Question

a bakery sells cakes for $23 and a dozen cupcakes for $18. in one day they sold 72 items and made a total of $1446. which system is an appropriate model of the problem? a. \\(\

$$\begin{cases} 23x + y = 72 \\\\ 18x + y = 1446 \\end{cases}$$

\\) b. \\(\

$$\begin{cases} x + y = 72 \\\\ 23x + 18y = 4600 \\end{cases}$$

\\) c. \\(\

$$\begin{cases} 23x + 18y = 72 \\\\ x + y = 1446 \\end{cases}$$

\\) d. \\(\

$$\begin{cases} 23x + y = 1446 \\\\ x + 18y = 72 \\end{cases}$$

\\) options: d, b, c, a

Explanation:

Step1: Define Variables

Let \( x \) be the number of cakes sold and \( y \) be the number of dozens of cupcakes sold.

Step2: Set Up Quantity Equation

The total number of items sold is 72. Since each cake is 1 item and each dozen cupcakes is 1 item, the equation for the total number of items is \( x + y = 72 \).

Step3: Set Up Cost Equation

Cakes sell for $23 each, so the total revenue from cakes is \( 23x \). Dozens of cupcakes sell for $18 each, so the total revenue from cupcakes is \( 18y \). The total revenue is $1446, so the cost equation is \( 23x + 18y = 1446 \)? Wait, no, wait. Wait, the options have B as \( x + y = 72 \) and \( 23x + 18y = 1446 \)? Wait, the original problem's option B is \( \begin{cases} x + y = 72 \\ 23x + 18y = 1446 \end{cases} \)? Wait, the user's image shows option B as:

B. \( \begin{cases} x + y = 72 \\ 23x + 18y = 1446 \end{cases} \)? Wait, the user's image: "B. \( \begin{matrix} x + y = 72 \\ 23x + 18y = 4600 \end{matrix} \)" Wait, no, the user's image:

Wait, the user's image:

"A. \( \begin{cases} 23x + y = 72 \\ 18x + y = 1446 \end{cases} \)

B. \( \begin{matrix} x + y = 72 \\ 23x + 18y = 4600 \end{matrix} \) Wait, no, maybe a typo? Wait, the total revenue is $1446, so 23x + 18y should be 1446. But option B has 4600? Wait, maybe the user made a typo, but let's check again.

Wait, the original problem: "made a total of $1446". So the correct cost equation is \( 23x + 18y = 1446 \), and quantity equation \( x + y = 72 \). So looking at the options, maybe option B has a typo, but among the options, let's check each:

Option A: \( 23x + y = 72 \) (wrong, y should be 18y for revenue) and \( 18x + y = 1446 \) (wrong, x is cakes, 23x, y is dozens, 18y).

Option B: \( x + y = 72 \) (correct quantity) and \( 23x + 18y = 1446 \)? Wait, the user's image shows option B as \( 23x + 18y = 4600 \)? That must be a typo. Wait, maybe the original problem's total was $1446, but maybe the option B has a typo, but among the given options, let's re-express:

Wait, the user's image:

Option B:

\( x + y = 72 \)

\( 23x + 18y = 4600 \) – no, that's not. Wait, maybe I misread. Wait, the user's problem: "made a total of $1446". So the correct system is \( \begin{cases} x + y = 72 \\ 23x + 18y = 1446 \end{cases} \). Now, looking at the options, which one is that?

Wait, the options:

A. \( \begin{cases} 23x + y = 72 \\ 18x + y = 1446 \end{cases} \) – wrong.

B. \( \begin{cases} x + y = 72 \\ 23x + 18y = 4600 \end{cases} \) – 4600 is wrong, but maybe a typo. Wait, maybe the original problem's total was $1446, but option B has 4600? No, that can't be. Wait, maybe I made a mistake. Wait, 23*72=1656, 18*72=1296, so total can't be 4600. So maybe the user's image has a typo, but among the options, the only one with \( x + y = 72 \) and \( 23x + 18y \) is option B, even though the total is wrong. But maybe the original problem's total was $1446, and option B has a typo. Alternatively, maybe I misread the options.

Wait, the user's image:

Option B:

\( x + y = 72 \)

\( 23x + 18y = 4600 \) – no. Wait, maybe the user's problem is correct, and I miscalculated. Wait, 23x + 18y = 1446, x + y = 72. Let's solve: from x + y = 72, y = 72 - x. Substitute into 23x + 18(72 - x) = 1446 → 23x + 1296 - 18x = 1446 → 5x = 150 → x = 30, y = 42. Then 23*30 + 18*42 = 690 + 756 = 1446. Correct. Now, looking at the options, which one has x + y = 72 and 23x + 18y = 1446? Option B has 23x + 18y = 4600, which is wrong. Wait, maybe the user's image has a typo, and option B is supposed to be 1446. Alternatively, maybe I misread the options.

Wait, the user's image:

Option B:

\( x + y = 72 \)

\( 23x + 18y = 4600 \) – no. Wait, maybe the original problem's total was $1446, but option B is the only one with x + y = 72 and 23x + 18y. So maybe the 4600 is a typo, and the correct answer is B.

Answer:

B. \(

$$\begin{cases} x + y = 72 \\ 23x + 18y = 1446 \end{cases}$$

\) (assuming 4600 is a typo)