Question

System of equations problem: From 1970-1990, the average annual per person consumption of whole milk in the US dropped from 103 to 43 quarts. the average annual consumption of low fat milk rose from 25 to 60 quarts. For which years did the consumption of lowfat milk exceed the consumption of whole milk?

Answer 1

whole milk
103-43=60
60÷20=3

low fat
60-25=35
35÷20=1 and 3/4

answer
1987

Answer 2

From 1987 to 1990

Explanation:

First we need to transform data into equations. Let's define x as years after 1970 (this means that x = 20 years represents 1990), y1 as average annual per person consumption of whole milk and y2 as the average annual consumption of low fat milk. Therefore, data can be expressed as

x y1 y2

0 103 25

20 43 60

Regression of the data as equation of a line gives

b1 = 103 (y-intercept for whole milk case)

b2 = 25 (y-intercept for low fat milk case)

m1 = (43-103)/20 = -3 (slope for whole milk case)

m2 = (60-25)/20 = 1.75 (slope for whole milk case)

Then, y1 = -3*x+103 and y2 = 1.75*x + 25. These two equations intersect in one point from which low fat milk exceed the consumption of whole milk. In that point y1 = y2, so

-3*x+103 = 1.75*x + 25

103 - 25 = 1.75*x + 3*x

78 = 4.75*x

x = 78/4.75

x = 16.42

In years that is 1970 + 16.42 = 1986.42