My last post discussed the mathematics of linear growth/decay from an analytic geometry point of view. If you have not read that post, you might find it helpful to read it before continuing. This post will focus on finding a linear equation that expresses a relationship between two variables by first creating a table of data-pairs. After your students inspect the table, encourage them to write the slope-intercept or the point-slope equation of a line that contains the data. Once the equation is found, it can be manipulated and used to answer a wide variety of questions. The general procedure for finding the linear relationship between independent variable **I** and dependent variable **D** is described below:

- Set up a table of data values so that the independent variable
**I**is in ascending order. - From the table, determine the
__steady rate of change__between**I**and**D**which equals slope**m**. - From the table, determine equation constant
**b**by finding**D**when**I**= 0. - The equation
**D = mI + b**expresses the relationship between**D**and**I**. - The equation
**I = (D – b)/m**expresses the__inverse__relationship between**I**and**D.**

If it is not possible to determine equation constant **b** from the table, pick any point **(p, q)** in the table. The equations **D – q = m(I – p)** or **D = mI – mp + q** express the linear relationship. Of course, if the data pairs in the table don’t show a __steady rate of change__, then the relationship between the variables is not linear.

Most discussions about finding the equation of a linear relationship don’t start by looking at data-pairs in a table. From my past experience, the table-data-pair approach works because it enables students to immediately see and understand the relationship. I routinely use this approach when writing computer programs. It helps me better understand the problem and helps to clarify my thinking. I have selected four familiar linear relation type math problems to demonstrate the table data-pairs approach.

**Problem 1:** Brian’s electricity provider charges him $0.14 per kWh (kilowatt-hour) of electricity, plus a basic connection charge of $17.50 per month. Let n = the number of kilowatt-hours of electricity Brian used in a month and c = the amount charged for n kilowatt-hours of electricity.

(a) Find an equation that expresses the relationship between the variables n and c.

(b) If Brian used 861 kWh of electricity in July, how much was he billed for July?

(c) If Brian was billed $155.82 for August, how many kWh of electricity did he use in August?

**Problem 1 Solution:** Use the information stated in the problem to construct a data-pair-table similar to the table shown in the text box below. The table immediately tells us that the steady rate of change constant m = $0.14/kWh, and c = 17.50 when n = 0. Using the slope-intercept form of the equation of a line, the equations c = 0.14n + 17.50 and n = (c – 17.50)/0.14 follow. If Brian used 861 kWh of electricity in July, he was billed $138.04 for July. If Brian was billed $155.82 for August, he used 988 kWh of electricity in August. (Refer to the text box below.)

**Problem 2:** Bubba’s doctor put Bubba on a strict diet that was designed to lose weight at a steady __average__ rate of 10 pounds every 4 weeks. When Bubba started the diet, he weighed 405 pounds. Let n = the number of weeks that Bubba is on the diet and w = Bubba’s expected weight after n weeks on the diet. (Expected weight approximately equals the actual weight on the diet.)

(a) Find an equation that expresses the relationship between the variables n and w.

(b) Find Bubba’s expected weight after 1 year (52 weeks) on the diet.

(c) At one point in his diet, Bubba weighed 335 pounds. Approximately how many weeks was he on the diet?

**Problem 2 Solution:** Use the information stated in the problem to construct a table of data-pairs similar to the table shown in the text box below. The table tells us that the steady weight loss rate m = -10/4 = -2.5 pounds per week and w = 405 pounds when n = 0. Using the slope-intercept form, the equations w = -2.5n + 405 and n = -(w – 405)/2.5 = -w/2.5 + 162 follow. Bubba’s expected weight after one year of the diet = 275 pounds. If Bubba weighed 355 pounds at one point in his diet, he had been on the diet about 20 weeks. (Refer to the text box below.)

**Problem 3:** A partial handicap chart for a bowling league is shown in the text box below. Bowlers who average more than 210 pins per game receive no handicap, and bowlers who average 210 or less pins per game receive a handicap as indicated in the chart. The variable A represents the bowler’s current average, and H represents the bowler’s handicap.

(a) Based on the data in the chart above, find a formula that expresses H as a function of A.

(b) Find a bowler’s handicap if his average score is 147. Round down to nearest integer.

(c) Find a bowler’s average score if he has a 100 pin handicap. Round down to nearest integer.

**Problem 3 Solution:** The table tells us that the steady rate of change constant m = 9/(-10) = -0.9. This tells us that the handicap drops 0.9 pins for every point increase in the bowler’s average score. Using the point-slope form of the equation of a line with the point (210, 9), the relationship between A and H can be expressed as H – 9 = -0.9(A – 210) or H = -0.9A + 198. Solving the equation for A, we have the equation A = (198 – H)/0.9. The handicap for a 147 bowler = -0.9*147 + 198 = 65.7 = 65 pins. If a bowler has a 100 pin handicap, his average bowling score = (198 – 100)/0.9 = 108.88889 = 108.

**Problem 4:** At 32^{0}F and 0^{0}C water freezes. At 212^{0}F and 100^{0}C water boils. Use the fact that the relationship between F and C is linear to find the following:

(a) Find an equation that expresses F as a function of C.

(b) Find an equation that expresses C as a function of F.

(c) Convert 68^{0}F (average room temperature) to degrees Celsius.

(d) Convert 98.6^{0}F (average body temperature) to degrees Celsius.

**Problem 4 Solution:** Use the information stated in the problem to construct a data-pair-table similar to the table shown in the text box below. The table immediately tells us that the steady rate of change m = 180/100 = 9/5 and F = 32 when C = 0. Using the slope-intercept form of the equation of a line, the equations F = 9C/5 + 32 and C = 5(F – 32)/9 follow. When F = 68^{0}F, C = 20^{0}C. When F = 98.6^{0}F, C = 37^{0}C.

**Comments and Suggestions:**

- The sample problems in this post are stated so that the questions are very direct and specific. All variables in the problems are well defined. The problems that students will typically encounter are stated in a less direct manner. Understanding the problem is the first rule of problem solving. Students need to carefully read the problem for understanding, and then write a clear description of the variables in the problem. If they learn to do this step well, they will understand the problem they are trying to solve and have much better chance of becoming a successful problem solver. When I ask struggling students what the problem is about or what the problem is asking them to find, invariably they can’t give me a simple answer. Of course, every experienced math teacher or tutor understands what I’m talking about.
- Initially present problems to students in very direct and specific manner. As they become more competent problem solvers, present problems in a less direct or general manner that is more representative of real-life problem solving scenarios.
- Before presenting the lesson, review the basics of solving for a variable in an equation.
- The manner in which the sample problem solutions are stated models how I would like my students to answer a question. Getting students to do this is not easy. Example: Refer to part (b) of problem 2. The response that I want from a student requires him/her to write a complete sentence. Requiring only a short response like “275 pounds” does not help students become better writers. At what age can we realistically expect students to respond with a complete sentence? I don’t know.
- Checking answers is another principle of problem solving. I tell my students that I always check my answer in order to catch a mistake. When writing this post, I caught a simple arithmetic error (and was able to fix it before you caught it).
- Remind more mathematically mature students that, in many cases, the equation we write to model a given problem situation is only an approximate model. In reference to problem (2) above, people on a diet won’t lose the exact same amount of weight in equal time periods. This is why least-squares regression equations are used to describe the relationship between two variables. Statisticians are forced to use least-squares regression equations to describe the relationship between two or more variables, at a specific level of confidence, because the statistical data was obtained from a random sample of some population.
- I use the handout
*Linear Growth and Decay Models*with my beginning algebra students, and as a review for more advanced students. To download the free student and teacher versions of the handout, go to mathteachersresource.com/instructional-content.html.