Class 9 Maths Ganita Manjari Chapter 2 Exercise 2.6 Solutions: Introduction to Linear Polynomials

Class 9 Maths Ganita Manjari Chapter 2 Exercise 2.6 Solutions cover graph-based questions from Introduction to Linear Polynomials. This exercise helps students draw graphs of linear equations and understand the role of a and b in equations of the form y = ax + b.

In this exercise, students draw and study lines such as y = 4x, y = −6x, y = 3x − 1 and y = −2x − 3. The main learning is to observe how a and b work in y = ax + b. The value of a controls the slope or steepness of the line, while b tells where the line cuts the y-axis. Through these Class 9 Maths Chapter 2 Exercise 2.6 Solutions, students also understand why lines with the same slope are parallel and how positive and negative slopes change the direction of a graph.

Ex 2.6 Class 9 Ganita Manjari Solutions – Ganita Manjari Class 9 Exercise 2.6 Solutions

Question 1.
Draw the graphs of the following sets of lines. In each case, reflect on the role of ‘a’ and ‘6’.
(i) y = 4x,y = 2x,y = x

Solution:

y = 4x, y = 2x, y = x
All lines are of the form y = ax (b = 0).
Observation:
All lines pass through the origin (0, 0).
The value of ‘a’ (slope) determines steepness.
Larger ‘a’ ⇒ steeper line.

(ii) y = – 6x, y = – 3x, y = – x

Solution:

y = – 6x, y = -3x, y =-x
All lines are of the form y = -ax (b = 0).
Observation:
All lines pass through the origin.
Negative ‘a’ means lines slope downward.
Larger magnitude of‘a’ ⇒ steeper downward slope.

(iii) y = 5x, y = -5x

Solution:

y = 5x, y = -5x
All lines are of the form y = ax (b = 0).
Observation: Here, both lines pass through the origin because the y-intercept is 0. y = 5x slopes upward, y = -5x slopes downward; same magnitude of a, same steepness but in opposite directions.

(iv) y = 3x – 1, y = 3x, y = 3x + 1

Solution:

y = 3x – 1, y = 3x, y = 3x + 1
All lines have same slope (a = 3).
Observation: Lines are parallel as slopes are the same (3).
Different values of ‘b’ shift the line. b = -1 line below origin, b = 0 passes through origin b = 1 line above origin.

(v) y = -2x – 3, y = -2x, y = 2x + 3

Solution:

For the lines:

y = -2x - 3
y = -2x
y = 2x + 3

Use the following values to plot the graphs:

For a linear equation:

y = ax + b

For y = -2x - 3:
a = -2, b = -3

For y = -2x:
a = -2, b = 0

For y = 2x + 3:
a = 2, b = 3

Role of a: The value of a determines the slope and direction of the line. When a is negative, the line slopes downward from left to right. When a is positive, the line slopes upward.

Role of b: The value of b determines where the line cuts the y-axis. Thus, the three lines cut the y-axis at -3, 0, and 3 respectively.

Related Resource:

• Exercise 2.1 – Degree, Coefficients and Constants
• Exercise 2.2 – Polynomial Values and Linear Equations
• Exercise 2.3 – Linear Patterns
• Exercise 2.4 – Linear Growth and Linear Decay
• Exercise 2.5 – Linear Relationships
• End-of-Chapter Exercises – Introduction to Linear Polynomials