Let's work through this step by step based on the questions presented in your problem.

a. What is the slope of the line?

The slope of a line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ We can use any two points from the table. Let’s use the first two points $(29, 97)$ and $(18, 64)$: $m = \frac{64 - 97}{18 - 29} = \frac{-33}{-11} = 3$ So, the slope $m = 3$.

b. Does it matter which points you use to calculate the slope of the line?

The slope should be consistent regardless of which two points you use because the points lie on a straight line. Let's verify this by calculating the slope using two different pairs of points:

Using $(18, 64)$ and $(14, 52)$: $m = \frac{52 - 64}{14 - 18} = \frac{-12}{-4} = 3$ The slope is again $3$.

Using $(14, 52)$ and $(-27, -71)$: $m = \frac{-71 - 52}{-27 - 14} = \frac{-123}{-41} = 3$ The slope is still $3$, confirming that it does not matter which points you use.

c. How can you use the slope and a point to write an equation?

We can use the point-slope form of the equation, which is: $y - y_1 = m(x - x_1)$ where $m$ is the slope, and $(x_1, y_1)$ is any point on the line. Let's use the point $(29, 97)$ and the slope $m = 3$: $y - 97 = 3(x - 29)$ Simplifying this equation: $y - 97 = 3x - 87$ $y = 3x - 87 + 97$ $y = 3x + 10$ Thus, the equation of the line in slope-intercept form is: $y = 3x + 10$

Would you like further details or clarifications? Let me know if you want more explanation on any part of this!

Here are five related questions to explore further:

1. What does the slope represent in terms of the relationship between $x$ and $y$ in this context?
2. How would the equation change if the slope were negative?
3. Can you use another pair of points to reverify the slope and equation?
4. How can you convert the equation back to point-slope form from slope-intercept form?
5. What would the graph of this line look like?

Tip:

When checking if points lie on the same line, consistently finding the same slope between any two points is a good indicator!