Question

1. for the three - part question that follows, provide your answer to each part in the given workspace. identify each part with a coordinating response. be sure to clearly label each part of your response as part a, part b, and part c.

a bakery charges $2 for each cupcake and $1 for each cookie. today, a customer bought a total of 60 cupcakes and cookies for $85. let x represent the number of cupcakes and y represent the number of cookies. the system of equations is graphed below.
part a: identify the constraints.
part b: explain what the constraints from part a represent.
part c: is there a viable solution? if so, name the ordered pair and describe what it means. if not, explain why.

Explanation:

Part A:

We know that the total number of cupcakes and cookies is 60, so $x + y=60$. Also, the total cost is $85$, and since cupcakes cost $2$ each and cookies cost $1$ each, we have $2x + y=85$. The non - negativity constraints are $x\geq0$ and $y\geq0$ as the number of cupcakes and cookies cannot be negative.

Part B:

The equation $x + y = 60$ represents the total quantity of cupcakes and cookies the customer bought, which is 60 items in total. The equation $2x + y=85$ represents the total cost of the purchase, where the cost of cupcakes ($2$ per cupcake times the number of cupcakes $x$) and the cost of cookies ($1$ per cookie times the number of cookies $y$) sums up to $85$. The non - negativity constraints $x\geq0$ and $y\geq0$ mean that we are dealing with real - world quantities of items that cannot be negative.

Part C:

We can solve the system of equations

$$\begin{cases}x + y=60\\2x + y=85\end{cases}$$

by subtracting the first equation from the second equation. $(2x + y)-(x + y)=85 - 60$. This simplifies to $2x + y-x - y=25$, so $x = 25$. Substitute $x = 25$ into $x + y=60$, we get $25+y=60$, then $y=60 - 25=35$. The ordered pair is $(25,35)$. It means the customer bought 25 cupcakes and 35 cookies. Since $x = 25\geq0$ and $y = 35\geq0$, it is a viable solution.

Answer:

Part A: $x + y=60$, $2x + y=85$, $x\geq0$, $y\geq0$
Part B: $x + y = 60$ represents total quantity, $2x + y=85$ represents total cost, non - negativity represents non - negative real - world quantities.
Part C: Yes, the ordered pair is $(25,35)$. It means the customer bought 25 cupcakes and 35 cookies.