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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.

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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

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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

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Important Formulas

General Linear Equation

Where:

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

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Equation Passing Through Origin

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Positive Slope

If (a > 0), graph rises upward.

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Negative Slope

If (a < 0), graph falls downward.

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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).

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

Solution

Equation 1

Points:

• (0,0)
• (1,4)

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Equation 2

Points:

• (0,0)
• (1,2)

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Equation 3

Points:

• (0,0)
• (1,1)

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Observation

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

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

Solution

Equation 1

Points:

• (0,0)
• (1,-6)

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Equation 2

Points:

• (0,0)
• (1,-3)

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Equation 3

Points:

• (0,0)
• (1,-1)

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Observation

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

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

Solution

Equation 1

Points:

• (0,0)
• (1,5)

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Equation 2

Points:

• (0,0)
• (1,-5)

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Observation

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

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

Solution

Equation 1

y-intercept = -1

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Equation 2

y-intercept = 0

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Equation 3

y-intercept = 1

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Observation

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

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

Solution

Equation 1

Slope = -2

y-intercept = -3

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Equation 2

Slope = -2

y-intercept = 0

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Equation 3

Slope = 2

y-intercept = 3

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Observation

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

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Final Understanding

Role of (a)

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

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Role of (b)

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

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Common Mistakes

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

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Exam Tips

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

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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)

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2. Which line has negative slope?

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

Answer

✅ C) (y=-3x)

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3. Parallel lines have:

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

Answer

✅ B) Same slope

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

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

Answer

✅ C) y-intercept

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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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