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Slope Formula

The slope of a line is the measure how steep the line is. The slope of a line is also known as gradient. It is usually denoted by letter "m". If a line goes towards upward direction, it is supposed to have slope greater than zero. On the other hand, if the line goes towards downward direction, it would be having a slope lesser than zero. Also, when the line is horizontal, its slope will naturally be zero.

Coordinate geometry is an important subject in mathematics. It studies about the position of the objects in space. The position is denoted by coordinates (x , y) in two dimensions and (x , y , z) in three dimensions. In coordinate geometry, a line segment is defined as the shortest distance between two points.

Some other concepts that come along with the concept of slope are slope-intercept and point slope form of a line. Again the concept intercept form includes x and y intercept. Finding slope is done with the help of the point slope formula. Thus all these topics are interrelated.

The easiest way of measuring the slope of line is to find the ratio of difference of y coordinates and x coordinates of the endpoints the line segment. The following graph demonstrates the slope formula which is given by ratio of change in y coordinate to change in x coordinate.

Slope of a Line

The slope can also be calculated by the means of the ratio of distance covered parallel to Y axis (called rise) to the distance covered parallel to X axis (run) between two points. It is demonstrated in the following image :

Measuring the Slope of Line

Slope formula is sometimes called as "Rise over Run". The slope formula of the straight line is Slope (m) =  (y2-y1)/(x2-x1). Can also be simply referred as m = rise / run.

Let us learn more about slope of a line, its formula, the way of calculating it and few examples based on it.

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Point Slope

Formula for Slope of a Line

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The slope of a line means the steepness or incline of the line. The higher the slope value the more is the steeper incline.The slope of a line including the x axis and y axis is represented by m and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.

The slope formula of the line is:

Slope (m) =  $\frac{(y_2 - y_1)}{(x_2 - x_1)}$

Slope Formula

Slope (m) = $\frac{Rise}{Run}$

(Rise = change in the y co-ordinate; Run = change in the x co-ordinate)

In the above given graph, the Rise is given for the points B (4,2) and C (4,4) and the Run is given for the points A (2,2) and B (4,2)

Rise = change in the y - coordinate = 4-2 =2

Run = change in the x- coordinate = 4-2 =2

Using the slope formula, we get :

Slope( m ) = $\frac{Rise}{Run}$ = $\frac{2}{2}$ =1

The slope of a straight line equation where the slope is given as $m$ and the intercept as $c$: 
                         y = mx + c

How to Find the Slope of a Line

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Steps to find slope of a line using slope formula:

Step 1: Find the difference between the y-coordinates.

Step 2: Find the difference between the x-coordinates.

Step 3: Divide the y-coordinate difference by the x-coordinate difference to get the slope.

Example 1: Determine the slope of a line, which contains the points A (-1, -2), B (-2, 1):

Solution: Here, $x_1$ = -1, $x_2$ = -2, $y_1$ = -2, $y_2$ = 1.

Slope of the line, m = $\frac{(y_2 - y_1)}{(x_2 - x_1)}$

= $\frac{(1 + 2)}{(-2 + 1)}$

= $\frac{3}{(-1)}$

Slope of the line (m) = -3

Slope Formula Examples

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Here are few other slope formula examples worked out for you with explanations:

Problem 1: Find the slope of the equation y=4x-3.

Solution: Y = 4x - 3

It is in the slope form y = mx + b.

So, slope(m) = 4

Problem 2: What will be the slope of the curve if the equation is 3y = 6x - 12?

Solution: 3y = 6x - 12

Divide by 3 on both sides,

y = 2x - 4 

By using the line formula y = [mx + c], slope(m) = 2

Problem 3 : Find the slope of a line passing through two points (4, 3) and (-2, -1).
Solution : Here, $x_{1}$ = 4, $y_{1}$ = 3

and $x_{2}$ = -2, $y_{2}$ = -1

The formula for finding slope if two points are given.

m = $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

= $\frac{-1 - 3}{-2 - 4}$

= $\frac{-4}{-6}$

= $\frac{2}{3}$

Work out these problems and get a clear understanding of finding the slope of a line. If you find any difficulties, just connect to our expert online tutor and thus gain quality information from the comfort of your home.

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