1. **State the problem:** We are given a table of values for $x$ and $y$ and need to find the linear equation that relates $y$ to $x$. 2. **Given table:** $$\begin{array}{c|c} x & y \\\hline 0 & 100 \\ 1 & 64 \\ 2 & 28 \\ 3 & -8 \end{array}$$ 3. **Identify the pattern:** The $y$ values decrease as $x$ increases. To find the slope $m$, use the formula for slope between two points: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 4. Calculate slope between points $(0,100)$ and $(1,64)$: $$m = \frac{64 - 100}{1 - 0} = \frac{-36}{1} = -36$$ 5. Verify slope with another pair, e.g., $(1,64)$ and $(2,28)$: $$m = \frac{28 - 64}{2 - 1} = \frac{-36}{1} = -36$$ 6. Since slope $m = -36$ is consistent, the linear function has the form: $$y = mx + b$$ 7. Use point $(0,100)$ to find $b$ (the y-intercept): $$100 = -36 \times 0 + b \implies b = 100$$ 8. **Final linear equation:** $$y = -36x + 100$$ This equation matches the table values and describes the linear relationship between $x$ and $y$.