Rise over Run Formula
The rise over run formula is another way of saying the "slope formula" for a straight line joining any two points. The difference between the ycoordinates of two points is called the rise. The difference between the xcoordinates of the same two points is called the run. The slope can be calculated by dividing the rise by run. Let us explore the rise over run formula (or slope formula) below.
What Is the Rise Over Run Formula?
Consider a line joining two points A(x_{1},y_{1}) and B(x_{2},y_{2}). The slope of this line is the ratio of the difference in y coordinates and the difference in x coordinates. The same applies to the rise over run formula as well.
Rise over run formula (or) slope (m) = \( \dfrac{y_2  y_1}{x_2  x_1} \) = \( \dfrac{\Delta y}{\Delta x} \)
Here,
 Rise = difference between the ycoordinates of points A and B = y_{2}  y_{1 }= Δy
 Run = difference between the xcoordinates of points A and B = x_{2}  x_{1 }= Δx
Let us see the applications of the rise over run formula in the solved examples section below.
Solved Examples using the Rise Over Run formula

Example 1: Find the slope of the line joining points A(0, 4) and B(1, 7) using the rise over run formula. Also, write the equation of this line in standard form.
Solution.
Slope (m) = \( \dfrac{y_2  y_1}{x_2  x_1} \) = \( \dfrac{7  4}{1  0} \) = 3
Using the slopeintercept form, equation of this line can be written as
y = mx + c
y = 3x + c
Point (0, 4) passes through the line, so substituting x = 0 and y = 4 in the above line equation we get,
4 = 3(0) + c
c = 4
Therefore, the line equation is y = 3x + 4
Converting the line into standard form
y = 3x + 4
Subtracting y on both sides we get,
3x  y + 4 = 0
Subtracting 4 on both sides we get,
3x  y = 4 (Standard Form)
Answer: The slope of the line joining A and B is 3. The standard form of this line equation is 3x  y = 4

Example 2: A square of side = 4 units has one of its corner points A(1, 1). Given that the square is in the first quadrant and its sides are parallel with the xaxis and the yaxis. Find the slopes of both diagonals of this square.
Solution.
This is the only possible square for the given conditions.
Since line joining A and B is parallel to the yaxis, this implies that the xcoordinates of both A and B are the same.
A = (1, 1) B = (1, y)
The distance between A and B is 4 units.
Using the distance formula \(AB = 4 = \sqrt{(1 1)^2 + (y1)^2}\) gives y = 5.
Therefore coordinates of B are (1, 5).
Similarly, coordinates of C are (5, 1) and coordinates of D are (5, 5).
Let m_{1} and m_{2} be the slopes of both the diagonals of the square. Let us find them using the rise over run formula.
m_{1} = \( \dfrac{5  1}{5  1} \) = 1 (for the diagonal AD)
m_{2} = \( \dfrac{1  5}{5  1} \) = 1 (for the diagonal BC)
Answer: Slopes of both the diagonals are m_{1 }= 1 and m_{2} = 1.