What is the linear equation to this problem?

Question

I need the linear equation in terms of the price and intrerpret the slope.

A manufacturer of pickup trucks has determined that 50,000 trucks per month can be sold at a price of $18,000 per truck. At a price of $17,000 the number of trucks sold per month would increase to 55,000.

What is the equation and interpret the slope.

Answer 1

Slope = rise/run = (55000-50000) / (17000-18000) = 5000/-1000 = -5

The slope is negative, which means the higher we price the trucks, the fewer we sell.

If we use the form y=mx +b for the line,

Then b = y - mx, where m is the slope, (-5)so B = 50000 +5(18000) = 140000

So Sales = 140000 - (5 * Price)


Just to check:

50,000 = 140000 - (5 * 18000) Yup
55,000 = 140000 - (5 * 17000) Yup

Note that the true behavior is only approximately linear. If we gave away trucks for free, we could give away a lot more than 140,000. If we sold the trucks for $28,000 each, we might sell one now and then.

Answer 2

There are two ways you can do this, depending on what you want to consider the dependent and independent variables. Let's assume that you're concerned with how much money you can make - that'll make the number of trucks the independent variable and the price per truck the dependent variable. In that case, you have two points:

(50000, 18000) and (55000, 17000)

Your slope, then, is (change in y)/(change in x) or

(17000 - 18000)/(55000 - 50000) =

-1000/5000 = -.2

This slope means that, for every additional truck you make beyond 50000, you'll drop your price by .2 dollars per truck. The negative is what tells you that the price is dropping.

Note, please, that there are other ways to work this problem, with corresponding differences in the interpretation of the slope. One way to start figuring out what the slope means is to look at the units. In this case, we have "dollars" over "dollars per truck". Since the slope is negative, that would translate to "lost dollars per dollars per truck". Other examples might be more clear.

Hope this helps.