Skip to main content
Naman Academy For Class 9
Chapter 13: Two Variables, One LineAll 4 Questions on This Page

Class 9 Maths Chapter 13 Exercise Set 13.3 Solutions

Step-by-step solutions for Class 9 Mathematics Ganita Manjari Part 1I Chapter 13 Exercise Set 13.3 (Questions 1–4). Covers slope of an incline (vertical rise over horizontal run), steepness ranking, slope of lines on coordinate grids (horizontal, downward, upward, vertical lines), undefined slope versus zero slope, and real-world wheelchair ramp calculations.

💡 Quick navigation: Tap any question number in the bar below to jump straight to that question and its step-by-step solution.← Chapter Overview
Jump:

Key Concepts for Exercise Set 13.3

Jump to Question 1
1

Definition of Slope (Steepness)

The slope of a line or incline measures its steepness, defined as the ratio of vertical change (rise) to horizontal change (run). A greater numerical slope corresponds to a steeper incline.

Slope = Vertical rise / Horizontal run = Δy / Δx
2

Direction and Sign of Slope

A line that rises from left to right has a positive slope (rise > 0). A line that falls from left to right has a negative slope (rise < 0).

Rising line: m > 0 | Falling line: m < 0
3

Slope of a Horizontal Line

Along any horizontal line, the y-coordinate remains constant, meaning the vertical change is zero while the horizontal change is non-zero. Hence, the slope of any horizontal line is exactly zero.

Horizontal line: Slope = 0 / (non-zero run) = 0
4

Slope of a Vertical Line

Along any vertical line, the x-coordinate is constant, meaning the horizontal change (run) is zero. Since division by zero is undefined in real mathematics, the slope of a vertical line is undefined.

Vertical line: Slope = (non-zero rise) / 0 ⇒ Undefined
5

Proportional Scaling of Incline Guidelines

When an accessibility guideline specifies a slope ratio (e.g. 1 cm rise to 12 cm run), any scaled ramp must maintain the same proportion: Horizontal length = Total vertical height × (Run / Rise ratio).

Horizontal length = Total rise × 12
Question 1
↑ Jump to Navigator

The following diagrams represent ski hills. Rank the hills in order of their steepness, from least to greatest.

Ski Hills: Rise and Run

Ski hills A (rise 60 m, run 70 m), B (rise 60 m, run 110 m), and C (rise 80 m, run 100 m).

Solution

We know that

Slope = Vertical riseHorizontal run

The greater the slope, the steeper the hill.

Hill A
Slope of A = 6070 = 67
Hill B
Slope of B = 60110 = 611
Hill C
Slope of C = 80100 = 45
Comparing the slopes,
611 < 45 < 67
Therefore, hill B is the least steep, followed by hill C, while hill A is the steepest.
Answer:B, C, A (from least to greatest steepness)
Question 2
↑ Jump to Navigator

The ramp at a loading dock rises 2.5 metres over a run of 4 metres. Find the slope of the ramp.

Loading-Dock Ramp

Loading dock ramp with a vertical rise of 2.5 m over a horizontal run of 4 m.

Solution

We know that

Slope = Vertical riseHorizontal run

Here,

Vertical rise = 2.5 metres
Horizontal run = 4 metres
Substituting these values,
Slope = 2.54
= 2540
= 58

The slope is positive because the ramp rises as we move from left to right.

Answer:Slope = 58 = 0.625
Question 3
↑ Jump to Navigator

Find the slope of the line l in each of the following diagrams.

Diagrams for Question 3

Coordinate grid diagrams (i), (ii), (iii), and (iv) showing line l.

  • (i)Horizontal line
  • (ii)Downward-sloping line
  • (iii)Upward-sloping line
  • (iv)Vertical line
Solution
(i) Horizontal line

The line is horizontal. As we move along it from left to right, there is no vertical change.

Using the slope formula,

Slope = Vertical rise / Horizontal run = 0 / non-zero run = 0

m = 0
(ii) Downward-sloping line

From the plotted points, we can take (−3, 3) and (3, −3).

Moving from (−3, 3) to (3, −3):

•Vertical change (rise) = −3 − 3 = −6
•Horizontal change (run) = 3 − (−3) = 6
Therefore,
Slope = Rise / Run
= −66
= −1

The slope is negative because the line goes downward from left to right.

m = −1
(iii) Upward-sloping line

The plotted points are (−4, −2) and (2, 2).

Moving from (−4, −2) to (2, 2):

•Vertical change (rise) = 2 − (−2) = 4
•Horizontal change (run) = 2 − (−4) = 6
Therefore,
Slope = Rise / Run
= 46
= 23

The slope is positive because the line goes upward from left to right.

m = 23
(iv) Vertical line

The line is vertical. As we move along it, the horizontal change (run) is zero.

Using the slope formula,

Slope = Vertical changeHorizontal change

The denominator is zero, so calculating the slope would require division by zero, which is not defined.

Slope is undefined

A vertical line does not have a slope of zero. A horizontal line has slope zero; a vertical line has undefined slope.

Note
The slope is sometimes informally described as infinity because the vertical change is non-zero while the horizontal change is zero. However, division by zero is undefined, so the slope of a vertical line is undefined, not infinity (∞).
Answer:(i) m = 0 (ii) m = −1 (iii) m = 23 (iv) Slope is undefined
Question 4
↑ Jump to Navigator

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?

Accessibility Ramp Alongside Stairs

Wheelchair ramp alongside a flight of stairs of height 18 cm with 1 cm : 12 cm slope guideline.

Solution

The given guideline says:

•Vertical rise = 1 cm
•Horizontal length = 12 cm

The total vertical height is 18 cm.

Since the vertical rise increases from 1 cm to 18 cm, the horizontal length must also increase by the same factor.

Therefore,
Horizontal length = 18 × 12
= 216 cm

The horizontal length is 216 cm, or 2.16 metres.

Answer:Horizontal length = 216 cm (or 2.16 m)

Need to practice other sections in Chapter 13?

Return to Chapter 13 (Two Variables, One Line)→