Linear Polynomials Class 9 PPT & Notes – Ganita Manjari Ch 2
Section 1 · Chapter overview
Our learning journey
- Algebraic expressions and polynomials
- Linear polynomials
- Exploring linear patterns
- Linear growth and decay
- Linear relationships (y = ax + b)
- Visualising lines on a graph
What is an algebraic expression?
Suppose x red boxes hold 4 pens each, y blue boxes hold 5 pencils each, and there are 3 free pens. The total is the expression 4x + 5y + 3.
- Terms are separated by + or − signs: 4x, 5y and 3
- Variables: x and y
- Coefficients: 4 and 5
- Constant: 3
Example: the decorated garden
A garden is fenced with wire fencing along its two lengths (Rs 100 per metre) and wooden fencing along its two widths (Rs 80 per metre), and seeds cost Rs 50 per square metre. For length ℓ m and width w m:
- Wire fencing (2 lengths) = 2ℓ × 100 = Rs 200ℓ
- Wooden fencing (2 widths) = 2w × 80 = Rs 160w
- Seeds over the area = 50 × ℓ × w = Rs 50ℓw
This expression uses two variables, ℓ and w.
Polynomials in one variable, and their degree
An expression with one variable and whole-number powers of it is a polynomial (a univariate polynomial). The degree is the highest power of the variable.
| Type | Degree | Example |
|---|---|---|
| Constant | 0 | 8 (which is 8 times x to the power 0) |
| Linear | 1 | 3z + 7 |
| Quadratic | 2 | x² + 5x + 1 |
| Cubic | 3 | 5y³ + y² + 2y − 1 |
This chapter is all about linear polynomials.
Section 2 · Linear polynomials and patterns
A linear polynomial in action
The perimeter of a square of side x is x + x + x + x = 4x, a linear polynomial (degree 1).
| Side | 1 | 1.5 | 2 | 2.5 | 3 |
|---|---|---|---|---|---|
| Perimeter | 4 | 6 | 8 | 10 | 12 |
Each time the side grows by 0.5, the perimeter grows by 2: a constant jump.
Example: the chess club fee
The joining fee is Rs 200 plus Rs 50 per match.
| Matches m | 1 | 2 | 3 | 4 | 5 | … | m |
|---|---|---|---|---|---|---|---|
| Amount (Rs) | 250 | 300 | 350 | 400 | 450 | … | 200 + 50m |
Total cost = Rs (200 + 50m), a linear polynomial in m. It adds a constant Rs 50 each match. If a player paid Rs 750, then 200 + 50m = 750, so 50m = 550 and m = 11 matches.
The secret of linear patterns
In a linear sequence the values change by a consistent (constant) difference with each step. For example, the perimeters 4, 6, 8 grow by +2 each time, and fees of Rs 250, 300, 350, 400 grow by +50 each time. Equal steps give a straight line on the number line.
From a linear polynomial to a linear equation
Equate a linear polynomial to a number and you get a linear equation. The sum of two numbers is 64 and one is 10 more than the other. Let the smaller be x, so the larger is x + 10:
The numbers are 27 and 37.
A polynomial is like a machine
Put in any x and you get exactly one output. This is called a function. For the function machine y = 2x + 3: input x = 4 gives 2 × 4 + 3 = 11, and input x = −6 gives 2 × (−6) + 3 = −9.
Section 3 · Linear growth and decay
Exploring linear patterns: the growing tile pattern
The tile pattern has 1, 3, 5, 7, 9, … tiles. The difference is always +2, so the relationship is linear. The stage number x and the number of tiles y are connected by y = 2x − 1.
- 15th stage: 2 × 15 − 1 = 29 tiles
- 26th stage: 2 × 26 − 1 = 51 tiles
- 47 tiles: 2x − 1 = 47, so x = 24, the 24th stage
Example: Bela’s pocket money (linear decay)
Bela starts with Rs 100 and spends Rs 5 each day. The amount left on day n is Rs (100 − 5n). Check: on day 12 she has 100 − 60 = Rs 40. This is a linear decay, because it falls by Rs 5 each day.
| Day | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Amount left (Rs) | 100 | 95 | 90 | 85 | 80 |
Linear growth: journey cost
The cost of a journey is C(d) = 100 + 60d (cost in Rs, distance d in km). A fixed rise of Rs 60 per km means linear growth.
| d (km) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| C (Rs) | 100 | 160 | 220 | 280 | 340 | 400 |
Cost for 15 km = 100 + 60 × 15 = Rs 1,000. The cost is Rs 700 when 100 + 60d = 700, so d = 10 km.
Linear decay: the water tank
For a cylindrical water tank, the height in metres is h(t) = 3 − 0.5t after t months. A fixed fall of 0.5 m per month means linear decay.
| t (months) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| h (m) | 3 | 2.5 | 2 | 1.5 | 1 |
Height after 5 months = 3 − 0.5 × 5 = 0.5 m.
Growth versus decay
| Linear growth | Linear decay |
|---|---|
| Increases by a constant amount each interval, e.g. 100 → 160 → 220 | Decreases by a constant amount each interval, e.g. 3 → 2.5 → 2 |
Section 4 · The linear relationship y = ax + b
What a and b mean
- In y = ax + b, x and y are two connected quantities
- a tells how fast y changes: it is the slope
- b is the starting value: it is the y-intercept
- The tile pattern is y = 2x − 1, so here a = 2 and b = −1
Example: finding a and b
10 GB of data gives a bill of Rs 350 and 20 GB gives Rs 550. Find y = ax + b.
Subtracting: 10a = 200, so a = 20
b = 350 − 200 = 150
So y = 20x + 150: a = 20 (Rs per GB) and b = 150 (the fixed monthly fee).
How to plot a line: y = 2x + 1
Pick two points. When x = 0, y = 1, so A (0, 1). When x = 3, y = 7, so B (3, 7). A point lies on the line only if its (x, y) fits the equation.
Guess the equation
- The points (0, 0), (1, 3), (3, 9) and (4, 12) lie on a line where every y is 3 times its x, so y = 3x. It passes through the origin
- The points (−3, 6), (−2, 4), (0, 0), (1, −2), (2, −4) and (3, −6) lie on a line where every y is −2 times its x, so y = −2x. A downward line means a negative relationship
Positive and negative slope
- A positive slope means linear growth: the line goes up
- A negative slope means linear decay: the line goes down
- The slope is the constant difference between consecutive terms: the tiles 1, 3, 5, 7 grow by +2 each step, so y = 2x − 1 has slope 2
- Compare y = 3x + 1 and y = −3x + 1: one rises (positive slope) and the other falls (negative slope), yet both cross the y-axis at the same point (0, 1)
Where a line cuts the y-axis
The line y = ax + b cuts the y-axis at (0, b), and b is the y-intercept. For example, y = 2x + 5 cuts at A (0, 5), y = x + 3 cuts at B (0, 3) and y = 3x − 2 cuts at C (0, −2). b = 3 means 3 units up; b = −2 means 2 units down.
Big ideas about y = ax + b
| If you… | Then… |
|---|---|
| Change b and keep a | The line slides up or down but stays parallel |
| Change a and keep b | The slope changes, and the line crosses the y-axis at the same point |
| Put b = 0 | The line y = ax passes through the origin |
Lines with the same slope a but different b are parallel. For Practice Set 2.6, draw y = 4x, 2x, x; y = −6x, −3x; y = 5x, −5x; and y = 3x − 1, 3x, 3x + 1 on graph paper, and note that a controls the steepness and direction, while b controls where the line crosses the y-axis.
Section 5 · Practice sets and exercises with solutions
Practice Set 2.2
| Question | Solution |
|---|---|
| Find the value of 5x − 3 if (i) x = 0, (ii) x = −1, (iii) x = 2 | (i) −3, (ii) 5(−1) − 3 = −8, (iii) 5(2) − 3 = 7 |
| Salil’s mother’s age is three times Salil’s age. After 5 years their ages add up to 70. Find their present ages. | Let Salil’s age be x, so his mother’s is 3x. (x + 5) + (3x + 5) = 70, so 4x = 60 and x = 15. Salil is 15 and his mother is 45 |
| The difference between two positive integers is 63 and their ratio is 2 : 5. Find them. | Let them be 2k and 5k. 5k − 2k = 63, so k = 21. The integers are 42 and 105 |
| Ruby has 3 times as many two-rupee coins as five-rupee coins, totalling Rs 88. How many of each? | Let there be x five-rupee coins and 3x two-rupee coins. 5x + 2(3x) = 88, so 11x = 88 and x = 8. She has 8 five-rupee coins and 24 two-rupee coins |
| A 300 ft fence is cut into two unequal pieces; the longer is four times the shorter. Find both. | x + 4x = 300, so x = 60. The pieces are 60 ft and 240 ft |
| The length of a rectangle is three more than twice its width, and its perimeter is 24 cm. Find its dimensions. | Length = 2w + 3. Perimeter 2(length + width) = 24, so 3w + 3 = 12 and w = 3. Width = 3 cm, length = 9 cm |
Practice Set 2.3: linear patterns
| Question | Solution |
|---|---|
| A student has Rs 500 and gets Rs 150 pocket money each month. Find a linear expression for the nth month. | Amount after n months = 500 + 150n |
| A rally starts with 120 members and 9 drop out each hour. Find the members at the end of the nth hour. | 120 − 9n |
| A rectangle has length 13 cm. Find the area for breadth 12, 10 and 8 cm, and the linear pattern. | Areas: 156, 130 and 104 cm². The pattern is 13b |
| A box has length 7 cm and breadth 11 cm. Find the volume for height 5, 9 and 13 cm, and the pattern. | Volumes: 385, 693 and 1001 cm³. The pattern is 77h |
Practice Set 2.4: growth and decay
| Question | Solution |
|---|---|
| A plant is 1.75 ft tall and grows 0.5 ft each month. Find the height after 7 months and an expression for h and t. | h = 1.75 + 0.5t. After 7 months: 1.75 + 3.5 = 5.25 ft. It is linear growth because it rises by a constant 0.5 ft each month |
| A phone bought for Rs 10,000 loses Rs 800 each year. Find its value after 3 years and an expression for v and t. | v = 10000 − 800t. After 3 years: Rs 7,600. It is linear decay because it falls by a constant Rs 800 each year |
| A village of 750 people gains 50 newcomers each year. Find the population after 6 years and an expression for P and t. | P = 750 + 50t. After 6 years: 1,050. It is linear growth |
| A Rs 600 recharge drops Rs 15 each day. Write b(x). | b(x) = 600 − 15x, where x is the number of days. It is linear decay |
Practice Set 2.5: finding a and b
| Question | Solution |
|---|---|
| 10 modules cost Rs 400 and 14 modules cost Rs 500. Find a and b in y = ax + b. | Subtracting 400 = 10a + b from 500 = 14a + b gives 4a = 100, so a = 25 and b = 150. y = 25x + 150 |
| A gym: 10 hours cost Rs 800 and 15 hours cost Rs 1100. Find a and b. | Subtracting gives 5a = 300, so a = 60 and b = 200. y = 60x + 200 |
| C = aF + b with 0 °C = 32 °F and 100 °C = 212 °F. Find a and b. | 0 = 32a + b and 100 = 212a + b. Subtracting gives 180a = 100, so a = 5/9 and b = −160/9 |
NCERT exercises
| Question | Solution |
|---|---|
| 1. Write a polynomial of degree 3 in x in which the coefficient of x² is −7. | For example x³ − 7x² + 2x + 1 |
| 2. Find 5x² − 3x + 7 if x = 1, and 4t³ − t² + 6 if t = a. | 5 − 3 + 7 = 9; and 4a³ − a² + 6 |
| 3. A number is multiplied by 5/2 and 2/3 is added to the product, giving −7/−12. Find the number. | (5/2)x + 2/3 = 7/12, so (5/2)x = −1/12 and x = −1/30 |
| 4. A positive number is 5 times another. If 21 is added to both, one new number becomes twice the other. Find the numbers. | 5x + 21 = 2(x + 21), so 3x = 21 and x = 7. The numbers are 7 and 35 |
| 5. Rs 800 and Rs 250 saved each month: the amount after (i) 6 months, (ii) 2 years. Express it as a linear pattern. | Amount = 800 + 250n. (i) 800 + 1500 = Rs 2,300; (ii) 24 months: 800 + 6000 = Rs 6,800 |
| 6. The digits of a two-digit number differ by 3. When the digits are interchanged and the new number is added to the original, we get 143. Find both numbers. | With digits a and b, the sum is 11(a + b) = 143, so a + b = 13. With a − b = 3, the digits are 8 and 5. The numbers are 85 and 58 |
| 7. Find the slopes and y-intercepts of (i) y = −3x + 4, (ii) 2y = 4x + 7, (iii) 5y = 6x − 10, (iv) 3y = 6x − 11. Are any parallel? | (i) slope −3, cuts the y-axis at (0, 4). (ii) y = 2x + 3.5: slope 2, (0, 3.5). (iii) y = 1.2x − 2: slope 6/5, (0, −2). (iv) y = 2x − 11/3: slope 2, (0, −11/3). (ii) and (iv) are parallel, as both have slope 2 |
| 8. y = (9/5)(x − 273) + 32 gives °F (y) from Kelvin (x). Find y when x = 313 K, and x when y = 158 °F. | (9/5)(40) + 32 = 104 °F. For 158: (9/5)(x − 273) = 126, so x − 273 = 70 and x = 343 K |
| 9. Work = constant force × distance. Write it as a linear equation with force 3 units, and find the work when the distance is 2 units. | w = 3d, a straight line through the origin. For d = 2, w = 6 |
| 10. The graph of a linear polynomial p(x) passes through (1, 5) and (3, 11). Find p(x) and where its graph cuts the axes. | Slope = (11 − 5)/(3 − 1) = 3, so p(x) = 3x + b with 5 = 3 + b, giving b = 2. p(x) = 3x + 2. It cuts the y-axis at (0, 2) and the x-axis at (−2/3, 0) |
| 11. p(x) = ax + b and q(x) = cx + d with p(0) = 5; p(x) − q(x) cuts the x-axis at (3, 0); and p(x) + q(x) = 6x + 4. Find p(x) and q(x). | p(0) = 5 gives b = 5. From p + q = 6x + 4: a + c = 6 and b + d = 4, so d = −1. p − q = (a − c)x + 6 is zero at x = 3, so a − c = −2. Then a = 2 and c = 4. p(x) = 2x + 5 and q(x) = 4x − 1 |
| 13. p(x) passes through (2, 3) and (6, 11); q(x) passes through (4, −1) and is parallel to p(x). Find p(x), q(x) and where each meets the x-axis. | Slope of p = (11 − 3)/(6 − 2) = 2, so p(x) = 2x − 1. q has the same slope and passes through (4, −1), so q(x) = 2x − 9. p meets the x-axis at (1/2, 0) and q at (9/2, 0) |
| 14. What do all linear functions f(x) = ax + a with a greater than 0 have in common? | f(x) = a(x + 1), so every such line passes through the point (−1, 0), and all of them have a positive slope |
Section 6 · Chapter summary
- An algebraic expression has terms, variables, coefficients and a constant
- A univariate polynomial has one variable, and its degree is the highest power
- A linear pattern has a constant difference between consecutive terms, e.g. 2, 5, 8, 11, …
- A linear relationship is y = ax + b, which is a straight line; a is the slope and b is the y-intercept, the point (0, b) where the line meets the y-axis
- When b = 0, the line y = ax passes through the origin
- A positive slope means linear growth and a negative slope means linear decay
- Lines with the same slope a but different b are parallel
Section 7 · Practice: multiple choice questions
Tap an option to check it at once. Open Check answer for the reason.
1In the expression 4x + 5y + 3, the constant term is: [1]
Check answer
Terms are 4x, 5y and 3. The coefficients are 4 and 5, and the constant is 3.
2The degree of the polynomial 2x² − 5x + 3 is: [1]
Check answer
The degree is the highest power of the variable, which is 2.
3The degree of the constant polynomial −9 is: [1]
Check answer
A constant can be written as −9 times x to the power 0, so its degree is 0.
4A polynomial of degree 1 is called: [1]
Check answer
For example 3z + 7 is a linear polynomial.
5In the decorated garden example, the total cost has how many variables? [1]
Check answer
Total cost = 200ℓ + 160w + 50ℓw uses the length ℓ and the width w.
6The perimeter of a square of side x is: [1]
Check answer
x + x + x + x = 4x, which has degree 1.
7In a linear pattern, the difference between consecutive terms is: [1]
Check answer
For example 4, 6, 8, 10 always grows by 2.
8The chess club fee is Rs (200 + 50m). A player who paid Rs 750 played how many matches? [1]
Check answer
200 + 50m = 750 gives 50m = 550, so m = 11.
9If x + (x + 10) = 64, the two numbers are: [1]
Check answer
2x + 10 = 64, so x = 27 and the other number is 37.
10For the function y = 2x + 3, the output for x = 4 is: [1]
Check answer
2 × 4 + 3 = 11.
11The number of tiles in the nth stage of the pattern 1, 3, 5, 7, … is: [1]
Check answer
The difference is 2, and the first term is 1, so y = 2n − 1.
12The number of tiles in the 15th stage of the same pattern is: [1]
Check answer
2 × 15 − 1 = 29.
13Bela has Rs 100 and spends Rs 5 each day. Her amount left on day n is: [1]
Check answer
On day 12 she has 100 − 60 = Rs 40.
14If the journey cost is C(d) = 100 + 60d, the cost for 15 km is: [1]
Check answer
100 + 60 × 15 = 100 + 900 = 1,000.
15A quantity that falls by a fixed amount at each step shows: [1]
Check answer
A fixed fall, as in a water tank losing 0.5 m a month, is linear decay.
16In y = ax + b, the letter a is the: [1]
Check answer
a tells how fast y changes, and b is the y-intercept.
17The line y = ax + b cuts the y-axis at: [1]
Check answer
Put x = 0, then y = b.
18The line y = 3x passes through: [1]
Check answer
When x = 0, y = 0, so it passes through (0, 0).
19The lines y = 2x + 5 and y = 2x − 3 are: [1]
Check answer
They have the same slope 2 but different values of b.
20The line y = −3x has: [1]
Check answer
The slope is −3, so y falls as x increases.
21Assertion (A): The tile pattern 1, 3, 5, 7, … is described by y = 2x − 1, a linear relationship.
Reason (R): The difference between consecutive terms of the pattern is constant (+2). [1]
Check answer
Both statements are true. A constant difference of 2 gives a linear relationship with slope 2, so R explains A.
22Assertion (A): Changing b in y = ax + b, with a kept the same, changes the slope of the line.
Reason (R): Changing b slides the line up or down so that it stays parallel. [1]
Check answer
A is false: the slope is a, and it does not change when b changes. R is true.
Section 8 · Practice: short answers
Q1Define the terms, variables, coefficients and constant in 4x + 5y + 3. [3]
Check answer
Q2What is the degree of a polynomial? Name the polynomials of degrees 0, 1, 2 and 3 with examples. [3]
Check answer
Q3Show that the perimeter of a square is a linear polynomial in its side. [2]
Check answer
Q4A chess club charges a joining fee of Rs 200 plus Rs 50 per match. Write the cost for m matches and find m if a player paid Rs 750. [3]
Check answer
Q5The sum of two numbers is 64 and one is 10 more than the other. Find them. [3]
Check answer
Q6For the pattern 1, 3, 5, 7, … find the rule, the 15th and 26th terms, and the stage with 47 tiles. [4]
Check answer
Q7Bela has Rs 100 and spends Rs 5 each day. Write the amount left on day n, and say whether it is growth or decay. [3]
Check answer
Q8Differentiate between linear growth and linear decay. [3]
Check answer
Q9What do a and b mean in y = ax + b? [3]
Check answer
Q10Ten GB of data costs Rs 350 and 20 GB costs Rs 550. Find y = ax + b. [3]
Check answer
Q11Without drawing, find the slope and y-intercept of 2y = 4x + 7, and say which line from y = −3x + 4, 5y = 6x − 10 and 3y = 6x − 11 is parallel to it. [4]
Check answer
Q12The graph of a linear polynomial passes through (1, 5) and (3, 11). Find the polynomial and where it cuts the axes. [4]
Check answer
Q13Why does a line with a negative slope show linear decay? [2]
Check answer
Q14What do all the lines f(x) = ax + a with a greater than 0 have in common? [3]
Check answer
Section 9 · Frequently asked questions
What is a linear polynomial?
What is the difference between a linear polynomial and a linear equation?
What do slope and y-intercept mean in y = ax + b?
How do I find a and b from two points?
When are two lines parallel?
Where can I download the Linear Polynomials Class 9 PPT?
More from Class 9 Maths
Browse all free Class 9 notes, PPTs and worksheets on the Class IX Maths page.
Save it on your phone and study offline. Free, no login needed.
⬇ Download PDFMore study material: Class IX Maths
- 🧩 Linear Polynomials Class 9 Worksheet Ch 2 (PDF+Ans)Worksheet
- 🖥️ Use of Coordinates Class 9 PPT & Notes – Ganita Manjari Ch 1PPT Slides
- 🖥️ World of Numbers Class 9 PPT & Notes – Ganita Manjari Ch 3PPT Slides
- 🖥️ Algebraic Identities Class 9 PPT Notes PDF – Ganita ManjariPPT Slides
- 📝 Propositions & Converses Class 9 Maths Ch 9 Notes + PPTNotes
- 🧩 Sequences and Progressions Class 9 Worksheet Ch 8Worksheet