To solve this, we assume a linear relationship between $x$ and $y$. Here's the process:

1. Identify the pattern:

   • For $x = 1, y = 1$.
   • For $x = 2, y = 5$.
   • For $x = 3, y = 9$.
   • For $x = 4, y = 13$.

   Notice the $y$-values increase by 4 as $x$ increases by 1.

2. Write the linear equation: The general equation of a line is $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.

   • Slope $m = \frac{\Delta y}{\Delta x} = \frac{5 - 1}{2 - 1} = 4$.
   • Substitute a known point, $(1, 1)$, into $y = 4x + c$: $1 = 4(1) + c \implies c = -3.$

   Thus, the equation is $y = 4x - 3$.

3. Find the missing value: For $x = 5$: $y = 4(5) - 3 = 20 - 3 = 17.$

Answer: The missing value in the table is $y = 17$ when $x = 5$.

Would you like a detailed explanation of any step? Here are five related questions:

1. How can we verify that $y = 4x - 3$ fits all points in the table?
2. How is the slope calculated in linear relationships?
3. What happens if the relationship is not linear?
4. How would this method change for a non-linear relationship?
5. How can we graph this equation to confirm the result?

Tip: Always double-check calculations for slope and intercept to ensure the equation matches all given points.