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.

`z=6-2.3(x-6)+0.4(y-6)`

`z=-2,3x+0.4y+17.4`

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`

`z=6-2.3(6-6)+0.4(6-6)=6-0-0=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.`

`z=6-2.3(7-6)+0.4(6-6)=6-2.3=3.7`

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

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