### Inverse Variation: Completing the TableGiven that $ x $ and $ y $ show inverse variation, complete the table.\[\begin{array}{|c|c|c|c|c|}\hlinex & 1 & 3 & \_ & 27 \\\hliney & \_ & \_ & 9 & \_ \\\hline\end{array}\]Inverse variation means that the product of $ x $ and $ y $ is constant. That is, $ x \times y = k $ for some constant $ k $.To complete the table, first determine the constant $ k $ using the given pairs:- For $ x = 1 $ and $ y = 27 $: \[ 1 \times 27 = 27 \implies k = 27 \]Now use $ k = 27 $ to find the missing values.1. When $ x = 3 $: \[ x \times y = 27 \implies 3 \times y = 27 \implies y = \frac{27}{3} = 9 \]2. When $ y = 9 $: \[ x \times 9 = 27 \implies x = \frac{27}{9} = 3 \]3. When $ x = 27 $: \[ x \times y = 27 \implies 27 \times y = 27 \implies y = \frac{27}{27} = 1 \]Now, fill in the table with the calculated values:\[\begin{array}{|c|c|c|c|c|}\hlinex & 1 & 3 & 9 & 27 \\\hliney & 27 & 9 & 3 & 1 \\\hline\end{array}\]This table demonstrates the principle of inverse variation between $ x $ and $ y $.

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Answered: Given that x and y show inverse…

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Given that x and y show inverse variation, complete the table. 3 27 27 9

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Equations and inequalities describe the relationship between two mathematical expressions.

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### Inverse Variation: Completing the Table

Given that $ x $ and $ y $ show inverse variation, complete the table.

\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 3 & \_ & 27 \\
\hline
y & \_ & \_ & 9 & \_ \\
\hline
\end{array}
\]

Inverse variation means that the product of $ x $ and $ y $ is constant. That is, $ x \times y = k $ for some constant $ k $.

To complete the table, first determine the constant $ k $ using the given pairs:
- For $ x = 1 $ and $ y = 27 $:
\[
1 \times 27 = 27 \implies k = 27
\]

Now use $ k = 27 $ to find the missing values.

1. When $ x = 3 $:
\[
x \times y = 27 \implies 3 \times y = 27 \implies y = \frac{27}{3} = 9
\]

2. When $ y = 9 $:
\[
x \times 9 = 27 \implies x = \frac{27}{9} = 3
\]

3. When $ x = 27 $:
\[
x \times y = 27 \implies 27 \times y = 27 \implies y = \frac{27}{27} = 1
\]

Now, fill in the table with the calculated values:

\[
\begin{array}{|c|c|c|c|c|}
\hline
x & 1 & 3 & 9 & 27 \\
\hline
y & 27 & 9 & 3 & 1 \\
\hline
\end{array}
\]

This table demonstrates the principle of inverse variation between $ x $ and $ y $.

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