1. **State the problem:** Given the table of values for $x$ and $y$, find the linear equation that relates $y$ to $x$. 2. **Identify the pattern:** From the table, as $x$ increases by 1, $y$ decreases by 8. This suggests a linear relationship of the form $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 3. **Calculate the slope $m$:** $$m = \frac{\Delta y}{\Delta x} = \frac{-31 - (-23)}{5 - 4} = \frac{-8}{1} = -8$$ 4. **Use a point to find $b$:** Using the point $(4, -23)$, $$-23 = -8 \times 4 + b$$ $$-23 = -32 + b$$ $$b = -23 + 32 = 9$$ 5. **Write the equation:** $$y = -8x + 9$$ 6. **Verify with another point:** For $x=5$, $$y = -8 \times 5 + 9 = -40 + 9 = -31$$ which matches the table. **Final answer:** $$y = -8x + 9$$