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Short Intro

Exercise 2.5 introduces the concept of linear relationships between two variables using equations of the form $y = ax + b$. These questions help students understand how real-life situations like internet bills, gym charges, and temperature conversion can be represented mathematically.

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Quick Information Box

Topic	Details
Chapter	Introduction to Linear Polynomials
Exercise	Exercise 2.5
Subject	Mathematics
Class	Grade 9
Main Concepts	Linear Relationships
Difficulty Level	Moderate
Useful For	School Exams & Olympiad Preparation

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Concepts Used (Topics Covered)

• Linear Relationships
• Linear Equations
• Slope and Intercept
• Variable Relationships
• Equation Formation
• Algebraic Substitution
• Real-life Mathematical Modelling
• Temperature Conversion

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Important Formulas

General Linear Equation

$y=ax+b$y=ax+b

Where:

• $a$= slope/rate of change
• $b$ = fixed value or intercept

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Celsius–Fahrenheit Relation

$C=aF+b$

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Solving Simultaneous Equations

Substitute one equation into another to find unknown values.

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Questions & Step-by-Step Solutions

Question 1

A learning platform charges a fixed monthly fee and an additional cost per digital learning module accessed. A student observes that when she accessed 10 modules, her bill was ₹400. When she accessed 14 modules, her bill was ₹500. If the monthly bill $y$y depends on the number of modules accessed $x$x, according to the relation $y=ax+b$y=ax+b, find the values of $a$a and $b$b.

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Solution

Given:

When $x=10$, $y=400$$$400 = 10a + b$$

When $x=14$, $y=500$$$500 = 14a + b$$

Subtract equations:$$500 – 400 = 14a – 10a$$$$100 = 4a$$$$a = 25$$

Substitute into first equation:$$400 = 10(25) + b$$$$400 = 250 + b$$$$b = 150$$

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Final Answer

✅ $a = 25$

✅ $b = 150$

Linear Relationship

$$y = 25x + 150$$

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Question 2

A gym charges a fixed monthly fee and an additional cost per hour for using the badminton court. A student observed that when she used the badminton court for 10 hours, her bill was ₹800. When she used it for 15 hours, her bill was ₹1100. If the monthly bill $y$y depends on the hours of use $x$x, according to the relation $y=ax+b$y=ax+b, find the values of $a$a and $b$b.

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Solution

Given:

When $x=10$, $y=800$$$800 = 10a + b$$

When $x=15$, $y=1100$$$1100 = 15a + b$$

Subtract equations:$$300 = 5a$$$$a = 60$$

Substitute into first equation:$$800 = 10(60) + b$$$$800 = 600 + b$$$$b = 200$$

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Final Answer

✅ $a = 60$a=60

✅ $b = 200$b=200

Linear Relationship

$$y = 60x + 200$$

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Question 3

Consider the relationship between temperature measured in degrees Celsius (°C) and degrees Fahrenheit (°F), which is given by:$$°C = a°F + b$$

Find $a$ and $b$, given that ice melts at $0°C$ and $32°F$, and water boils at $100°C$ and $212°F$.

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Solution

Given:

When $F=32$, $C=0$$$0 = 32a + b$$

When $F=212$, $C=100$$$100 = 212a + b$$

Subtract equations:$$100 = 180a$$$$a = \frac{100}{180}$$$$a = \frac{5}{9}$$

Substitute into first equation:$$0 = 32\left(\frac{5}{9}\right) + b$$$$b = -\frac{160}{9}$$

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Final Answer

✅ $a = \frac{5}{9}$​

✅ $b = -\frac{160}{9}$​

Linear Relationship

$C=\frac{5}{9}(F-32)$

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Common Mistakes

❌ Forgetting to substitute values correctly
❌ Sign errors while subtracting equations
❌ Confusing fixed charge with variable charge
❌ Incorrect fraction simplification

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Exam Tips

✔ Write equations clearly before solving
✔ Use elimination or substitution method carefully
✔ Verify answers by substituting values back
✔ Remember slope represents rate of change

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Practice MCQs

1. Which equation represents a linear relationship?

A) $y=x^2+1$
B) $y=3x+2$
C) $y=2^x$
D) $y=x^3$

Answer

✅ B) $y=3x+2$

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2. In $y=ax+b$, what does $a$ represent?

A) Constant term
B) Slope
C) Variable
D) Intercept

Answer

✅ B) Slope

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3. In $y=ax+b$, what does $b$ represent?

A) Slope
B) Variable
C) y-intercept
D) Equation degree

Answer

✅ C) y-intercept

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4. Which relation converts Fahrenheit to Celsius?

A) $C=F+32$
B) $C=\frac{5}{9}(F-32)$
C) $C=2F$
D) $C=F-100$

Answer

✅ B) $C=\frac{5}{9}(F-32)$

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FAQ Section

Q1. What is a linear relationship?

A linear relationship is a relationship between two variables represented by a straight-line equation.

Q2. What is the standard form of a linear equation?

The standard form is:$$y=ax+b$$

Q3. What does slope mean?

Slope represents the rate at which one quantity changes compared to another.

Q4. What is y-intercept?

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

Q5. Why are linear equations important?

They help represent real-life situations mathematically.

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