QuestionFind the linear function for $Z$ in terms of $X$ and $Y$ given the sales effects: $Z = -2.8X + 0.5Y + C$ .

Studdy Solution

STEP 1

Assumptions1. The sales of brand $Z$ is affected by the sales of brand $X$ and brand $Y$ . . Each \ $1,000,000 increase in sales of brand$ X $causes a \$,800,000 decline in sales of brand$ Z $.<br />3. Each \$1,000,000 increase in sales of brand$ Y $results in an increase of \$0.5,000,000 in sales of brand$ Z $.<br />4. The relationship between the sales of brands$ X $,$ Y $, and$ Z$ is linear.

STEP 2

We can represent the relationship between the sales of brands $X$ , $Y$ , and $Z$ with a linear equation of the form $Z = aX + bY + c$ , where $a$ , $b$ , and $c$ are constants.

STEP 3

Given that each \ $1,000,000 increase in sales of brand$ X $causes a \$2,800,000 decline in sales of brand$ Z $, we can say that for every unit increase in$ X $,$ Z $decreases by2.8 units. Therefore,$ a = -2.8$.

STEP 4

Similarly, given that each \ $1,000,000 increase in sales of brand$ Y $results in an increase of \$0.,000,000 in sales of brand$ Z $, we can say that for every unit increase in$ Y $,$ Z $increases by0. units. Therefore,$ b =0.$.

STEP 5

Since there is no information given about any fixed amount of sales for brand $Z$ that is independent of the sales of brands $X$ and $Y$ , we can assume that $c =0$ .

STEP 6

Substitute the values of $a$ , $b$ , and $c$ into the linear equation to get the linear function for $Z$ in terms of $X$ and $Y$ .
$Z = -2.8X +0.5Y +0$

STEP 7

implify the equation to get the final linear function for $Z$ in terms of $X$ and $Y$ .
$Z = -2.X +0.5Y$ The linear function for $Z$ in terms of $X$ and $Y$ is $Z = -2.X +0.5Y$ .

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