Slope Formula

What is the Slope?

The slope of a line calculates the "steepness" of a line. It isusually denoted by the letter m. It is the change in y divide by the change in x along the line.

The slope of a line shows how slant the line is. is the rise over the run, that is, how much the line rises vertically compared with how much it runs horizontally. Being able to find the slope of a line, or using the slope to find points on the line, is an important skill used in economics, geoscience, accounting/finance and other fields.Slope of a line is also defined as the ratio of rise over run 


Slope = rise/run


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The x and y coordinates of the lines are used to calculate the slope of the lines. It is the ratio of the change in the y-axis to the change in the x-axis.

The formula to calculate slope is given as,


Slope of a Line m = [y2 - y1 / x2 - x1]


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.

Let us study What is the slope formula?

Finding the Slope using Two points on the Line

To find the slope, you have to divide the difference of y-coordinates of 2 end-points on a line by the difference of x-coordinates of the same endpoints. 

Suppose A(x1 y1) and B(x2 y2) then their slope will be

 Slope(m) = y2-y1 / x2-x1 

Here, x1 and x2 are x-coordinates and y1 and y2 are y coordinates on X-axis and Y-axis respectively.

The slope of the line can be a positive or negative value.

The subscripts of x and y are only used to identify the two points. They are not values or exponents, the points can be given any names.

From the above figure, the slope of the straight line joining the points A (x1, y1) and B (x2, y2) is   


Slope of Line (m) = [y2 - y1 / x2 - x1]


That is, 

\[\frac{\text{Change in y Coordinates}}{\text{Change in x Coordinates}}\]

If the general equation of a straight line is given as

ax + by + c  =  0,   

then, the formula for the slope of the line is 

m = - coefficient of x / coefficient of y 

m = -a/b


Slope Intercept Formula

Linear equations are “straight line” equations that have simple variable expressions with terms without exponents on them. You are dealing with a straight line equation, if you come across an equation with only x and y. To find the equation of a line and y-intercept in the steepness of the line, we use slope intercept formula.

Slope Intercept Formula is


y = mx + b


The values used in formula are as follows:

  • m is the slope of the line

  • b is the y-intercept of the line


Rules for Calculating the Slope of Line

Method 1:

Step 1: Find two points on the line.

Step 2: Count the rise. The number of units counts up or down to get from one point to the next. Record this number as your numerator.

Step 3:Count the run. The number of units counts left or right to get to the point. Record this number as your denominator.

Step 4:Simplify your fraction if possible.


Method 2:

To find out the slope of a line we need only two points from that line, (x1, y1) and (x2, y2). 

There are three steps for calculating the slope of a straight line.

Step1: Identify two points on the line.

Step2: Select one to be (x1,y1) and the other to be(x2,y2)

Step3: Use the slope of the line formula to calculate the slope.

Some of the important points to remember to find the slope of the line. They are as follows:

  • The slope formula can give a positive or negative result. 

  • If the slope is a positive value, the line is in a rising state. 

  • If the slope is a negative value, the line is descending. 

  • Vertical lines have no slope.

  • Horizontal lines have a zero slope.

  • Parallel lines have equal slopes. 

  • Perpendicular lines have negative reciprocal slopes. 


Solved Examples

Example 1: Find the slope of the line whose coordinates are (2,6) and (5,1)?

Solution:

We have,

(x1, y1) = (2, 6)

(x2, y2) = (5, 1)

The slope of a line formula is m = (y2 − y1 / x2 − x1)

m = (1 − 6/ 5 − 2)

m = −5/3

m = − 2


Example 2: If the slope of a line passing through the points (4, x) and (2, -7) is 3, then what is the value of x?

Solution:

We have

Slope = m = 3

Points:

(x1, y1 ) = (4, x)

(x2, y2) = (2, -7)

We know that,

Slope (m) = (y2– y1 )/(x2– x1)

3 = (-7 - x)/(2 – 4)

3 = (-7 - x)/(-2)

-7 – x = 3(-2)

-7 – x = -6

x = -7 + 6 = -1

Therefore, the value of x = -1.

FAQ (Frequently Asked Questions)

1. What is Slope Intercept Form?

Answer: There are two types of intercept, x-intercept and y-intercept. X-intercept, means the line passes through the x-axis with coordinates (x, 0). And y-intercept, means the line passes through the y-axis with coordinates (0, y). To find x or y intercepts, observe where the line on the graph cuts the x or y axis respectively.

The y-intercept is the point at which the line crosses the y-axis.The x-intercept is the point at which the line crosses the x-axis.


Slope Intercept Form

y = mx + b, where m is the gradient or the slope and b is the y-intercept.

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1. Slope of the line when θ is the angle between the lines.

Let θ be the angle made by the line the figure given below illustrates this. 

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Then slope of a line formula is given as 

m = tan θ. 

If two points A (x₁,y₁) and B (x₂,y₂) lie on the given line such that x₁≠ x₂ then the slope of the line AB is given as:


tan θ = (y₂ - y₁ )/( x₂ - x₁)


Where θ is the angle made by the line AB with the positive direction of the x-axis. θ lies between 0° and 180°

It must be noted that θ = 90° is only possible when the line is parallel to y-axis i.e. at 

x₁ = x₂,then the slope of the line is undefined.