Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (2023)

In mathematics, the slope or gradient of a line is a number that describes both the direction X and Y and the steepness of the line. Slope is denoted by ‘m‘. Slope is calculated by finding the ratio of the “vertical change” to the “horizontal change” between any two distinct points on a line. Sometimes the ratio is expressed as a quotient giving the same number for every two distinct points on the same line.

  • A line if it is increasing and it goes up from left to right. The slope is positive, m > 0.
  • A line if it is decreasing and it goes down from left to right. The slope is negative, m < 0.
  • If the line is horizontal the slope is zero and it is a constant function.
  • If a line is vertical, then the slope is undefined.

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (1)

In mathematical language, the slope m of the line is

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (2)

The slope m of the line is related to its angle of incline θ by the tangent function, m = tan(θ)

Thus, a 45-degree rising line has a slope of +1 and a 45-degree falling line has a slope of -1.

Slope Intercept Form

The linear equation written in the form as

y = mx + c

In the slope-intercept form where:

M is the slope and c is the y-intercept

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (3)

Examples of Linear equation in slope-intercept form:

y = 2x + 2

y = 7x – 5

y = 12x + 9

The Coefficients in Slope-Intercept Form

Slope-intercept form’s advantage is that it gives two main features of the line:

  • The slope is m
  • The y-coordinate of the y-intercept is b. the y-intercept of the line at (0,b)

For example: a line y = 3x + 1 has slope m as 3 and y – intercept at (0,1)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (4)

This form gives slope (m) and the y-intercept is the reason why it is called as slope-intercept form.

One Point-Slope Form

If you know the slope m of line and one point (x1, y1) at the line then you can write the equation as y – y1 = m (x – x1) of the line in slope form.

y – y1 = m (x – x1)

This equation is useful, when we know one point on the line and the slope of the line (m) and want to get the other points on the line.

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (5)

This equation is useful, when we know one point on the line and the slope of the line (m) and want to get the other points on line.

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (6)

Starting with slope,

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (7)

Rearranging like this y – y1 = m (x-x1)

For example:

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (8)

Slope,m = 8/2 = 4

(Video) Check if points are in a Straight Line [LeetCode 1232] | Math | Approach and Intuition

y – y1 = m (x – x1)

We know slope m = 4 and point (x1, y1) = (4,2)

Now putting these value in equation

y – 2 = 4 (x – 4)

So; y – 2 = 4x – 16

y = 4x – 16 +2

y = 4x – 14

Two Point-Slope Form

Two-point which is given and goes through two-point, says (x1, y1) and (x2, y2). We call this two-point form of the equation of the line.

And two-point form equation given as

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (9)

This equation used when there is two point on line.

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (10)

Since point A, B and C lies on same line

Slope of AC = slope AB

Now using the two-point slope formula:

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (11)

Multiplying both side by (x – x1)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (12)

Example:

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (13)

Two given points (x1, y1) and (x2, y2) are A (2,3) and B (5,7)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (14)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (15)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (16)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (17)

3y-9 = 4x-8

3y = 4x+1

Normal Form of Slope

The equation of the line whose length of the perpendicular from the origin is p and the angle made by the perpendicular with the positive x-axis is given by α is given by:

x cos α + y sin α = p

This is known as the normal form of the line.

In case of the general form of the line Ax + By + C = 0 can be represented in normal form as:

(Video) 0105NLCktAnalysis

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (18)

From this we can say thatSlope of a Straight Line | Class 11 Maths - GeeksforGeeks (19)andSlope of a Straight Line | Class 11 Maths - GeeksforGeeks (20)

Also, it can be inferred that,

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (21)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (22)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (23)

From the general equation of a straight line Ax + By + C = 0, we can conclude the following:

• The slope is given by -A/B, given that B ≠ 0.

• The x-intercept is given by -C/A and the y-intercept is given by -C/B.

• It can be seen from the above discussion that:

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (24)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (25)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (26)

• If two points (x1, y1) and (x2, y2) are said to lie on the same side of the line Ax + By + C = 0, then the expressions Ax1+ By1 + C and Ax2 + By2 + C will have the same sign or else these points would lie on the opposite sides of the line.

Calculating Slope From Graph

For calculating the slope, the formula is given as:

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (27)

Where,

m is the slope of the line.

x1, x2 are the coordinates of x-axis, and

y1, y2 are the coordinates of y-axis.

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (28)

Example 1:

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (29)

Solution:

We have to find Δx and Δy (change in x and change in y)

So change in Δx is 6 and change is Δy -3

Now slope m is

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (30)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (31)

Graphing a Line using given Point and Slope:

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (32)

(Video) Finding the gradient and and equation of a line given two points

  1. Plot the given point.
  2. Use the slope formula to identify the rise and the run.
  3. Starting at the given point, count out the rise and run to mark the second point.
  4. Connect the points with a line.

Example 2: Graph the line passing through the point (1, -2) and slope m isSlope of a Straight Line | Class 11 Maths - GeeksforGeeks (33)

Solution:

Plot the given point (1, -2)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (34)

Now, use the slope formula to identify the rise and run

We have given slope m isSlope of a Straight Line | Class 11 Maths - GeeksforGeeks (35)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (36)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (37)

So, rise is equal to 3 and run is equal to 4

Starting at the point we plotted, count out the rise and run to mark the second point. We count 3 units up and 4 unit’s right.

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (38)

Then we connect the points with a line draw arrow at the ends to show it continues.

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (39)

We can check our line by starting at any point and counting up 3 and to the right 4. We should get to another point on the line.

Calculating Slope from Tables

For calculating the slope,

  1. Identify the change in each consecutive pair of Y – values in the table.
  2. Identify the change in each consecutive pair of X – values in the table.
  3. Write ratio showing this corresponding vertical change and the horizontal change.

Example:

xy
25
310
415
520

Solution:

  1. Identify change in each consecutive pair of y so change in the y is 5, 5 and 5.
  2. Identify change in each consecutive pair of x so change in the x is 1, 1 and 1.
  3. Now writing the ratio using slope formulaSlope of a Straight Line | Class 11 Maths - GeeksforGeeks (40),Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (41)andSlope of a Straight Line | Class 11 Maths - GeeksforGeeks (42).
  4. Simplifying each value we getSlope of a Straight Line | Class 11 Maths - GeeksforGeeks (43),Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (44)andSlope of a Straight Line | Class 11 Maths - GeeksforGeeks (45). So slope from table isSlope of a Straight Line | Class 11 Maths - GeeksforGeeks (46).

Example: Find the point of slope if points are (4, 2) and (8, 12).

Solution:

Given

Point A (4,2) and point B (8,12)

Coordinate x1 and y1 is 4 and 2

Coordinate x2 and y2 is 8 and 12

Now using slope formula for given two point

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (47)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (48)

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (49)

SLOPE REVIEW:

What is slope?

Slope is measure as the steepness of a line.

Slope of a Straight Line | Class 11 Maths - GeeksforGeeks (50)Change in y and change in x.

(Video) Check If It Is a Straight Line (LeetCode May Challenge Day 8) | Programming Tutorials

Sample Problems

Question 1. Find the slope of points (1,2) and (2,3).

Solution:

m = (y2 – y1)/(x2 – x1)

= (3 – 2)/(2 – 1)

= 1

Question 2. Find the value of x if slope is 2 and points are (2,2) and (x,6).

Solution:

m = (y2 – y1)/(x2 – x1)

= (6 – 2)/(x – 2)

4 = 2(x-2)

x-2 = 2

x = 4

Question 3. Find the value of y if slope is 3 and points are (2,13) and (4,y).

Solution:

m = (y2 – y1)/(x2 – x1)

= (y – 13)/(4 – 2)

y – 13 = 3(2)

y – 13 = 6

y = 6 + 13 = 19

Question 4. Find the line passing from coordinates (2,5) and slope of line is 5.

Solution:

Slope m = 4

y – y1= m (x – x1)

We know slope m = 5 and point (x1, y1) = (2,5)

Now putting these value in equation

y – 5 = 5 x (x – 2)

So; y – 5 = 5x – 10

y = 5x – 10 + 5

y = 5x -5


My Personal Notesarrow_drop_up

(Video) Mathematics 2: Calculus

Videos

1. #12-Equation of Straight Line using Determinant | Class 11 Maths | IIT JEE Mains and Advanced |
(Sushmit Pandey)
2. How to find image of Equation of a st line by a vector
(Rabih Khater)
3. REDUCTION OF GENERAL EQUATION OF STRAIGHT LINE INTO INTERCEPT FORM AND SLOPE FORM
(gpc alwar)
4. Maths Straight Lines part 5 (Forms of equation of lines- Horizontal,vertical) CBSE class 11
(LearnoHub - Class 11, 12)
5. Practice 9 1c Determine whether given point lies on the Straight Line
(anba1250)
6. Leetcode 1232: Check If It Is a Straight Line
(Algorithms Casts)
Top Articles
Latest Posts
Article information

Author: Moshe Kshlerin

Last Updated: 12/27/2022

Views: 5965

Rating: 4.7 / 5 (57 voted)

Reviews: 80% of readers found this page helpful

Author information

Name: Moshe Kshlerin

Birthday: 1994-01-25

Address: Suite 609 315 Lupita Unions, Ronnieburgh, MI 62697

Phone: +2424755286529

Job: District Education Designer

Hobby: Yoga, Gunsmithing, Singing, 3D printing, Nordic skating, Soapmaking, Juggling

Introduction: My name is Moshe Kshlerin, I am a gleaming, attractive, outstanding, pleasant, delightful, outstanding, famous person who loves writing and wants to share my knowledge and understanding with you.