Why Study Math? Linear Equations and the Point-Slope Form

As we saw in the article "Why Study Math? - Linear Equations and Slope-Intercept Form," linear equations or functions are some of the more basic ones studied in algebra and basic mathematics. Here we are going to take a look at and examine another common way of writing linear equations: the point-slope form.

As the name implies, the point-slope form for the equation of a line depends on two things: the slope, and a given point on the line. Once we know these two things, we can write the equation of the line. In mathematical terms, the point-slope form of the equation of the line which passes through the given point (x1, y1) with a slope of m, is y - y1 = m(x - x1). (The 1 after the x and y is actually a subscript which allows us to distinguish x1 from x and y1 from y.)

To show how this form is used, take a look at the following example: Suppose we have a line which has slope 3 and passes through the point (1, 2). We can easily graph this line by locating the point (1, 2) and then use the slope of 3 to go 3 units up and then 1 unit to the right. To write the equation of the line, we use a clever little device. We introduce the variables x and y as a point (x,y). In the point-slope form y - y1 = m(x - x1), we have (1, 2) as the point (x1, y1). We then write y - 2 = 3(x - 1). By using the distributive property on the right hand side of the equation, we can write y - 2 = 3x - 3. By bringing the -2 over to the right side, we can write
y = 3x -1. If you have not already recognized it, this latter equation is in slope-intercept form.

To see how this form of the equation of a line is used in a real world application, take the following example, the information of which was taken from an article that appeared in a newspaper. It turns out that temperature affects running speed. In fact, the best temperature for running is below 60 degrees Fahrenheit. If a person ran optimally at 17.6 feet per second, he or she would slow by about 0.3 feet per second for every 5 degree increase in temperature above 60 degrees. We can use this information to write the linear model for this situation and then calculate, let us say, the optimal running pace at 80 degrees.

Let T represent the temperature in degrees Fahrenheit. Let P represent the optimal pace in feet per second. From the information in the article, we know that the optimal running pace at 60 degrees is 17.6 feet per second. Thus one point is (60, 17.6). Let's use the other information to determine the slope of the line for this model. The slope m is equal to the change in pace over the change in temperature, or m = change in P/change in T. We are told that the pace slows by 0.3 feet per second for every increase in 5 degrees above 60. A decrease is represented by a negative. Using this information we can calculate the slope at -0.3/5 or -0.06.

Now that we have a point and the slope, we can write the model which represents this situation. We have P - P1 = m(T - T1) or P - 17.6 = -0.06(T - 60). Using the distributive property we can put this equation into slope-intercept form. We obtain P = -0.06T + 21.2. To find the optimal pace at 80 degrees, we need only substitute 80 for T in the given model to get 16.4.

Situations like these show that math is really used to solve problems that occur in the world. Whether we are talking about optimal running pace or maximal profits, math is the key to unlocking our potential toward understanding the world around us. And when we understand, we are empowered. What a nice way to exist!