Slope, Modeling, and Linear Relationships
In previous sections, we learned how to write formulas, or equations, in order to represent real-
life situations. We call this mathematical modeling. We can use mathematical modeling to solve
problems in many areas such as science, business, and medicine.
Before we can set up an equation as a mathematical model, we must determine what the variable
will be and organize all of the information in terms of that variable.
For example, let’s assume that all of the SLAC lab employees will receive a $150 Christmas
bonus. If the college must pay $3,300 to cover the cost of the Christmas bonuses, how many
employees must be working in the SLAC lab?
We want to know how many employees work in the SLAC lab, so we will set e to be our
variable; therefore, we know that e represents the number of employees in the lab. Now, we
organize the other information in terms of the variable e. If each employee receives $150, then
the total amount of money paid will be $150e. (The information given in this problem is simple,
but in some cases it will be helpful to set up a table in order to organize the information.)
Now, we can set up the model and solve for the unknown variable.
$3,300 = $150e
e = 22 (Divide by $150.)
There are 22 employees in the SLAC lab. We can check our answer by substituting 22 in for e.
We can summarize the steps we used by the following guidelines.
Modeling with Equations:
1. Determine the variable. The desired quantity will usually be found in the last sentence of
the problem. The notation for this value (x,a,c,etc.) can be anything you choose to call it,
but it is helpful if it, in some way, represents the desired quantity.
2. Express all unknown quantities in terms of the variable. In some cases, it may be helpful
to make a table with the information.
3. Set up the model. In equation form, organize the expressions listed in Step 2.
4. Solve the equation and check your answer. Be sure to express the answer in sentence
form, clearly stating the answer to the question in the problem.
The slope of a line tells you the “steepness” of the line (or the rate of change at which the line is
increasing or decreasing). The line is increasing if the slope is positive and decreasing if the
slope is negative.
In order to find the slope of a line, you must have the coordinates for two points on the line. The
slope is then determined by dividing the change in the y-coordinates (referred to as the rise) by
the change in the x-coordinates (referred to as the run).
Math 0303
Student Learning Assistance Center - San Antonio College
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