To determine which expression is equivalent to [tex]\(\left(\frac{r}{s}\right)(6)\)[/tex], we need to first understand the functions [tex]\( r(x) \)[/tex] and [tex]\( s(x) \)[/tex] and evaluate [tex]\(\frac{r(x)}{s(x)}\)[/tex] at [tex]\(x=6\)[/tex].
Given:
[tex]\[ r(x) = 3x - 1 \][/tex]
[tex]\[ s(x) = 2x + 1 \][/tex]
Now evaluate these functions at [tex]\(x=6\)[/tex]:
1. Calculate [tex]\( r(6) \)[/tex]:
[tex]\[ r(6) = 3 \cdot 6 - 1 = 18 - 1 = 17 \][/tex]
2. Calculate [tex]\( s(6) \)[/tex]:
[tex]\[ s(6) = 2 \cdot 6 + 1 = 12 + 1 = 13 \][/tex]
Now form the ratio [tex]\(\frac{r(6)}{s(6)}\)[/tex]:
[tex]\[ \frac{r(6)}{s(6)} = \frac{17}{13} \][/tex]
We then compare this with the given choices:
- [tex]\(\frac{3(6)-1}{2(6)+1}\)[/tex]
- [tex]\(\frac{(6)}{2(6)+1}\)[/tex]
- [tex]\(\frac{36-1}{26+1}\)[/tex]
- [tex]\(\frac{(B)-1}{(B)+1}\)[/tex]
Let's evaluate the first option to ensure if it matches:
[tex]\[ \frac{3(6)-1}{2(6)+1} = \frac{18-1}{12+1} = \frac{17}{13} \][/tex]
This is exactly what we obtained when we calculated the ratio [tex]\(\frac{r(6)}{s(6)}\)[/tex].
Checking the other options:
- [tex]\(\frac{(6)}{2(6)+1}\)[/tex] simplifies to [tex]\(\frac{6}{13}\)[/tex], which is not equal to [tex]\(\frac{17}{13}\)[/tex].
- [tex]\(\frac{36-1}{26+1}\)[/tex] simplifies to [tex]\(\frac{35}{27}\)[/tex], which is not equal to [tex]\(\frac{17}{13}\)[/tex].
- [tex]\(\frac{(B)-1}{(B)+1}\)[/tex] does not provide a specific value; it is a general form depending on B and is therefore not specific to our problem.
Hence, the expression equivalent to [tex]\(\left(\frac{r}{s}\right)(6)\)[/tex] is:
[tex]\[
\frac{3(6)-1}{2(6)+1}
\][/tex]
