Brand Z's annual sales are affected by the sales of related products X and Y as follows: Each $1 million increase in sales of brand X causes a $2.3 million decline in sales of brand Z, whereas each $1 million increase in sales of brand Y results in an increase of $0.4 million in sales of brand Z. Currently, brands X, Y, and Z are each selling $6 million per year. Model the sales of brand Z using a linear function. (Let z = annual sales of Z (in millions of dollars), x = annual sales of X (in millions of dollars), and y = annual sales of Y (in millions of dollars).)

Expert Answers

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I will first write the linear equation and then explain each part of it.



Now to explain the equation in the first line (the second line is just simplified first line).

First part `6` is the initial value of `z.`

Second part `-2.3(x-6)` is there because each time `x` increases by `1,` `z` decreases by `2.3` and since the starting point for `x` is `6` we have `(x-6).`

Third part is the same as the second part the only difference is that instead of decreasing by `2.3,` `z` increases by `0.4,` hence `+0.4.`

Let's now check our equation.

First let's put in the initial state i.e. `x=6,y=6`


So we get `z=6` which is what we were supposed to get.

Let's check what happens if `x` increases from `6` to `7.`


So `z` is decreased by 2.3 which is what was supposed to happen.  

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