Writing and Solving a System of Equations

from 1970-1990, the average annual per person consumption of whole milk in the U.S. 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 low fat milk exceed the consumption of whole milk?

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Writing and Solving a System of Equations

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Dec 06, 2010
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starstarstarstarstar 		Writing and Solving a System of Equations
by: Karin

For this problem, you will need to write a system of equations. You will need to write an equation for the consumption of whole milk and one for low fat milk.

Before you can write the equation, we must identify two points for each. Since the consumption for whole milk states that it decreased from 103 to 43 quarts in 1990, we can identify the two points as:

Let 0 represent the year 1970

(0, 103) & (20,43)

In 1970(0 year), the consumption was 103 quarts. 20 years later in 1990, the consumption was 43 quarts.

So, now you will need to write an equation based on the linear regression of these two points.

First you will need to find the slope using the slope formula.

43 - 103
_________
20-0

The slope = -3.

We know that the y intercept is the point where x = 0. Our point (0,103) is the y-intercept. Therefore, the y-intercept, or b in the equation is 103.

So, now we have a slope of -3 and a y intercept of 103. The equation for whole milk is:
y = -3x + 103

Now we need to repeat the process for low fat milk.

First identify your two points.

(0, 25) (20, 60)

The consumption was 25 quarts in 1970 and 60 quarts 20 years later in 1990.

So, we need to find the slope of these two points.

60 - 25
--------
20 -0

The slope = 1.75

Again, since the y-intercept is the point when x = 0, we know that the y-intercept is 25. (0,25)

So, we have a slope of 1.75 and a y-intercept of 25.

y = 1.75x + 25

Whole milk: y = -3x + 103
Low Fat Milk: y = 1.75x + 25

We know that whole milk is consumed more until a certain point. This point is the intersection of these two lines, or the solution. This is the point where the two types of milk are consumed in the same amounts.

The years following this point, will be the years in which low fat milk is consumed more. So, let's find the solution or the point of intersection.

I will use the substitution method to solve.

I will substitute the whole milk equation into the low fat milk equation.

-3x + 103 = 1.75x + 25

Now lets solve for x.

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

103 = 4.75x + 25

103 - 25 = 4.75x + 25 - 25

78 = 4.75x

78/4.75 = 4.75x/4.75

16.42 = x

We know that 16.42 years after 1970, the milk consumption will be the same. This means that 17 years after 1970, low fat milk will be consumed more.

Let's check:

First we'll check 16 years later which is 1986.

Whole milk: y = -3x + 103
y = -3(16) + 103
y = 55 quarts for whole milk

Low Fat Milk: y = 1.75x + 25
y = 1.75(16) +25
y = 53 quarts for low fat milk.

So, in 1986, whole milk was still consumed more.

What happens in 1987, 17 years later?

Whole milk: y = -3x + 103
y = -3(17) + 103
y = 52 quarts for whole milk

Low Fat Milk: y = 1.75x + 25
y = 1.75(17) +25
y = 54.75 quarts for low fat milk.

Low fat milk is consumed more.

Therefore, in the years 1987- 1990, low fat consumption is more than whole milk consumption.

There are a lot of steps and I know that it can be confusing, but I hope that I've been able to help.

Karin

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