT Solutions For Class 9 Maths Chapter 4 Linear Equations in Two Variables
Q.
The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.
Q.
Q.
Which one of the following options is true, and why? y = 3x + 5 has
(i) a unique solution,
(ii) only two solutions,
(iii) infinitely many solutions
Q.
Q.
Find the value of k, if x = 2, y = 1 is a solution of the equation 2x + 3y = k.
Q.
Draw the graph of each of the following linear equations in two variables:
(i) x + y = 4
(ii) x – y = 2
(iii) y = 3x
(iv) 3 = 2x + y
Q.
Give the equations of two lines passing through (2, 14). How many more such lines are there, and why?
Q.
If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a.
Q.
The taxi fare in a city is as follows: For the first kilometre, the fare is Rs 8 and for the subsequent distance it is Rs 5 per km. Taking the distance covered as x km and total fare as Rs y, write a linear equation for this information, and draw its graph.
Q.
From the choices given below, choose the equation whose graphs are given in Fig. 4.6 and Fig. 4.7.
For Fig. 4. 6 For Fig. 4.7
(i) y = x (i) y = x + 2
(ii) x + y = 0 (ii) y = x – 2
(iii) y = 2x (iii) y = –x + 2
(iv) 2 + 3y = 7x (iv) x + 2y = 6
Q.
Yamini and Fatima, two students of Class IX of a school, together contributed Rs 100 towards the Prime Minister’s Relief Fund to help the earthquake victims. Write a linear equation which satisfies this data. (You may take their contributions as Rs x and Rs y.). Draw the graph of the same.
Q.
Give the geometric representations of y = 3 as an equation
(i) in one variable
(ii) in two variables
Q.
Write four solutions for each of the following equations
(i) 2x + y = 7
(ii) πx + y = 9
(iii) x = 4y
Q.
If the work done by a body on application of a constant force is directly proportional to the distance travelled by the body, express this in the form of an equation in two variables and draw the graph of the same by taking the constant force as 5 units. Also read from the graph the work done when the distance travelled by the body is
(i) 2 units (ii) 0 unit
Q.
In countries like USA and Canada, temperature is measured in Fahrenheit, whereas in countries like India, it is measured in Celsius.
Here is a linear equation that converts Fahrenheit to Celsius:
F = (9/5) C + 32
(i) Draw the graph of the linear equation above using Celsius for x-axis and Fahrenheit for y-axis.
(ii) If the temperature is 30°C, what is the temperature in Fahrenheit?
(iii) If the temperature is 95°F, what is the temperature in Celsius?
(iv) If the temperature is 0°C, what is the temperature in Fahrenheit and if the temperature is 0°F, what is the temperature in Celsius?
(v) Is there a temperature which is numerically the same in both Fahrenheit and Celsius? If yes, find it.
Q.
Give the geometric representations of 2x + 9 = 0 as an equation
(i) in one variable
(ii) in two variables
NCERT Solutions For Class 9 Maths Chapter 4 Linear Equations in Two Variables
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Q. Find five rational numbers between 3/5 and 4/5.
A. Take a common denominator 50:
3/5 = 30/50 and 4/5 = 40/50
Five rational numbers between them are: 31/50, 32/50, 33/50, 34/50, 35/50.
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Q. State whether the following statements are True or False. Give reasons.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
A.
(i) True — Natural numbers 1, 2, 3, … are included in whole numbers 0, 1, 2, 3, ….
(ii) False — Integers include negative numbers like −1, −2, … which are not whole numbers.
(iii) False — Rational numbers such as 1/2 and 3/4 are not whole numbers.
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Q. State whether the following statements are True or False. Justify your answers.
(i) Every irrational number is a real number.
(ii) Every point on the number line is of the form m where m is a natural number.
(iii) Every real number is an irrational number.
A.
(i) True — Irrational numbers are a subset of real numbers.
(ii) False — The number line contains fractions, negative numbers and irrational numbers too, not only natural numbers.
(iii) False — Real numbers include both rational and irrational numbers.
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Q. How can the number 5 be represented on the number line?
A. Start from 0 and move 5 units to the right on the number line. The point obtained represents 5.
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Q. Write the following in decimal form and state the type of decimal expansion:
(i) 36/100 (ii) 1/11 (iii) 4 1/8 (iv) 3/13 (v) 2/11 (vi) 329/400
A.
(i) 36/100 = 0.36 — Terminating
(ii) 1/11 = 0.09̅ — Non-terminating recurring
(iii) 4 1/8 = 4 + 1/8 = 4.125 — Terminating
(iv) 3/13 = 0.230769̅ — Non-terminating recurring
(v) 2/11 = 0.18̅ — Non-terminating recurring
(vi) 329/400 = 0.8225 — Terminating
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Q. If 1/7 = 0.142857̅, predict the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 without long division.
A. The digits repeat in a cyclic pattern:
2/7 = 0.285714̅
3/7 = 0.428571̅
4/7 = 0.571428̅
5/7 = 0.714285̅
6/7 = 0.857142̅
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Q. Express the following in the form p/q, where q ≠ 0:
(i) 0 (ii) 0.47̅ (iii) 0.001̅
A.
(i) 0 = 0/1
(ii) 0.47̅ = 47/99
(iii) 0.001̅ = 1/999
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Q. Express 0.9999… in the form p/q. Are you surprised?
A. Let x = 0.9999…
10x = 9.9999…
Subtracting: 10x − x = 9 ⇒ 9x = 9 ⇒ x = 1
So, 0.9999… = 1 (This is mathematically exact.)
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Q. What is the maximum number of digits in the repeating block of the decimal expansion of 1/17?
A. Maximum length of the repeating block is 17 − 1 = 16.
So, at most 16 repeating digits.
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Q. What condition must the denominator q satisfy for p/q (in lowest form) to have a terminating decimal expansion?
A. The prime factors of q must be only 2 and/or 5.
That is, q = 2^m × 5^n.
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Q. Classify the following numbers as rational or irrational:
(i) √23 (ii) √225 (iii) 0.3796 (iv) 7.478478… (v) 1.101001000100001…
A.
(i) √23 — Irrational
(ii) √225 = 15 — Rational
(iii) 0.3796 — Rational (terminating)
(iv) 7.478478… — Rational (recurring)
(v) 1.101001000100001… — Irrational (non-terminating, non-recurring)
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Q. Visualise 4.26̅ on the number line up to 4 decimal places.
A. 4.26̅ = 4.262626…
Up to 4 decimal places: 4.2626
Mark 4.2626 on the number line.
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Q. Classify the following numbers as rational or irrational:
(i) 2 − √5 (ii) 3 + √23 − √23 (iii) 2/7777 (iv) √12 (v) 2π
A.
(i) Irrational
(ii) 3 + √23 − √23 = 3 — Rational
(iii) Rational
(iv) √12 = 2√3 — Irrational
(v) 2π — Irrational
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Q. Simplify:
(i) (√3 + √3)(√2 + √2)
(ii) (√3 + √3)(√3 − √3)
(iii) (√5 + √2)^2
(iv) (√5 − √2)(√5 + √2)
A.
(i) 4√6
(ii) 0
(iii) 7 + 2√10
(iv) 3
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Q. π is defined as c/d. Why is π irrational?
A. c and d are real lengths (circumference and diameter).
π is irrational means π cannot be written as p/q where p and q are integers.
So there is no contradiction.
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Q. Represent 9.3 on the number line.
A. Divide the segment from 9 to 10 into 10 equal parts. The third mark from 9 is 9.3.
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Q. Rationalise the denominators:
(i) 1/√7 (ii) 1/(√7 − √6) (iii) 1/(√5 + 2) (iv) 1/(√7 − 2)
A.
(i) √7/7
(ii) √7 + √6
(iii) √5 − 2
(iv) (√7 + 2)/3
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Q. Find: (i) 64^(1/2) (ii) 32^(1/5) (iii) 125^(1/3)
A. 8, 2, 5
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Q. Find: (i) 9^(3/2) (ii) 32^(2/5) (iii) 16^(3/4) (iv) 125^(−1/3)
A. 27, 4, 8, 1/5
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Q. Simplify:
(i) 2^(2/3) × 2^(1/5)
(ii) (1/3^3)^7
(iii) 11^(1/2) × 11^(1/4)
(iv) 7^(1/2) × 8^(1/2)
A.
(i) 2^(13/15)
(ii) 3^(−21)
(iii) 11^(3/4)
(iv) 2√14
FAQs: Class 9 Maths Chapter 4 – Linear Equations in Two Variables
Q1. Is this chapter important for Class 9 exams?
Yes, it is a basic and scoring chapter in algebra.
Q2. What is a linear equation in two variables?
An equation of the form ax + by + c = 0, where a, b, c are real numbers.
Q3. Are graph-based questions asked?
Yes, plotting straight lines on the Cartesian plane is common.
Q4. Is this chapter useful for Class 10 Maths?
Yes, it forms the base for systems of linear equations in Class 10.
Q5. How do NCERT Solutions help?
They provide NCERT-aligned, exam-ready explanations with clear steps and graphs.