Class 9 Mathematics

Chapter 13 Solutions — Two Variables, One Line

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Solutions

Overview

Step-by-step NCERT solutions for Two Variables, One Line (Chapter 13, NCERT Class 9 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Two Variables, One Line textbook chapter.

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What these solutions cover

All 42 questions in Two Variables, One Line are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise Set 13.1

Write a linear equation in two variables in which a = 3, b = 0 and c = −1/5.
2.
Complete the following table after expressing the given linear equations in standard form:
- (i) y − 15 = √2 x
- (ii) 3y − 2x = 0
- (iii) 5x = 3y
- (iv) x = 8
- (v) 3y = 1
The cost of a notebook is twice the cost of a pen. Cost of notebook = ₹t, cost of pen = ₹p. Charlie wrote t = 2p, whereas Meera wrote p = 2t. Which is correct? (ii) In a cricket match, two Indian batsmen together scored 176 runs. Manisha expressed this as x + y = 176. Is this correct?

Exercise Set 13.2

Verify if the ordered pair (4, 3) is a solution of 5x − 6y = 2. Explain your reasoning.
2.
Find any two solutions for each of the following equations:
- (i) 7x − 3y = 21
- (ii) 2x + 3y = 5
In the equations given below, m and n are unknown constants: 2mx + 3y = 7; 4x + ny = −10. If (2, −1) is the solution of both equations, find the values of m and n.
4.
Find two solutions which lie in different quadrants for each of the following linear equations. Identify the quadrants in which the points lie:
- (i) 5x + 3y = 7
- (ii) 5x − 3y = 7
- (iii) −5x + 3y = 7
- (iv) −5x − 3y = 7
Consider the graph of the equation 3x − 7y = 21 shown, passing through A(7, 0) and B(0, −3). Does the point C(2, 3) lie on the line? Does it satisfy the equation? Can points that do not lie on the line satisfy the equation?
6.
State whether the following sentences are True or False. Justify your answer:
- (i) A linear equation in two variables has only one solution.
- (ii) The graph of a linear equation in two variables always passes through the origin.
- (iii) A linear equation in two variables can never have rational solutions.
- (iv) x = 3 is a valid linear equation in two variables.
- (v) The equation 2x + 3y = 7 has…
Compare the solutions of 3x + 4y = 7 and 6x + 8y = 14. Argue that they have the same set of solutions. (ii) Show that the equations ax + by = c and kax + kby = kc, with k ≠ 0, have the same set of solutions.

Exercise Set 13.3

The following diagrams represent ski hills A, B and C. Hill A: rise = 60 m, run = 70 m. Hill B: rise = 60 m, run = 110 m. Hill C: rise = 80 m, run = 100 m. Rank the hills in order of their steepness, from least to greatest.
The ramp at a loading dock rises 2.5 metres over a run of 4 metres. Find the slope of the ramp.
3.
Find the slope of the line l in each of the following diagrams:
- (i) Line l is horizontal
- (ii) Line l goes from upper-left to lower-right, reading from the graph it passes through (−1, 1) and (1, −1)
- (iii) Line l has positive slope, passing through approximately (−2, −1) and (0, 1)
- (iv) Line l is vertical.
An accessibility ramp for wheelchairs is to be made alongside the staircase. The guidelines given for the slope of the ramp is 1 cm vertical rise to 12 cm horizontal length. What should be the horizontal length of the ramp if the total height of the stairs is 18 cm?

Exercise Set 13.4

1.
On two different days a family buys movie tickets and snack boxes. Cost of one movie ticket = ₹x, cost of one snack box = ₹y. Frame a pair of linear equations:
- (i) Day 1: 2 movie tickets and 3 snack boxes for ₹850.
- (ii) Day 2: 4 movie tickets and 1 snack box for ₹1100.
2.
Two friends, Sahil and Meena, travelled by taxi. Fixed charge for one taxi trip = ₹x, additional charge per km = ₹y. Frame a pair of linear equations:
- (i) Sahil travelled 6 km and paid ₹122.
- (ii) Meena travelled 8 km and paid ₹160.
In a sports meet, adult tickets cost ₹150 each and children's tickets cost ₹100 each. Total attendance = 200, total ticket revenue = ₹25,000. Let x = number of adult tickets, y = number of children's tickets. Frame a pair of linear equations.

Exercise Set 13.5

1.
Form a pair of linear equations for each of the following problems and find their solutions:
- (i) The sum of two integers is +5 and their difference is −21.
- (ii) The difference between two numbers is 26 and one number is three times the other.
- (iii) The coach of a cricket team buys 7 bats and 6 balls for ₹8880. Later, she buys 3 bats and 5 balls for ₹4000. Find the cost of each.
- (iv) Taxi: fixed…
Form a pair of linear equations and find their common solutions graphically. 10 students of Grade 9 took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls.
3.
Use the ratios a₁/a₂, b₁/b₂, c₁/c₂ to determine whether the lines representing the following pairs of linear equations intersect at a point, are parallel or are coincident:
- (i) 5x − 4y + 8 = 0; 7x + 6y − 9 = 0
- (ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0
- (iii) 6x − 3y + 10 = 0; 2x − y + 9 = 0
4.
Which of the following pairs of linear equations have solutions? If they have solutions, find them graphically:
- (i) x + y = 5, 2x + 2y = 10
- (ii) x − y = 8, 3x − 3y = 16
- (iii) 2x + y − 6 = 0, 4x − 2y − 4 = 0
- (iv) 2x − 2y − 2 = 0, 4x − 4y − 5 = 0
Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
6.
Given the linear equation 2x + 3y − 8 = 0, write another linear equation in two variables so that the graphs of the pair formed represent
- (i) intersecting lines
- (ii) parallel lines
- (iii) coincident lines.
Problem from Mahāvīrāchārya's Gaṇita sāra saṅgraha (c. 850 CE): The price of 9 citrons and 7 fragrant wood-apples together is 107; and the price of 7 citrons and 9 fragrant wood-apples together is 101. Find the price of each citron and each fragrant wood-apple.
5 pencils and 7 pens together cost ₹50, whereas 7 pencils and 5 pens together cost ₹46. Find the cost of each pencil and pen.
Find the height of the stool. (The figure shows: a cat sitting on the stool reaches a combined height of 85 cm; the stool is 25 cm taller than the cat.)

End-of-Chapter Exercises

1.
The graph of the line y = 3x passes through (0, 0) and (2, 6).
- (i) Identify the slope from the graph.
- (ii) What is the y-intercept?
- (iii) A water tank is filled so that water level y cm after x minutes follows y = 3x.
- (a) How high after 5 minutes?
- (b) How many minutes to reach 21 cm?
- (iv) Without a new graph, does (4, 10) lie on this line?
- (v) Where does y = 3x intersect the line x = 3?
2.
Temperature conversion: F = (9/5)C + 32.
- (i) Draw the graph with Celsius on x-axis and Fahrenheit on y-axis.
- (ii) Temperature at 30°C in Fahrenheit.
- (iii) Temperature at 95°F in Celsius.
- (iv) Temperature at 0°C in Fahrenheit and at 0°F in Celsius.
- (v) Is there a temperature numerically the same in both scales?
Solve the following system of equations graphically: 2x + y = 6, 2x − y − 2 = 0.
Find the point of intersection of the lines shown on the cover page.
Give a formula to find the x-intercept of the line y = mx + c.
A person is choosing between two mobile plans. Plan A: ₹50 monthly fee + ₹0.20 per minute. Plan B: ₹30 monthly fee + ₹0.30 per minute. For how many minutes is Plan A cheaper? For how many minutes is Plan B cheaper? Find the number of minutes at which both plans cost the same.
7.
How many lines exist that
- (i) have a given slope?
- (ii) have a given slope and pass through a given point?
For what values of p does the pair of equations have a unique solution? 4x + py + 8 = 0; 2x + 2y + 2 = 0.
Find the values of a and b for which the following system has infinitely many solutions: (a + b)x − 2by = 5a + 2b + 1; 3x − y = 14.
Find the value of k for which the following system represents a pair of coincident lines: x + 2y = 3; (k − 1)x + (k + 1)y = k + 3.
Robot 1 starts from origin, moves 3 right and 4 up repeatedly. Robot 2 starts from (10, 0), moves 1 right and 2 up repeatedly. Do their paths intersect? (ii) Robot 1 starts from (3, 0), moves 5 right and 5 up. Robot 2 starts from (7, 0), moves 5 right and 3 down. Do their paths intersect?
★ At a certain time, Jacob's watch reads 'a' minutes after two o'clock. Fifteen minutes later, it reads 'b' minutes after three o'clock. He noticed that a is six times greater than b. What time was it when he looked at his watch the second time?
★ The sum of the digits of a two-digit number is 15. The number obtained by interchanging the digits exceeds the given number by 9. Find the number.
★ In a cyclic quadrilateral ABCD, ∠A = (x + 7)°, ∠B = (y + 8)°, ∠C = (3y + 23)°, ∠D = (4x + 12)°. Find all four angles.
★ A train moving with uniform speed for a certain distance takes 6 hours less if its speed is increased by 6 km/h. It would have taken 6 hours more had its speed been decreased by 4 km/h. Find the distance and the speed of the train.
★ The age of a father is equal to the sum of the ages of his four children. After 20 years, the sum of the ages of the children will be twice the age of the father. Find the age of the father.

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