To determine which equation best represents the relationship between the number of shirts purchased ([tex]\(s\)[/tex]) and the cost per shirt ([tex]\(c\)[/tex]), we need to test each equation with the given [tex]\(s\)[/tex] and [tex]\(c\)[/tex] values.
The given table:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
s & 3 & 6 & 9 & 12 \\
\hline
c & 9.5 & 9 & 8.5 & 8 \\
\hline
\end{array}
\][/tex]
Let's test each of the proposed equations:
Option A: [tex]\(c = 10 - \frac{s}{6}\)[/tex]
1. For [tex]\(s = 3\)[/tex]:
[tex]\[
c = 10 - \frac{3}{6} = 10 - 0.5 = 9.5
\][/tex]
[tex]\(c\)[/tex] matches the table value.
2. For [tex]\(s = 6\)[/tex]:
[tex]\[
c = 10 - \frac{6}{6} = 10 - 1 = 9
\][/tex]
[tex]\(c\)[/tex] matches the table value.
3. For [tex]\(s = 9\)[/tex]:
[tex]\[
c = 10 - \frac{9}{6} = 10 - 1.5 = 8.5
\][/tex]
[tex]\(c\)[/tex] matches the table value.
4. For [tex]\(s = 12\)[/tex]:
[tex]\[
c = 10 - \frac{12}{6} = 10 - 2 = 8
\][/tex]
[tex]\(c\)[/tex] matches the table value.
Since the values match for all given [tex]\(s\)[/tex], Option A is a likely candidate. However, we should confirm by checking the other options.
Option B: [tex]\(c = 15 - s\)[/tex]
1. For [tex]\(s = 3\)[/tex]:
[tex]\[
c = 15 - 3 = 12
\][/tex]
[tex]\(c\)[/tex] does not match the table value of 9.5.
Since the first calculation does not match the table, this option is incorrect.
Option C: [tex]\(c = s - 0.5\)[/tex]
1. For [tex]\(s = 3\)[/tex]:
[tex]\[
c = 3 - 0.5 = 2.5
\][/tex]
[tex]\(c\)[/tex] does not match the table value of 9.5.
Since the first calculation does not match the table, this option is incorrect.
Option D: [tex]\(c = \frac{3s}{2}\)[/tex]
1. For [tex]\(s = 3\)[/tex]:
[tex]\[
c = \frac{3 \times 3}{2} = \frac{9}{2} = 4.5
\][/tex]
[tex]\(c\)[/tex] does not match the table value of 9.5.
Since the first calculation does not match the table, this option is incorrect.
After evaluating all options, we see that only Option A correctly represents the data given in the table for all values of [tex]\(s\)[/tex]. Therefore, the equation that represents this relationship algebraically is:
Option A: [tex]\(c = 10 - \frac{s}{6}\)[/tex]
