A multi-year analysis was begun in 1987 to determine the percentage of American women claiming to be on a weight-loss diet. 8 years into the study, 17\% of wom

Question

Use this information to find the equation of the line that models this data.
Let \, y \, = percentage of women on a diet, and let \, x \, = number of years into the study.
Give the point-slope form and the slope-intercept form.

 

What is the point-slope form?

What is the slope-intercept form?

 

Use your equation to predict the percentage of American women on a diet in 2007.
Simplify your answer. 

 

 

Answer 1

Hi Erin,

 

Let's begin by identifying the two different slope forms for the equation of a straight line:

    1.  Point Slope form:  y - y₁ = m(x - x₁)

             This form is used when we know one point

on the line (x₁,y₁) and the slope (m).

    2.  Slope Intercept form:  y = mx + b

             This form is used when we know the slope (m) and the y-intercept (b).

 

Next, let's define the two points on our line from the given information in the problem:

    1.  (x₁,y₁) = (0,0) 

             0% of American women on a diet at 0 years into the study.

    2.  (x,y) = (8,17)

             17% of American women on a diet at 8 years into the study.

 

Next. let's calculate the slope using our two points:

    m = (y - y₁) / (x - x₁)

    m = (17 - 0) / (8 - 0)

    m = 17/8

 

Next, let's calculate the y-intercept (b):

    y = mx + b  [Slope Intercept form]

          m = 17/8  [calculated above]

          x = 0  [x coordinate of 1st point above]

          y = 0  [y coordinate of 1st point above]

    0 = (17/8)(0) + b

    0 = 0 + b

    b = 0

 

Now let's write the equation of our line using both slope forms for the equation of a straight line:

    1.  y - y₁ = m(x - x₁)  [Point Slope form]

                      m = 17/8  [calculated above]

                      (x₁,y₁) = (0,0)   [1st point above]

         y - 0 = 17/8(x - 0)

         y = (17/8)x  [Point Slope form]

    2.  y = mx + b  [Slope Intercept form]

              m = 17/8  [calculated above]

              b = 0  [calculated above]

         y = (17/8)x + 0

         y = (17/8)x  [Slope Intercept form]

 

We can see that BOTH slope forms for the equation of a straight line gave us the SAME equation for the SAME line, which is what we would expect.

 

Finally, let's use our equation to calculate the predicted percentage of American women on a diet in the year 2007:

    2007 - 1987 (20 years into the study)

    y = 17/8(x)  [our equation]

           x = 20 (2007 - 1987)

    y = 17/8(20)

    y = 340/8

    y = 42.5% of American women on a diet in 2007 

 

We can verify that our answer is correct by realizing that 20 years is 2.5 times the length of 8 years and that 42.5% is also 2.5 times the amount of 17%.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.