To find the linear equation for this table, let's determine the slope and y-intercept for the points provided.

Step 1: Find the Slope (m)

The slope, $m$, can be calculated using the formula: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points $(4, -8)$ and $(5, -10)$: $m = \frac{-10 - (-8)}{5 - 4} = \frac{-10 + 8}{1} = \frac{-2}{1} = -2$

So, the slope $m = -2$.

Step 2: Find the y-intercept (b)

The equation of a line is $y = mx + b$. Using the slope we found and one of the points, say $(4, -8)$, we can substitute into the equation to find $b$: $-8 = -2(4) + b$ $-8 = -8 + b$ $b = 0$

Step 3: Write the Equation

With $m = -2$ and $b = 0$, the linear equation is: $y = -2x$

Answer

The equation that gives the rule for this table is: $y = -2x$

Would you like a detailed breakdown or have any questions about this?

Here are some additional questions that expand on this concept:

1. How would the equation change if the slope were positive?
2. What if the y-intercept was a different value, say 3? How would the graph and equation change?
3. How can you determine if a set of points is linear by examining the table values?
4. What does the slope represent in real-world applications?
5. How would you write the equation if the relationship were not linear?

Tip: For any set of points in a table, you can find the slope by examining the rate of change in $y$ values as $x$ increases.