Building Linear Models

Learning Outcomes

  • Use prescribed strategies to build linear models
  • Use intercepts and data points to build a linear model
  • Use a diagram to build a model

tip for success

When following the steps given below for solving real-world situations, it is helpful to keep these tips in mind.

  • Read the problem carefully, more than once if needed.
  • Look for the question being asked. That’s usually where you’ll find the unknown.
  • Separate the information given in the problem into chunks and look for relationships between the pieces to translate the information into expressions, equations, or inequalities.

When modeling scenarios with linear functions and solving problems involving quantities with a constant rate of change, we typically follow the same problem solving strategies that we would use for any type of function. Let’s briefly review them:

  1. Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
  2. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
  3. Determine what we are trying to find, identify, solve, or interpret.
  4. Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
  5. When needed, write a formula for the function.
  6. Solve or evaluate the function using the formula.
  7. Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
  8. Clearly convey your result using appropriate units, and answer in full sentences when necessary.

Building Linear Models

Now let’s take a look at the student from the introduction who wants to spend the summer in Seattle. In her situation, there are two changing quantities: time and money. The amount of money she has remaining while on vacation depends on how long she stays. We can use this information to define our variables, including units.

  • Output: M, money remaining, in dollars
  • Input: t, time, in weeks

So, the amount of money remaining depends on the number of weeks. Hence, amount of money remaining is a function of time: M(t)

We can also identify the initial value and the rate of change.

  • Initial Value: She saved $3,500, so $3,500 is the initial value for M.
  • Rate of Change: She anticipates spending $400 each week, so –$400 per week is the rate of change, or slope.

Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because she is spending money each week.

Pictoral of M(t) = -400t + 3500, with -400 highlighted as the slope, and 3500 highlighted as the intercept

The rate of change is constant, so we can start with the linear model [latex]M(t)=mt+b[/latex]. Then we can substitute the intercept and slope provided.

To find the x-intercept, we set the output to zero and solve for the input.

[latex]\begin{array}{l}0=-400t+3500\hfill \\ t=\frac{3500}{400}\hfill \\ t=8.75\hfill \end{array}[/latex]

The x-intercept is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Emily will have no money left after 8.75 weeks.

When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before she saved $3,500, but the scenario discussed poses the question once she saved $3,500 because this is when her trip and subsequent spending starts. It is also likely that this model is not valid after the x-intercept, unless Emily will use a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is [latex]0\le t\le 8.75[/latex].

In the above example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given and use it appropriately to build a linear model.

Using a Given Intercept to Build a Model

Some real-world problems provide the y-intercept, which is the constant or initial value. Once the y-intercept is known, the x-intercept can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The y-intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is –$250 per month. We can then use slope-intercept form and the given information to develop a linear model.

[latex]\begin{array}{l}f\left(x\right)=mx+b\hfill \\ f\left(x\right)=-250x+1000\hfill \end{array}[/latex]

Now we can set the function equal to 0 and solve for x to find the x-intercept.

[latex]\begin{array}{l}0=-250x+1000\hfill \\ 1000=250x\hfill \\ 4=x\hfill \\ x=4\hfill \end{array}[/latex]

The x-intercept is the number of months it takes her to reach a balance of $0. The x-intercept is 4 months, so it will take Hannah four months to pay off her loan.

Using a Given Input and Output to Build a Model

Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output.

How To: Given a word problem that includes two pairs of input and output values, use the linear function to solve a problem.

  1. Identify the input and output values.
  2. Convert the data to two coordinate pairs.
  3. Find the slope.
  4. Write the linear model.
  5. Use the model to make a prediction by evaluating the function at a given x value.
  6. Use the model to identify an x value that results in a given y value.
  7. Answer the question posed.

Example: Using a Linear Model to Investigate a Town’s Population

A town’s population has been growing linearly. In 2004 the population was 6,200. By 2009 the population had grown to 8,100. Assume this trend continues.

  1. Predict the population in 2013.
  2. Identify the year in which the population will reach 15,000.

Try It

A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It costs $0.25 to produce each doughnut.

  1. Write a linear model to represent the cost C of the company as a function of x, the number of doughnuts produced.
  2. Find and interpret the y-intercept.

Try It

A city’s population has been growing linearly. In 2008, the population was 28,200. By 2012, the population was 36,800. Assume this trend continues.

  1. Predict the population in 2014.
  2. Identify the year in which the population will reach 54,000.

Using a Diagram to Model a Problem

It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometric shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.

tip for success

It simply cannot be stressed too often; sketching a diagram or trying a few sample equations is an essential part of doing mathematics. Paths that don’t pan out and wrong turns are common and natural. We need to make mistakes in math in order to rule out possibilities that don’t lead to success. Don’t be afraid to sketch a diagram or take a wrong turn. It’s all part of the mathematical process.

Example: Using a Diagram to Model Distance Walked

Anna and Emanuel start at the same intersection. Anna walks east at 4 miles per hour while Emanuel walks south at 3 miles per hour. They are communicating with a two-way radio that has a range of 2 miles. How long after they start walking will they fall out of radio contact?

Q & A

Should I draw diagrams when given information based on a geometric shape?

Yes. Sketch the figure and label the quantities and unknowns on the sketch.

Example: Using a Diagram to Model Distance between Cities

There is a straight road leading from the town of Westborough to Agritown 30 miles east and 10 miles north. A certain distance down this road, it junctions with a second road, perpendicular to the first, leading to the town of Eastborough. If the town of Eastborough is located 20 miles directly east of the town of Westborough, how far is the road junction from Westborough?

Try It

There is a straight road leading from the town of Timpson to Ashburn 60 miles east and 12 miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located 22 miles directly east of the town of Timpson, how far is the road junction from Timpson?