102 H1.01: Overview

Topic H. Linear Formulas—Word Problems

Objectives:

1. Solve applications problems using linear models and interpret the results.
2. Re-define the input variable if needed to make the model easier to interpret in a useful manner.

Terminology:

We use the following instructions interchangeably:

Write a formula to express the linear relationship.

Write an equation to express the linear relationship.

Write an algebraic linear model to express the relationship.

Overview:

To solve the applications problems in this topic, follow these steps.

1. Variables:
   1. What are the names of the two variables?
   2. What are the units of each variable?
2. Prediction:
   1. Which variable will we predict in the main question? (That’s the output variable, i.e. y-variable.)
   2. What letters will we use for each of the variables in our linear model?
   3. Are there any limits on the values of either variable? If so, what? Write them in mathematical notation.
   4. What are some points, that is, values of both variables for more than one point? Write these in appropriate ordered pair notation, with the x-variable first and the y-variable second.
3. Is a linear model appropriate? How do you know? (Use one of these three methods: (1) the problem says that a linear model should be used, (2) graph the data given and see that it forms a straight line, (3) the problem says that a certain increase in the input variable will always give the same increase in the output variable.)
4. Slope:
   1. Find the slope.
   2. Write a sentence to interpret the slope, using the units of each variable in the sentence.
5. Find the formula for the line.
6. Write a sentence to interpret the y-intercept, using the units of each variable in the sentence. If this value is out of the range of the acceptable values for the y-variable, comment on that.
7. Use the formula for the line to answer the each prediction question. Do the algebra and write a sentence, naming the variable and the units, interpreting your answer.
8. Graphing and checking your algebraic work.
   1. Make a graph appropriate to answer the prediction questions.
      (If you make the graph from only the information in the original problem, then you can check your work in developing the model by determining whether the results of the model are consistent with the graph. If you graph the formula you obtained, then you should check to be sure that the graph is consistent with the information given in the original problem.)
   2. Use the graph to answer the prediction questions.       Did you find the same answers as when you used the formula?
   3. Do your answers to the prediction questions make sense? Explain something you looked at to see if they are reasonable.