Slope Formula with Solved Examples

Slope Formula

The slope formula is used to calculate the inclination or steepness of a line. It finds application in determining the slope of any line by finding the ratio of the changes in the y axis to x axis.

The Slope of a line is defined as the change in the y coordinate with respect to the change in the X coordinate of that line.

What is the Slope Formula?

The slope formula refers to the formula used to calculate the steepness of a line and determines how much it is inclined. To calculate the slope of the line the x and y coordinates of the points lying on the line can be used.

The formula to calculate slope is given as

m=(y₂– y₁₎ )(x₂– x₁)= Δy/Δx 

Where, m is the slope of the line,

              x₁, x₂ or the coordinate of the x-axis, 

              y₁, y₂ are the coordinates of the y axis

Derivation of Slope Formula

The x and y coordinates of the line are used to calculate the slope of the line. The net change in y coordinate is  ∆y, while the net change in the x coordinate is ∆x.

So the change in y coordinate with respect to the change in x coordinate can be written as,

   m= ∆y / ∆x                      

Where, m is the slope. 

             ∆y is the changed in y coordinates 

             ∆x is the change in the x coordinates

We know that tan ϴ  is also the slope of the line where is the angle made by the line with the positive direction of the x axis

We know that tan ϴ is also the slope of the line bar is the angle made by the line with the positive direction of the x axis

tan θ= heightbase given by the following formula: 

   y₂-y₁/x₂– x₁

The slope equation is, m=tan ϴ = ∆y / ∆x  

           From the graph we observe

                ∆y=   y₂-y₁

                ∆x=   x₂-x₁

          The slope formula is given us 

           Slope: m =   y2-y1/x2-x1

Slope Equation

The slope formula can be used to determine the slope of any line. The equation that can be used in finding this slope can be written as.

            m=rise / run=tan ϴ Δy / Δx =   y2-y1/x2-x1

where, m is the slope.

              is the angle made by the line with the positive x- axis.

Also the equation of slope of any line using the line equation can be given us 

              y = mx + b

Where, m is there slope of the line. 

              b is the y intercept of the line. 

Solved Examples

Example 1: Find the slope of the line whose coordinates are (2,9) and (4,1).

Solution:

The slope formula is m=   y2-y1/x2-x1.

Gn:      (x1,y1)=(2,9) and x2 ,y2=(4,1)

                                              m = (1-9)/(4-2)

                                              m=-4

Example 2: Determine the value of b if the Slope of a line passing through the point (b,7) and (8,-5) is 6.

Solution:

To Find: the value of b.

Given: slope = m = 6,

Points (x1,y1)= (b,7), and x2 ,y2 = (8,-5).

Slope m = y2-y1/x2-x1

6 = (-5-7)/(8-b)

6 = (-12)/ (8-b)

-2= (8-b)

b = 10

The value of b = 10

Example 3: If the angle made by the line with the positive y axis is 30°, then what is the value of the slope of the line?

Solution: 

Angle made by a line with the positive y axis = 30°.

If the line makes an angle of 30° from the positive y axis then it makes an angle of 120° from the positive x axis therefore the value of the slope of the line is tan120°= -√3

The value of the slope of the line = -√3.