Select the correct answer.

The total cost of a flower bouquet depends on the types of flowers in the arrangement. A bouquet containing 6 roses, 3 lilies, and 2 carnations costs \[tex]$26.75. A bouquet containing 3 roses, 4 lilies, and 5 carnations costs \$[/tex]25.50. A bouquet containing 1 of each flower costs \$6.75.

Which system of equations represents this situation, where [tex]r[/tex] represents the number of roses, [tex]l[/tex] represents the number of lilies, and [tex]c[/tex] represents the number of carnations?

A.
[tex]\[
6r + 3l + 2c = 25.5 \\
3r + 4l + 5c = 26.75 \\
3r = 6.75
\][/tex]

B.
[tex]\[
\begin{aligned}
6r + 3l + 2c & = 26.75 \\
3r + 4l + 5c & = 25.5 \\
r + l + c & = 6.75
\end{aligned}
\][/tex]

C.
[tex]\[
\begin{aligned}
6r + 3l + 2c & = 25.5 \\
3r + 4l + 5c & = 26.75 \\
r + l + c & = 6.75
\end{aligned}
\][/tex]

D.
[tex]\[
\begin{aligned}
6r + 3l + 2c & = 26.75 \\
3r + 4l + 5c & = 25.5 \\
3r & = 6.75
\end{aligned}
\][/tex]



Answer :

To determine which system of equations represents the situation accurately, we need to translate the given information into mathematical equations. We are given the costs of three different bouquets:

1. A bouquet with 6 roses, 3 lilies, and 2 carnations costs \$26.75.
2. A bouquet with 3 roses, 4 lilies, and 5 carnations costs \$25.50.
3. A bouquet with 1 rose, 1 lily, and 1 carnation costs \$6.75.

We need to represent these facts with a system of linear equations where \( r \) represents the cost of one rose, \( l \) represents the cost of one lily, and \( c \) represents the cost of one carnation.

Let's write down the equations for each bouquet:

1. For the first bouquet:
[tex]\[ 6r + 3l + 2c = 26.75 \][/tex]

2. For the second bouquet:
[tex]\[ 3r + 4l + 5c = 25.50 \][/tex]

3. For the third bouquet:
[tex]\[ r + l + c = 6.75 \][/tex]

Now we compare these equations to the given choices:

A.
[tex]\[ 6r + 3l + 2c = 25.50 \\ 3r + 4l + 5c = 26.75 \\ 3r = 6.75 \][/tex]

B.
[tex]\[ \begin{aligned} 6r + 3l + 2c & = 26.75 \\ 3r + 4l + 5c & = 25.50 \\ r + l + c & = 6.75 \end{aligned} \][/tex]

C.
[tex]\[ 6r + 3l + 2c = 25.50 \\ 3r + 4l + 5c = 26.75 \\ r + l + c = 6.75 \][/tex]

D.
[tex]\[ \begin{aligned} 6r + 3l + 2c & = 26.75 \\ 3r + 4l + 5c & = 25.50 \\ 3r & = 6.75 \end{aligned} \][/tex]

Comparing these, we see that the correct system matches the equations we wrote down:

B.
[tex]\[ \begin{aligned} 6r + 3l + 2c & = 26.75 \\ 3r + 4l + 5c & = 25.50 \\ r + l + c & = 6.75 \end{aligned} \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{B} \][/tex]