To determine which system of equations correctly models the situation given in the problem, we will break down the information provided and form the appropriate equations step-by-step.
1. Understand the variables and their meaning:
- Let [tex]\( B \)[/tex] be the number of bagels sold.
- Let [tex]\( M \)[/tex] be the number of muffins sold.
2. Translate the given conditions into equations:
- Bagels sold for [tex]$0.75 each.
- Muffins sold for $[/tex]1.50 each.
- Jenna sold three times as many bagels as muffins.
- Jenna made a total of $112.50 from selling bagels and muffins.
3. Formulate the equations:
- The condition that Jenna sold three times as many bagels as muffins can be written mathematically as:
[tex]\[
B = 3M
\][/tex]
Alternatively, we can rearrange this equation as:
[tex]\[
B - 3M = 0
\][/tex]
- The condition regarding the total sales amount can be formulated as:
[tex]\[
0.75B + 1.50M = 112.50
\][/tex]
4. Compare the formulated equations with the given options:
- A. [tex]\(\left\{\begin{array}{l}3 B - M = 0 \\ 0.75 B + 1.50 M = 112.5\end{array}\right.\)[/tex]
- Equation 1: [tex]\( 3B - M = 0 \)[/tex] does not match our rearranged form [tex]\( B - 3M = 0 \)[/tex].
- B. [tex]\(\left\{\begin{array}{l}B - 3 M = 112.50 \\ 0.75 B + 1.50 M = 0\end{array}\right.\)[/tex]
- This does not fit any of our conditions both in terms of sales amount and logical quantities.
- C. [tex]\(\left\{\begin{array}{l}B - 3 M = 0 \\ 0.75 B + 1.50 M = 112.50\end{array}\right.\)[/tex]
- Equation 1: [tex]\( B - 3M = 0 \)[/tex] matches our equation from the condition of bagels being three times the muffins.
- Equation 2: [tex]\( 0.75B + 1.50M = 112.50 \)[/tex] matches our equation for the total sales amount.
- D. [tex]\(\left\{\begin{array}{l}B + 3 M = 0 \\ 0.75 B - 1.50 M = 112.50\end{array}\right.\)[/tex]
- Neither of these equations makes sense in the given context. The equations do not align with our formulated equations.
Thus, the system of equations which correctly models the given situation is provided in option C:
[tex]\[
\left\{\begin{array}{l}
B - 3 M = 0 \\
0.75 B + 1.50 M = 112.50
\end{array}\right.
\][/tex]
