Introduction to Linear Polynomials – Class 9 Maths Chapter 2 Notes & NCERT PDF

Class 9MathematicsChapter 2NCERT book: Ganita Manjari - Mathematics Class 9

Notes and a simple summary of Chapter 2 of the NCERT Class 9 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 2: Introduction to Linear Polynomials

Introduction

The chapter begins with everyday situations that produce algebraic expressions:

Raju’s boxes: x red boxes of 4 pens and y blue boxes of 5 pencils, plus 3 free pens, give 4x + 5y + 3. Here 4x, 5y and 3 are terms; x and y are variables (the book’s new word for letter-numbers); 4 and 5 are coefficients; 3 is a constant.

Fencing a garden of length l and width w gives the cost 200l + 160w + 50lw rupees.

A 20 cm wire bent into rectangles: if the length is x cm, the width is (10 – x) cm and the area is x(10 – x) = 10x – x².

Key Ideas

2.1 Polynomials and degree

An expression in one variable and its powers is a univariate polynomial. Its degree is the highest power of the variable:

5y³ + y² + 2y – 1 → degree 3 (cubic)

x² + 5x + 1 → degree 2 (quadratic)

3z + 7 → degree 1 (linear)

8 = 8x⁰ → degree 0 (constant)

2.2 Linear polynomials

A linear polynomial has degree 1. Its key feature: the difference between successive values at integers is constant.

Perimeter of a square of side x is 4x.

A chess club charges ₹200 to join plus ₹50 per match: 200 + 50m. The amount rises by exactly ₹50 per match.

Equating a linear polynomial to a constant gives a linear equation, e.g. x + (x + 10) = 64 → x = 27, so the numbers are 27 and 37.

A polynomial acts like an input–output machine (a function): for 2x + 3, input 4 gives 11 and input –6 gives –9.

2.3 Linear patterns

Square tiles growing 1, 3, 5, 7, … : stage n has 2n – 1 tiles.

Bela’s pocket money: ₹100 – ₹5 a day → 100 – 5n.

An auto fare of ₹25 for the first 2 km and ₹15 per km after that → 15n – 5 for n ≥ 2.

A linear pattern is a sequence in which the difference between consecutive terms is constant.

2.4 Linear growth and linear decay

Growth: C(d) = 100 + 60d — the journey cost rises by a fixed ₹60 per km.

Decay: h(t) = 3 – 0.5t — the water level falls by a fixed 0.5 m each month.

2.5 Linear relationships y = ax + b

Given two data points, we can find a and b. A data plan charges ₹350 for 10 GB and ₹550 for 20 GB:

350 = 10a + b and 550 = 20a + b give a = 20 and b = 150, so y = 20x + 150. The 20 is the cost per GB; the 150 is the fixed monthly fee.

2.6 Visualising linear relationships

Two points are enough to draw a line; e.g. y = 2x + 1 passes through (0, 1) and (3, 7).

y = ax always passes through the origin; a is the slope. If a > 1 the line is steeper than y = x; if a < 1 it is less steep.

Positive slope means linear growth; negative slope means linear decay.

In y = ax + b, b is the y-intercept: the line cuts the y-axis at (0, b).

Changing b with a fixed gives parallel lines (y = 2x – 1, y = 2x + 1, y = 2x + 5).

Worked Examples

Example 1 — Evaluate. 7s² – 4s + 6 at s = –3: 7(9) + 12 + 6 = 81.

Example 2 — Ages. Salil’s mother is 3 times his age; in 5 years their ages add to 70.

(3s + 5) + (s + 5) = 70 → 4s = 60 → s = 15. Salil is 15, his mother 45.

Example 3 — Linear relationship. A gym bill is ₹800 for 10 hours and ₹1100 for 15 hours.

Subtracting: 5a = 300 → a = 60; then b = 800 – 600 = 200. So y = 60x + 200.

Example 4 — Line through two points. p(x) passes through (1, 5) and (3, 11).

Slope = (11 – 5)/(3 – 1) = 3, so p(x) = 3x + b; 5 = 3 + b → b = 2. p(x) = 3x + 2, cutting the y-axis at (0, 2) and the x-axis at (–2/3, 0).

Example 5 — Kelvin to Fahrenheit. y = (9/5)(x – 273) + 32. At 313 K: (9/5)(40) + 32 = 104 °F. For 158 °F: (9/5)(x – 273) = 126 → x – 273 = 70 → 343 K.

Figure it Out — Hints

Hexagon matchsticks: the first hexagon needs 6 sticks and each new one adds 5, giving 5n + 1 sticks. Stage 15 needs 76. 200 sticks is impossible, since 199 is not a multiple of 5.

Parallel lines: 2y = 4x + 7 and 3y = 6x – 11 both have slope 2 after dividing, so they are parallel.

Two-digit numbers: if the digits are a and b, the number plus its reverse is 11(a + b) = 143, so a + b = 13; with the digits differing by 3, the numbers are 85 and 58.

f(x) = ax + a: every such line passes through (–1, 0).

Maths Words

Variable (चर)

Coefficient (गुणांक)

Constant term (अचर पद)

Polynomial (बहुपद)

Degree (घात)

Linear polynomial (रैखिक बहुपद)

Quadratic / cubic (द्विघात / त्रिघात)

Linear equation (रैखिक समीकरण)

Linear pattern (रैखिक प्रतिरूप)

Linear growth / decay (रैखिक वृद्धि / ह्रास)

Slope (ढाल / प्रवणता)

y-intercept (y-अंतःखंड)

Function (फलन)

Common Mistakes

Calling 10x – x² linear — it has degree 2, so it is quadratic.

Thinking a constant like 8 has no degree — its degree is 0.

Mixing up slope and intercept in y = ax + b: a is the slope, b is the y-intercept.

Using the auto-fare rule 15n – 5 for n < 2 — the first 2 km are a flat ₹25.

Forgetting to divide through before reading the slope, e.g. 2y = 4x + 7 has slope 2, not 4.

Maths in Daily Life

Taxi and auto fares, phone and data plans, gym memberships, savings that grow by a fixed amount each month, the depreciating value of a phone, falling water levels in summer, and temperature conversion are all linear relationships.

💡 Key Learning Points

✓Identify terms, variables, coefficients, constants and the degree of a one-variable polynomial.
✓Recognise linear polynomials by their constant difference between successive integer values.
✓Model linear patterns, linear growth and linear decay with expressions like 2n – 1, 100 + 60d and 3 – 0.5t.
✓Find a linear relationship y = ax + b from two data points and interpret a and b in context.
✓Draw straight-line graphs and read the slope a and y-intercept b; lines with equal slopes are parallel.

👨‍🏫 Teaching Tips

→Start with Raju’s pen and pencil boxes to name terms, variables, coefficients and constants before any definitions.
→Build the square-tile pattern with real tiles or counters and let students discover 2n – 1 for themselves.
→Use an ‘input–output machine’ card game: one student gives x, another computes 2x + 3.
→Collect a real phone or taxi tariff from students’ families and fit y = ax + b to two data points.
→Plot y = ½x, y = x, y = 2x on one sheet to let students see how the slope controls steepness.
→Contrast growth and decay by graphing C(d) = 100 + 60d and h(t) = 3 – 0.5t side by side.
→Have students build the matchstick hexagon pattern to test the rule 5n + 1.

📋 Assessment Questions

Can the student state the degree and name the type (constant, linear, quadratic, cubic) of a polynomial?
Can the student write a linear expression for a real situation such as the chess-club fee 200 + 50m?
Can the student solve for a and b from two data points and explain what each means?
Can the student read the slope and y-intercept from equations like 3y = 6x – 11?
Can the student distinguish linear growth from linear decay on a graph?

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