Question

A farmer finds there is a linear relationship between the number of bean stalks, n, she plants and the yield, y, each plant produces. When she plants 30 stalks, each plant yields 31 oz of beans. When she plants 36 stalks, each plant produces 29 oz of beans. Find a linear relationship in the form y=mn+b that gives the yield when n stalks are planted.

Answer 1

`y = -(1)/(3)m + 41`

Explanation:

Linear equation:

A linear function has the following format:

`y = mn + b`

In which m is the slope and b is the y-intercept.

When she plants 30 stalks, each plant yields 31 oz of beans. When she plants 36 stalks, each plant produces 29 oz of beans.

This means that these two points belong to the line: (30,31), (36,29).

Finding the slope:

When we have two points, the slope is given by the change in the output divided by the change in the input.

Change in the output: 29 - 31 = -2

Change in the input: 36 - 30 = 6

Slope:

`m = (-2)/(6) = -(1)/(3)`

Thus

`y = -(1)/(3)m + b`

Finding b:

We take one of the points and replace on the equation.

(30,31) means that
`m = 30, y = 31`. Thus

`y = -(1)/(3)m + b`

`31 = -(1)/(3)(30) + b`

`31 = -10 + b`

`b = 41`

Thus

`y = -(1)/(3)m + 41`