To solve this problem, we need to verify the linear relationship between $x$ and $y$. The relationship is of the form:

$y = mx + c$

where $m$ is the slope and $c$ is the y-intercept.

Step 1: Find the slope $m$

The slope $m$ can be calculated using any two consecutive points: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points $(1, 15)$ and $(2, 23)$: $m = \frac{23 - 15}{2 - 1} = \frac{8}{1} = 8$

So, the slope $m = 8$.

Step 2: Find the missing value for $y$ when $x = 5$

We use the slope and the given relationship: $y = mx + c$

To find $c$, substitute $x = 1$ and $y = 15$ into the equation: $15 = 8(1) + c \implies c = 15 - 8 = 7$

The equation becomes: $y = 8x + 7$

Substitute $x = 5$: $y = 8(5) + 7 = 40 + 7 = 47$

Final Answer:

The missing value is: $\boxed{47}$

Would you like further details or have any questions? Here are some related questions to extend the concept:

1. How can we generalize this method for other types of relationships (e.g., quadratic or exponential)?
2. Can you verify the linear relationship using all the given data points?
3. What happens if the slope between points is not constant? How would you solve the problem?
4. How do you interpret the y-intercept $c$ in real-world scenarios?
5. What are alternative methods for solving problems involving linear equations?

Tip: Always verify the linearity of data by checking that the slope $\frac{\Delta y}{\Delta x}$ is constant between all points.