When trying to write equations of lines, there are two major areas to consider. First, the slope of the line. Second, what point does the line go through? Armed with these two pieces of information, a very special equation describing the line can be written.
The slope of a line is how much vertical rise happens over a horizontal distance. It is a measure of "steepness". Some common words associated with slope are the rise and the run. Think about a driveway, how many feet does the driveway rise as the horizontal runs? How about the pitch of a roof? How many feet vertically does the roof rise compared to each foot horizontally? If the driveway is too steep, then the car may not make it up. If the roof does not have enough slope, then snow and water might not run off and cause damage to the roof. These are just a few examples of how slope is used in some real world applications. So how is the slope of a line calculated?
Looking left to right on the graph, suppose the point \(A(x_i,y_i)\) is the initial point and \(B(x_f,y_f)\) is the final point. The rise is how much change in the \(y\)-coordinates, and the run is the change in the \(x\)-coordinates. By change, the mathematical operation for that is subtraction. Hence the formula for the slope is given by: $$m=\large{\frac{y_f-y_i}{x_f-x_i}}$$ The letter \(m\) is commonly used to denote the slope of the line. Take care to put the \(y\)-values in the numerator and the \(x\)-values in the denominator. A positive slope will move up, whereas a negative slope will move down.
Find the slope of the line that passes through \((-3,7)\) and \((12,-5)\).
To emphasize correct use of the formula, the coordinates are color coded: \((\color{green}{-3},\color{blue}{7})\) and \((\color{green}{12},\color{blue}{-5})\).
| Plug into formula: | \(m=\large{\frac{\color{blue}{7-\,-5}}{\color{green}{-3-12}}}\) |
| \(m=\large{\frac{12}{-15}}\) | |
| Simplify if possible: | \(m=\large{\frac{4}{-5}}\) |
This slope is negative, so the line moves down \(4\) units for every \(5\) units to the right.
The most common equation of a line is called the slope intercept form. If \(m\) represents the slope of the line, and the point \((0,b)\) is the \(y\)-intercept, then $$y=mx+b$$ This equation has two main purposes: #1 it provides a quick way to graph the line and #2 it is an easy equation to write down provided the \(y\)-intercept is known.
The following video provides an example of graphing from slope intercept form and also writing an equation on a line from a given graph.