Please help find point-slope form and slope intercept with real life problem about dieting

Question

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 women were found to be on a diet, while 18 years into the study, 10.5% of women were found to be on a diet.

 

Find the point-slope form

Find the slope-intercept form

 

Use your equation to predict the percentage of American women on a diet in 2007.
Simplify your answer. The % symbol is already provided, so you do not need to enter it.

 

I asked this question already on this website and the only response I got was wrong, so I really need some help with this because my homework is due tomorrow and this is my last question!

 

Please help! Thanks!

Answer 1

Hi Erin,

 

The point-slope form of a linear equation:

    (y₂ - y₁) = m(x₂ - x₁)

                   point (x₁,y₁) = (8,17)

                   point (x₂,y₂) = (18,10.5)

                   slope (m) = (y₂ - y₁) / (x₂ - x₁)

                   m = (10.5 - 17) / (18 - 8)

                   m = -6.5/10

                   m = -0.65

Using slope (m) and point (x₁,y₁):

    y - 17 = -0.65(x - 8)  [point-slope answer]

 

The slope-intercept form of a linear equation:

    y = mx + b

Using point (x₁,y₁) and slope (m) to calculate

y-intercept (b):          

    17 = -0.65(8) + b

    17 = -5.2 + b

    b = 17 + 5.2

    b = 22.2

Using slope (m) and y-intercept (b):

    y = -0.65x + 22.2  [slope-intercept answer]

 

The percentage of women on diet (y) in 2007:

    x = 2007 - 1987

    x = 20 years into study

    y = -0.65x + 22.2 (slope-intercept equation)

    y = -0.65(20) + 22.2

    y = -13 + 22.2

    y = 9.2% of women on diet in 2007

 

Regards, Jordan.     

Answer 2

According to the study, there is a relationship between time and percentage of women found on a diet over time.

 

Time is always independent.  This makes the percentage of women found on a diet as time progresses the dependent variable.

 

Let x = time

Let y = percentage of women found on diet

 

 

It looks like you have a linear relationship between the two quantities.  We can convert the numbers you have given into coordinate points to find the function that represents this relationship in point-slope form and slope-intercept form.

 

Point 1 = (8, 17)

Point 2 = (18, 10.5)

 

 

Point - slope form:

 

(y - y₁) = m(x - x₁)

 

 

m is the slope.   x₁ and y₁ are the values of point 1.

 

Find the slope between these points.

 

m = (10.5 - 17) / (18 - 8)

m = -6.5 / 10

m = -0.65

 

Plug this value of m into the point-slope form.

 

y - 17 = -0.65(x - 8)

 

 

Slope-intercept form is   y=mx+b.   We can use the point-slope form to get the new form.

 

y = -0.65(x - 8) + 17

 

y = -0.65x + 5.2 + 17

 

y = -0.65x + 22.2

 

 

Now, we use this function to find the percentage of women found on diet in the year 2007.

 

The x coordinate represents the number of years after the study began.  So to find the x value for the year 2007, subtract 1987 from 2007.

 

 

2007 - 1987 = 20

 

 

So evaluate y when x=20 to find the percentage of women on a diet in the year 2007.

Answer 3

Covert the data to points where x the years since 1987 and y is the percentage on a diet.  Therefore your 6 data points are (8, 17) and (18, 10.5).  The slope of the line is found by (y₂-y₁)/(x₂-x₁)  or (10.5 - 17)/(18-8)= -6.5/10 = -.65 or -13/20.  The point/slope form of a linear equation is y-y₁ = m(x-x₁) where m is the slope and (x₁,y₁) is one point.  So:

 

y - 17 = (-13/20)(x-8)  we can simplify this into the point-slope form:

 

y = (-13/20)x -26/5 + 17 

 

y = (13/20)x +59/5