To determine if the function is linear, we need to check if the rate of change between each pair of $(x, y)$ values is constant.

Step 1: Calculate the Rate of Change (Slope) Between Each Pair of Points

The slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$

Calculate the slope between each pair of consecutive points:

1. Between $(25, 23.35)$ and $(30, 25.4)$: $\frac{25.4 - 23.35}{30 - 25} = \frac{2.05}{5} = 0.41$

2. Between $(30, 25.4)$ and $(35, 27.45)$: $\frac{27.45 - 25.4}{35 - 30} = \frac{2.05}{5} = 0.41$

3. Between $(35, 27.45)$ and $(40, 29.5)$: $\frac{29.5 - 27.45}{40 - 35} = \frac{2.05}{5} = 0.41$

4. Between $(40, 29.5)$ and $(45, 31.55)$: $\frac{31.55 - 29.5}{45 - 40} = \frac{2.05}{5} = 0.41$

Since the slope is constant (0.41) between each pair of points, the function is indeed linear.

Step 2: Find the Equation of the Line

Now that we have verified that the function is linear, we can use the slope-intercept form $y = mx + b$ to find the equation of the line.

The slope $m$ is $0.41$.

To find the y-intercept $b$, we can use one of the points, such as $(25, 23.35)$: $y = mx + b$ $23.35 = (0.41)(25) + b$ $23.35 = 10.25 + b$ $b = 23.35 - 10.25 = 13.1$

Thus, the equation of the line is: $y = 0.41x + 13.1$

Conclusion

Since the function is linear, the equation that can be used to predict the next entry is: $y = 0.41x + 13.1$

Additional Questions

1. How would the equation change if the slope between points varied slightly?
2. Can we use this equation to make predictions for values of $x$ much larger than 45?
3. What is the significance of the y-intercept in this context?
4. How does the linearity of the function affect its graphical representation?
5. How can we verify the accuracy of our predicted next value?

Tip: To quickly check for linearity in a set of values, calculate the slope between pairs and look for consistency.