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The slope-intercept form is one of several ways you can write the equation of a line. In this lesson, you will learn about the definition and equation of the slope-intercept form and work through some real-world examples.
The Cost Equation and Cupcakes
A friend of mine opened a cupcake store called Wild About Cupcakes. She has a storefront where she bakes all the cupcakes and sells them. She hired two employees to help her bake and sell. She asked me to help her determine the daily cost for her business.
After some analysis, I figured out that, whether she bakes a single cupcake or not, there is a fixed daily cost of $550. This amount includes employee salaries, rent, etc. There is also a variable cost that changes with the number of cupcakes baked. The variable cost is $1.25 per cupcake. This amount includes the cost of ingredients, liners, etc. If x is the number of cupcakes baked and y is the total daily cost then:
y = 1.25x + 550
This table shows sample values for x and y.
x y
0 $550.00
1 $551.25
2 $552.50
3 $553.75
4 $555.00
100 $675.00
250 $862.50
The graph for y = 1.25x + 550 looks like this:
Graph of y = 1.25x + 550
Graph of Daily Cost
So what do the table and the graph tell us? We know:
Each additional cupcake produced incurs an incremental cost of $1.25.
With no cupcakes produced (x = 0), there is still a cost of $550 (y = 550).
The graph of y = 1.25x + 550 crosses the y-axis at the point (0, 550).
Definition and Equation
The equation of the total daily cost function for Wild About Cupcakes is in the form:
y = mx + b
where m = 1.25 and b = 550. The graph of the equation y = mx + b (where m and b are real numbers) is a line with slope, m, and y-intercept, b. This form of the equation of a line is called the slope-intercept form.
The slope of a line (m), is a measure of its steepness. As we have seen in the Wild About Cupcakes example, it is also a measure of how much y changes for every unit increase in x. Given any two points on a line, (x1, y1) and (x2, y2), the formula to calculate the slope is:
m = rise / run
or
(y2 – y1) / (x1 – x2)
The y-intercept of a line (b), is the y-coordinate of the point where the graph of the line crosses the y-axis. As we have seen in the Wild About Cupcakes example, it is the value of y when x = 0.
Example Problem
Let’s look at an example problem.
Two points on line L are (0, -1) and (3, -3). What is the slope-intercept form of the equation of L? We can work out the answer together!
The slope-intercept form of the equation of a line is y = mx + b, where m is the slope of the line, and b is the y-intercept. Since we are given two points, we can calculate the slope m as follows:
Calculation of Slope
 
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