The table in the image represents a linear relationship between $x$ and $y$. From the given values:

$x = 1, 2, 3, 4, 5$ $y = 15, 23, 31, 39$

To find the missing value of $y$ when $x = 5$, we need to check if the relationship between $x$ and $y$ is consistent, i.e., if the difference between consecutive $y$-values is constant. Let's calculate the differences:

$23 - 15 = 8$ $31 - 23 = 8$ $39 - 31 = 8$

The difference is consistently 8. Therefore, this is indeed a linear relationship, and the equation for the line can be expressed as:

$y = mx + b$

Where $m$ is the slope (which is the difference between consecutive $y$-values), and $b$ is the y-intercept. From the calculations, the slope $m = 8$.

Now, to find the equation of the line, we use one of the known points, such as $(x = 1, y = 15)$:

$y = 8x + b$

Substitute $x = 1$ and $y = 15$ into the equation to find $b$:

$15 = 8(1) + b$ $15 = 8 + b$ $b = 15 - 8 = 7$

So, the equation of the line is:

$y = 8x + 7$

Now, to find the value of $y$ when $x = 5$:

$y = 8(5) + 7 = 40 + 7 = 47$

Thus, the missing value for $y$ when $x = 5$ is 47.

Would you like further details or have any questions? Here are 5 related questions you might find helpful:

1. How do you derive the equation of a line from two points?
2. How can you check if a relationship is linear?
3. What does the slope of a line represent in a real-world scenario?
4. How do you calculate the y-intercept when given a point and the slope?
5. What is the difference between linear and non-linear relationships?

Tip: When working with linear equations, always check the consistency of the differences between $y$-values to confirm that the relationship is linear.