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:
The slope equation is, m=tan ϴ = ∆y / ∆x
From the graph we observe
The slope formula is given us
Slope: m = y2-y1/x2-x1
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.
Example 1: Find the slope of the line whose coordinates are (2,9) and (4,1).
The slope formula is m= y2-y1/x2-x1.
Gn: (x1,y1)=(2,9) and x2 ,y2=(4,1)
m = (1-9)/(4-2)
Example 2: Determine the value of b if the Slope of a line passing through the point (b,7) and (8,-5) is 6.
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)
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?
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.