Question

1. for a wedding, dana bought several dozen roses and several dozen carnations. the roses cost $15 per dozen, and the carnations cost $9 per dozen. dana bought a total of 16 dozen flowers and paid a total of $200. how many roses did she buy? write and solve write and solve a system of equations (2 equations) that represent the problem.

Explanation:

Step1: Define variables

Let \( x \) be the number of dozen roses and \( y \) be the number of dozen carnations.

Step2: First equation (total dozen flowers)

The total number of dozen flowers is 16, so \( x + y = 16 \).

Step3: Second equation (total cost)

Roses cost $15 per dozen, carnations $9 per dozen, total cost $200. So \( 15x + 9y = 200 \).

Step4: Solve the system

From \( x + y = 16 \), we get \( y = 16 - x \). Substitute into the second equation:

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Wait, this seems off. Wait, maybe miscalculation. Let's redo step 4:

Substitute \( y = 16 - x \) into \( 15x + 9y = 200 \):

\( 15x + 9(16 - x) = 200 \)

\( 15x + 144 - 9x = 200 \)

\( 6x + 144 = 200 \)

\( 6x = 200 - 144 = 56 \)

\( x = 56 / 6 = 28 / 3 ≈ 9.33 \). But the number of dozen should be a whole number? Maybe the problem has a typo? Wait, maybe I made a mistake. Wait, total cost $200, 16 dozen. Let's check: if x=9 (9 dozen roses, 7 dozen carnations): 915 +79=135+63=198. If x=10: 1015 +69=150+54=204. So 200 is between. So maybe the problem is correct, and the answer is 28/3 dozen roses, which is 112 roses (since 28/3 dozen *12=112).

Answer:

The system of equations is \(

$$\begin{cases} x + y = 16 \\ 15x + 9y = 200 \end{cases}$$

\), and the number of dozen roses is \( \frac{28}{3} \) (or 112 roses).