To write the linear equation that represents the data in the table, we need to determine the relationship between $x$ and $y$. Here's the table provided:

Step 1: Find the slope (rate of change)

The formula for the slope ($m$) between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points $(5, -22)$ and $(6, -34)$: $m = \frac{-34 - (-22)}{6 - 5} = \frac{-34 + 22}{1} = \frac{-12}{1} = -12$

So, the slope is $-12$.

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

The general form of a linear equation is: $y = mx + b$ We can use one of the points, say $(5, -22)$, and the slope $m = -12$ to solve for $b$.

$-22 = (-12)(5) + b$ $-22 = -60 + b$ $b = -22 + 60 = 38$

Step 3: Write the equation

Now that we have $m = -12$ and $b = 38$, the equation is: $y = -12x + 38$

Thus, the linear equation that gives the rule for this table is: $y = -12x + 38$

Would you like any further details or clarifications?

Here are five related questions to deepen your understanding:

1. How would the equation change if the slope were positive instead of negative?
2. How can you check if a set of points follows a linear pattern using a graph?
3. What does the slope of a line represent in real-world scenarios?
4. How can you determine the y-intercept just by looking at a graph?
5. What is the significance of the y-intercept in different types of linear equations?

Tip: When writing a linear equation, always verify both the slope and y-intercept by substituting a point from the table back into the equation.