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Setting Up Linear Models

Many real-world situations can be described (modelled) by a linear function. When setting up a linear model, we can use either form for the equation of a line, but most of the time we will use the slope-intercept form: y = mx + b.

Understanding Word Problems

In order to set up a model, it is important to be able to decipher what the information you are given in a word problem represents. There are several pieces of information that the problem may provide:

  • The slope of the line.
    • Also known as “rate of change”.
    • Called “marginal costs” in business applications.
  • The y-intercept.
    • Often given as an “initial condition”… in this case, the corresponding ordered pair is (x, y) where x = 0.
    • Called the “fixed cost” in business applications.
  • An ordered pair (ie., a point that the line must go through).

You will need 2 of the above pieces of information in order to set up the linear model. It is important to be able to pick out and properly use the important information that is provided in the question.

When working with linear models, variables other than x and y are commonly used. Therefore, it is important to first assign a letter to the quantities that change within the problem. This ensures that other people understand your work.

Example: Going to a movie costs $12 per person. Set up a linear model to represent the cost as a function of the number of people that go to the movie.

Solution: Let p represent the number of people that go to the movie (independent variable), and let C represent the total cost (dependent variable). We will have an equation of the form:

C = mp + b

We know the cost per person is 12, so this is our slope. We know the y-intercept is zero since the cost is zero if no people attend. So, we have:

C = 12p

Example: A cup of coffee is initially 95oC and the temperature is decreasing at a constant rate of 2oC per minute until it levels off at room temperature. Determine the equation of the line while the coffee is cooling.

Solution: Assign variables and determine which variable is dependent and which is independent.

Let T represent the temperature in degrees Celsius (dependent).

Let t represent time in minutes (independent).

T = mt + b

The temperature is initially 95oC. This gives us the ordered pair, (0, 95) and our T-intercept. Therefore b = 95.

The rate of change gives us our slope. Notice the rate is decreasing and therefore has a negative value. Therefore, m = -2.

T = -2t + 95

Note that this equation is only valid while the coffee is cooling. In this case, a linear model is an approximation anyway. If you continue to study more complicated functions, you’ll find an exponential function best describes the cooling of a cup of coffee. 

Setting Up Linear Models Example 1:

Setting Up Linear Models Example 2: