NCERT Class 9 Maths Exercise 2.6 Solutions – Visualizing Linear Relationships

Short Intro

Exercise 2.6 introduces students to the graphical representation of linear equations. In this exercise, students learn how to draw straight-line graphs, identify slopes, compare parallel lines, and understand the role of coefficients in equations of the form (y=ax+b). These step-by-step solutions simplify graph plotting and linear relationship concepts for Class 9 learners.

Quick Information Box

Topic Details
Chapter Introduction to Linear Polynomials
Exercise Exercise 2.6
Subject Mathematics
Class Grade 9
Main Concepts Graphs of Linear Equations
Difficulty Level Moderate
Useful For School Exams & Foundation Maths

Concepts Used (Topics Covered)

  • Linear Relationships
  • Straight Line Graphs
  • Coordinate Geometry
  • Slope of a Line
  • y-Intercept
  • Parallel Lines
  • Positive and Negative Slopes
  • Graph Plotting

Important Formulas

General Linear Equation

image

Where:

  • (a) = slope of line
  • (b) = y-intercept

Equation Passing Through Origin

image

Positive Slope

If (a > 0), graph rises upward.

Negative Slope

If (a < 0), graph falls downward.

Questions & Step-by-Step Solutions

Question 1

Draw the graphs of the following sets of lines. In each case, reflect on the role of (a) and (b).

(i) (y = 4x, y = 2x, y = x)

Solution

Equation 1

image

Points:

  • (0,0)
  • (1,4)

Equation 2

image

Points:

  • (0,0)
  • (1,2)

Equation 3

image

Points:

  • (0,0)
  • (1,1)

Observation

  • All graphs pass through origin because (b=0).
  • Larger slope means steeper graph.
  • (y=4x) is steepest.

(ii) (y=-6x, y=-3x, y=-x)

Solution

Equation 1

image

Points:

  • (0,0)
  • (1,-6)

Equation 2

image

Points:

  • (0,0)
  • (1,-3)

Equation 3

image

Points:

  • (0,0)
  • (1,-1)

Observation

  • Negative slopes produce downward graphs.
  • All pass through origin.
  • Greater magnitude of slope gives steeper line.

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

Solution

Equation 1

image

Points:

  • (0,0)
  • (1,5)

Equation 2

image

Points:

  • (0,0)
  • (1,-5)

Observation

  • Same steepness because slopes have same magnitude.
  • One rises upward, the other falls downward.

(iv) (y=3x-1, y=3x, y=3x+1)

Solution

Equation 1

image

y-intercept = -1

Equation 2

image

y-intercept = 0

Equation 3

image

y-intercept = 1

Observation

  • All lines have same slope = 3.
  • Therefore, all are parallel lines.
  • Different values of (b) shift the graph vertically.

(v) (y=-2x-3, y=-2x, y=2x+3)

Solution

Equation 1

image

Slope = -2

y-intercept = -3

Equation 2

image

Slope = -2

y-intercept = 0

Equation 3

image

Slope = 2

y-intercept = 3

Observation

  • First two lines are parallel because slopes are same.
  • Third line rises upward because slope is positive.

Final Understanding

Role of (a)

  • Determines slope or steepness.
  • Positive (a) → upward line.
  • Negative (a) → downward line.

Role of (b)

  • Determines y-intercept.
  • Changes vertical position of graph.
  • Same slope but different (b) gives parallel lines.

Common Mistakes

❌ Plotting incorrect coordinates
❌ Confusing slope with y-intercept
❌ Using wrong sign for negative slopes
❌ Joining points inaccurately

Exam Tips

✔ Always make coordinate table first
✔ Plot at least two correct points
✔ Use ruler for straight lines
✔ Remember: same slope ⇒ parallel lines

Practice MCQs

1. Which equation passes through origin?

A) (y=2x+1)
B) (y=3x)
C) (y=x-5)
D) (y=4x+2)

Answer

✅ B) (y=3x)

2. Which line has negative slope?

A) (y=5x)
B) (y=2x+1)
C) (y=-3x)
D) (y=x)

Answer

✅ C) (y=-3x)

3. Parallel lines have:

A) Same intercept
B) Same slope
C) Same equation
D) Same coordinates

Answer

✅ B) Same slope

4. In y=ax+b, b represents:

A) slope
B) coefficient
C) y-intercept
D) x-coordinate

Answer

✅ C) y-intercept

FAQ Section

Q1. What is a linear graph?

A straight-line graph representing a linear equation is called a linear graph.

Q2. What is slope?

Slope measures the steepness of a line.

Q3. What is y-intercept?

It is the point where the graph cuts the y-axis.

Q4. When are two lines parallel?

Two lines are parallel when their slopes are equal.

Q5. Why do some graphs pass through origin?

Graphs pass through origin when y-intercept is zero.

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