# Slope of a Line

** Slope of a Line **is the measure of the steepness of a line or a surface or a curve whichever is the point of consideration. The slope of a Line is a fundamental concept in the stream of

**or**

**calculus****or we can say the slope of a line is fundamental to the complete mathematics subject. The understanding of slope helps us solve many problems in mathematics, physics, or engineering.**

**coordinate geometry**In this article, we will learn about the** slope of a line** in detail,

**with its various methods of calculations, and also the**

**slope of a straight line**

**equation for the slope of a line.****Table of Content**

- What is a Slope?
- What is the Slope of a Line?
- Slope of a Line Formula
- Slope of a Line Equation
- How to find the Slope of a Line?
- Positive and Negative Slope
- Slope of Horizontal Line
- Slope of Vertical Line
- Slope of Perpendicular Lines
- Slope of Parallel Lines
- Equation of Line in Slope Intercept Form
- Equation of Line Using Slope
- Equation of Line in Normal Form
- Angle between Two Lines
- Sample Problems
- FAQs

## What is a Slope?

In mathematics, the slope is the measure of the steepness of a line or a surface which means the slope of a line tells us how steep a line or surface is inclined with the positive x-axis. The higher the slope of a curve or a line, the greater the amount of steepness it will show in the graph. Generally, the slope is defined in two-dimensional coordinates as the ratio of the change in the y-coordinate with respect to the change in the x-coordinate.

## What is the Slope of a Line?

Similar to the general slope, ** the slope of a line or the slope of a straight line** is the measure of the steepness of a line and is mathematically given as the ratio of the change in the y-coordinate to the change in the x-coordinate of the line. Other than this mathematical interpretation, the slope has a physical interpretation that tells us about the line that is in which direction the line will tilt. A higher positive slope shows less tilt in the positive x direction and a lower positive slope shows a higher tilt in the positive x direction. For the negative slope, the higher the absolute value of the negative slope higher the tilt towards the negative x direction.

## Slope of a Line Formula

The formula of the slope is given by the ratio of change in the y coordinate to the change in the x coordinate, which can be written mathematically can as

Slope = Change in y coordinate/Change in x coordinate = Î”y/Î”x

If the inclination of the line with a positive x-axis is Î¸, then the slope is given as follows:

m = tan Î¸

## Slope of a Line Equation

The slope of a line is given as

y – y_{1}= m(x – x_{1})

â‡’ y = mx + C where m is slope and C is intercept

## How to find the Slope of a Line?

There are various methods to find the slope of any line, which can be used appropriately on the basis of the given conditions. These methods are:

- Calculation of Slope between Two Points
- Calculation of Slope from Graph
- Calculation of Slope from Table

### Calculation of Slope between Two Points

If the given points are (x_{1}, y_{1}) and (x_{2}, y_{2}) then the slope of a line passing through both points is given by:

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

**Answer:**

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

Coordinate x

_{1}and y_{1}is 4 and 2Coordinate x

_{2}and y_{2}is 8 and 12Thus,

â‡’ m = (12 – 2)/(8 – 4)

â‡’ m = 10/4 = 2.5

### Calculation of Slopes from Graph

Calculation of the slope from a Graph can be achieved using the following steps:

Mark two points on the line with their coordinates.Step 1:

Use the Formula for the Slope between two points to calculate the slope.Step 2:

**Example: Find the slope of the following line in the graph.**

**Solution:**

We have to find Î”x and Î”y (change in x and change in y)

So, change in Î”x is 6 and change is Î”y -3

Now slope m is given as follows:

m = Î”y/Î”x

â‡’ m = 6/-3 = -2

**Calculation of Slope from Table**

**Calculation of Slope from Table**

Calculation of the Slope from the Table can be done using the following steps:

Choose two values of x and its corresponding values of y from the table.Step 1:

Calculate the change in x value and change in y value.Step 2:

Calculate the slope using the formula,Step 3:Slope = change in y-values/change in x-values.

**Example 1: Calculate the slope between x = 1 and x = 3 of the following table.**

x-value |
y-value |
---|---|

1 |
5 |

2 |
7 |

3 |
9 |

4 |
11 |

5 |
13 |

**Answer:**

Change in x-values = 3 – 1 = 2

Change in y-values = 9 – 5 = 4

As we know,

Slope = change in y-values/change in x-valuesâ‡’ Slope = 4/2 = 2

So, the slope between x=1 and x=3 in this table is 2.

**Example 2: Calculate the slope of the following table.**

x |
y |
---|---|

2 | 5 |

3 | 10 |

4 | 15 |

5 | 20 |

**Solution:**

Identify change in each consecutive pair of y so change in the y is 5, 5 and 5.

Identify change in each consecutive pair of x so change in the x is 1, 1 and 1.

Now writing the ratio using slope formula

5/1, 5/1 and 5/1.So, slope from table is 5.

## Positive and Negative Slope

A line is said to have a ** positive slope** if it is making less than the right angle with the positive x-axis and a line is said to have a

**if it makes more than the right angle with the positive x-axis. In other words, a line with a positive slope looks tilted forward in the direction of the positive x-axis, and a line with a negative slope looks tilted backward in the direction of the negative y-axis.**

**negative slope**## Slopes of Different Lines

There can be various different lines that can be named such as:

- Horizontal Line
- Vertical Line
- Perpendicular Lines
- Parallel Lines

Let’s discuss the Slope of these various different lines as follows:

## Slope of Horizontal Line

The line that is parallel to the x-axis is called a horizontal line and the slope of a Horizontal Line is 0 as there is no change in the y-coordinate throughout the line for any change in the x-coordinate. Since the slope of the Horizontal Line is Zero, it is also called a Zero Slope line. Thus mathematically we can represent this as:

Slope of Horizontal Line = 0/change in x-coordinate = 0

## Slope of Vertical Line

The line parallel to the y-axis is called a vertical line and the slope of a Vertical Line is not defined as there is no change in the x-coordinate throughout the line for any change in the y-coordinate. The Vertical Line is also called Undefined Slope Line. Thus mathematically we can represent this as:

Slope of Vertical Line = Change in y-coordinate/0 = Not Defined

## Slope of Perpendicular Lines

The slope of Perpendicular Lines are inversely proportional to each other and their product is -1. In other words, if we have two lines with slopes m_{1} and m_{2}, then the condition for those two lines to be perpendicular is:

m_{1 }= -1/m_{2 }OR

m_{1}Ã— m_{2 }= -1

## Slope of Parallel Lines

The slope of Parallel Lines is the same as both the lines are at the same inclined with the positive x-axis. In other words, if the slope of one line is m then the slope of a line parallel to that line is also m.

## Equation of Line in Slope Intercept Form

The equation of a line in the slope-intercept form is given as follows:

y = mx + cWhere,

is the slope of the line, andmis the y-intercept cut by the line.c

## Equation of Line Using Slope

If a line is passing through a point (x_{1}, y_{1}) and its slope is m, then the equation of a line is given as follows:

y – y_{1}= m(x – x_{1})Where x and y represent all the coordinates of the line.

We can also write the same equation using the two points from which the line is passing. If the line passes through (x_{1}, y_{1}) and (x_{2}, y_{2}) then its equation is given by:

**Example 1: Find the equation of a line given in the graph.**

**Answer:**

Slope of the graph is, m = 8/2 = 4

and we know the equation of line passing through (x

_{1}, y_{1}) with slope m is given by

y â€“ y_{1}= m (x â€“ x_{1})Thus, equation of line (4,2) with slope 4 is

y â€“ 2 = 4 (x â€“ 4)

â‡’ y â€“ 2 = 4x â€“ 16

â‡’ y = 4x â€“ 16 +2

â‡’ y = 4x â€“ 14

**Example 2: Find the equation of the line given in the graph.**

**Answer:**

Two given points (x

_{1}, y_{1}) and (x_{2}, y_{2}) are A (2,3) and B (5,7)â‡’ y-3= {(7 -3)/(5-2)} (x-2)

â‡’

â‡’

â‡’ 3y-9 = 4x-8

â‡’ 3y = 4x+1

## Equation of Line in Normal Form

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 Î± is given by:

x cos Î± + y sin Î± = pThis is known as the

.normal form of the line

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

From this we can say that and

Also, it can be inferred that,

â‡’

â‡’

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 â‰ 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:

- If two points (x
_{1}, y_{1}) and (x_{2}, y_{2}) are said to lie on the same side of the line Ax + By + C = 0, then the expressions Ax_{1}+ By_{1}+ C and Ax_{2}+ By_{2}+ C will have the same sign or else these points would lie on the opposite sides of the line.

## Angle between Two Lines

When two lines with slopes m_{1} and m_{2 }intersect they form angles. The relation between the slope of the intersecting lines and the angle so formed is given as follows:

tanÎ¸ =

if > 0, acute angle is formed between the lines

if < 0, obtuse angle is formed between the lines

**Read More,**

## Sample Problems on Slope of a Line

**Problem 1: Find the slope of points (1,2) and (2,3).**

**Solution:**

As slope is given as m = (y

_{2}– y_{1})/(x_{2}– x_{1})â‡’ m = (3 – 2)/(2 – 1)

â‡’ m = 1

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

**Solution:**

m = (y

_{2}– y_{1})/(x_{2}– x_{1})â‡’ 2 = (6 – 2)/(x – 2)

â‡’ 4 = 2(x-2)

â‡’ x-2 = 2

â‡’ x = 4

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

**Solution:**

m = (y

_{2}– y_{1})/(x_{2}– x_{1})â‡’ 3 = (y – 13)/(4 – 2)

â‡’ y – 13 = 3(2)

â‡’ y – 13 = 6

â‡’ y = 6 + 13 = 19

**Problem 4: Find the line passing from coordinates (2,5) and the slope of a line is 5.**

**Solution:**

Slope m = 4

y â€“ y

_{1}= m (x â€“ x_{1})We know slope m = 5 and point (x

_{1}, y_{1}) = (2,5)Now putting these value in equation

â‡’ y â€“ 5 = 5 x (x â€“ 2)

â‡’ y â€“ 5 = 5x â€“ 10

â‡’ y = 5x â€“ 10 + 5

â‡’ y = 5x -5

## FAQs on Slope of a Line

### 1. What is the Slope of a Line?

Slope is the measure of the steepness of a line which tells us how much a line is inclined with the positive x-axis.

### 2. What is the slope of a straight line?

Slope of a straight line is same as slope of a line. As a line is always a straight line.

### 3. How is the Slope of a Line calculated?

To calculate the slope of a line, we just need to take two point on the line and calculate the ratio of change in y-coordinate to change in x-coordinate.

### 4. What does a Positive and Negative Slope represent?

Positive Slope represents that graph is increasing in nature while Negative Slope represents that in graph line is going down from left to right.

### 5. Can the Slope of a Line be greater than 1?

Yes slope of a line can be greater than 1, as one of the formula for is slope m = tan Î¸, and range of tan Î¸ is the complete set of real numbers.

### 6. What is the Slope-Intercept form of a Line?

The slope-intercept form of a line is given as follows:

y = mx + cWhere,

is the slope of the line, andmis the y-intercept cut by the line.c

### 7. What is the Point-Slope form of a Line?

The point slope form is called point slope form, as it is defined as a line is passing through a point (x

_{1}, y_{1}) and its slope is m, and is given by:

y – y_{1}= m(x – x_{1})

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