To find the linear equation that fits the table, we need to determine the relationship between the $x$-values and the $y$-values.

The points from the table are:

• $(1, 11)$
• $(2, 9)$
• $(3, 7)$
• $(4, 5)$

Step 1: Find the slope $m$

The formula for the slope 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 $(1, 11)$ and $(2, 9)$: $m = \frac{9 - 11}{2 - 1} = \frac{-2}{1} = -2$ So, the slope $m = -2$.

Step 2: Find the y-intercept $b$

The equation of a line is in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. To find $b$, we can substitute one of the points into the equation. Let's use $(1, 11)$: $11 = (-2)(1) + b$ $11 = -2 + b$ $b = 13$

Final Equation:

The linear equation is: $y = -2x + 13$

Would you like more details or have any questions about this solution?

Here are 5 related questions:

1. How do you interpret the slope of $-2$ in this context?
2. Can we use another point to verify the linear equation?
3. What happens to the y-values if the x-values increase beyond 4?
4. How do we determine if a table represents a linear function?
5. Could you find the equation if the points were not in a linear relationship?

Tip: When writing a linear equation, always verify the result by plugging in more than one point from the table to ensure the equation is correct.