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 19000 shirts can be sold at a price of $ 34 , while 24000 shirts can be sold at a price of $ 14 . Give a linear equation in the form p = m n + b that gives the price p they can charge for n shirts

Answer 1

p = -0.004n + 110 is the linear equation

Solution:

Let "p" be the price per shirt

Let "n" be the number of shirts

We have to write a linear equation in the form:

p = mn + b

Where, m is the slope of line and b is the y intercept

In particular, historical data shows that 19000 shirts can be sold at a price of $ 34

While, 24000 shirts can be sold at a price of $ 14

Therefore, two points are:

(19000, 34) and (24000, 14)

Find the slope m using the points

`m = (y_2-y_1)/(x_2-x_1)`

Here,

`(x_1, y_1) = (19000, 34)\\\\(x_2, y_2) = (24000, 14)`

Therefore, slope is given as:

`m = (14-34)/(24000-19000)\\\\m = (-20)/(5000)\\\\m = -0.004`

Here negative slope means, the more shirts, the lower the price

Now that we have the slope and a point, we can find the equation

`p - y_1 = m(n - x_1)\\\\p - 34 = -0.004(n - 19000)\\\\p - 34 = -0.004n + 76\\\\p = -0.004n + 76 + 34\\\\p = -0.004n + 110`

Thus the linear equation is found