Question

A clothing business finds there is a linear relationship between the number of shirts, n, it can sell and the price, p, it can charge per shirt. In particular, historical data shows that 1000 shirts can be sold at a price of $30 , while 3000 shirts can be sold at a price of $22 . Find a linear equation in the form p=mn+b that gives the price p they can charge for n shirts.

Answer 1

Answer: p = -0.004n+34

Explanation:

Given that we need to derived the linear equation of the form;

p = mn+b ....1

Where p is the p is the price and n is the number of shirts they can sell

We need to substitute the values of p and n for the two cases to determine the slope m and constant b.

Case 1:

n = 1000 and p = $30

Substituting into the equation 1, we have:

30 = 1000m +b .....2

Case 2:

n = 3000 and p = $22

Substituting into the equation 1

22 = 3000m + b ......3

Substracting equation 2 from 3, we have

22-30 = 3000m-1000m +b-b

-8 = 2000m

m = -8/2000

m = -0.004

Substituting the value of m into equation 2

30 = 1000(-0.004) + b

b = 30 + 1000(0.004)

b = 30 + 4 = 34

b = 34

Therefore substituting the values of m and b into equation 1, we have our linear equation:

p = -0.004n+34

Answer 2

p = -0.004n+34

Explanation:

The slope of the linear equation can be found by using the two given points (30, 1000) and (22, 3000):

`m = (p_1-p_0)/(n_1-n_0)= (30-22)/(1000-3000)\\m=-0.004`

Applying the point (30, 1000) to the general form of a linear equation with m =-0.004, gives us the linear relationship between price (p) and number of shirts (n):

`(p-p_0) = m(n-n_0)\\p-30=-0.004(n-1000)\\p=-0.004n+34`