To determine which relationships have the same constant of proportionality between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] as the equation [tex]\( 3y = 27x \)[/tex], we need to first simplify the original equation and then compare it with the given options.
### Step-by-Step Solution:
1. Simplify the Given Equation:
The given equation is [tex]\( 3y = 27x \)[/tex]. We can simplify this to find the constant of proportionality.
[tex]\[
3y = 27x
\][/tex]
Divide both sides by 3:
[tex]\[
y = 9x
\][/tex]
This tells us that the constant of proportionality [tex]\( k \)[/tex] is 9.
2. Analyze Each Option:
Option A: [tex]\( y = 9x \)[/tex]
- This is already in the form [tex]\( y = kx \)[/tex], where [tex]\( k = 9 \)[/tex].
- This matches our constant of proportionality.
Option B: [tex]\( 2y = 18x \)[/tex]
- Divide both sides by 2 to simplify:
[tex]\[
y = 9x
\][/tex]
- This also gives us [tex]\( y = kx \)[/tex] with [tex]\( k = 9 \)[/tex].
- This matches our constant of proportionality.
Option C:
- We have a table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
3 & \frac{1}{3} \\
6 & \frac{2}{3} \\
9 & 1 \\
\hline
\end{array}
\][/tex]
- Let's check each pair to see if [tex]\( \frac{y}{x} \)[/tex] is constant and equal to 9.
[tex]\[
\frac{\frac{1}{3}}{3} = \frac{1}{9}, \quad \frac{\frac{2}{3}}{6} = \frac{1}{9}, \quad \frac{1}{9} = \frac{1}{9}
\][/tex]
- None of these ratios equal 9. Hence, this option does not have the same constant of proportionality.
Option D:
- There is no information given for Option D.
Option E:
- We have a table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 18 \\
4 & 27 \\
6 & 36 \\
\hline
\end{array}
\][/tex]
- Let's check each pair to see if [tex]\( \frac{y}{x} \)[/tex] is constant and equal to 9.
[tex]\[
\frac{18}{2} = 9, \quad \frac{27}{4} = 6.75, \quad \frac{36}{6} = 6
\][/tex]
- Only the first pair has [tex]\( y = 9x \)[/tex]. The others do not match the constant of proportionality.
### Conclusion:
The options that have the same constant of proportionality as [tex]\( y = 9x \)[/tex] are Options A and B.
[tex]\[
\boxed{1, 2}
\][/tex]
