Let's solve the problem based on the image provided. We need to write the linear equation that gives the rule for the table, which relates values of $x$ and $y$.

The table shows the following pairs:

• $(1, -31)$
• $(2, -13)$
• $(3, 5)$
• $(4, 23)$

Step 1: Calculate the slope ($m$)

To find the slope, we can use the formula for the slope between two points, $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Let's take two points, for example, $(1, -31)$ and $(2, -13)$: $m = \frac{-13 - (-31)}{2 - 1} = \frac{-13 + 31}{1} = \frac{18}{1} = 18$ The slope $m = 18$.

Step 2: Find the y-intercept ($b$)

We can now use the slope-intercept form of a line, which is $y = mx + b$, to find the $y$-intercept. Using the slope $m = 18$ and one of the points, say $(1, -31)$, we substitute into the equation: $y = mx + b$ $-31 = 18(1) + b$ $-31 = 18 + b$ $b = -31 - 18 = -49$

Step 3: Write the equation

Now that we have the slope $m = 18$ and the y-intercept $b = -49$, the equation of the line is: $y = 18x - 49$

This is the linear equation that gives the rule for the table.

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. How do you find the slope if you are given two different points?
2. Can you derive the equation using another pair of points from the table?
3. What happens to the equation if the slope is negative instead of positive?
4. How can you verify if a point lies on the line based on the equation $y = 18x - 49$?
5. What does the slope and y-intercept represent in the context of this problem?

Tip: Always double-check your calculations when finding the slope and intercept to ensure accuracy in your final equation!