25. (See Week 1 Lecture page The Equation of a Line 1 for a starting point.)

As part of the second stage of a drug trial, volunteers are given one of two dosages, in order to determine therapeutic levels. Volunteers who were given 6.5 mg had blood concentrations of 2.7 parts per million (ppm), while volunteers who were given 10 mg had blood concentrations of 5.1 ppm.

a. Determine the equation of a line that models this data, where x is the dosage given and y is the blood concentration.

b. Graph this line.

c. If the desired concentration in the bloodstream is 7.5 ppm, what should be the therapeutic dosage?

d. If the toxic concentration in the bloodstream is 11.8 ppm, what is the maximum safe dosage?

The Equation of a Line 1

Let's say that (x₁, y₁) and (x, y) are two points lying on the line. Applying the definition of slope m, we have

m=y-y1/x-x1

Multiplying both sides by (x x₁) and rearranging, we have

y y₁ = m(x x₁)

We refer to this as the point-slope form of the line, since the equation is written using the coordinates of a single point on the line (x₁, y₁), and its slope m.

Now, lets assume that the point (x₁, y₁) is actually the y-intercept, which occurs at some point (0, b). Substituting these coordinates into the point-slope form of the line, we have

y b = m(x 0)

or

y = mx + b

We refer to this as the slope-intercept form of the line, since the equation is written using the slope m and the y-intercept (0, b). These two forms of the equation of a line are completely equivalent; you can use whichever is most convenient in a given situation.

Example:

Given the points (4, 3) and (2, 7), determine the equation of the line on which these points lie.

Solution:

Since we dont know the y-intercept, lets begin by using the point-slope form of the line y y₁ = m(x x₁) We'll select (4, 3) as the point (x₁, y₁) whose coordinates well use. We calculate the slope as Plugging these values into the format, we have y 3 = -2(x 4) which is one possible form of the line. If we applied the distributive law on the right, then added three to both sides, we would have y 3 = -2x + 8 y = -2x + 11 which is the slope-intercept form of this line.