1. **State the problem:** We are given a table of values for $x$ and $y$ and need to determine if these values satisfy a linear equation of the form $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 2. **Recall the formula for slope:** The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Calculate the slope between consecutive points:** - Between $(0,7)$ and $(1,1)$: $$m = \frac{1 - 7}{1 - 0} = \frac{-6}{1} = -6$$ - Between $(1,1)$ and $(2,-5)$: $$m = \frac{-5 - 1}{2 - 1} = \frac{-6}{1} = -6$$ - Between $(2,-5)$ and $(3,-11)$: $$m = \frac{-11 - (-5)}{3 - 2} = \frac{-6}{1} = -6$$ - Between $(3,-11)$ and $(4,-17)$: $$m = \frac{-17 - (-11)}{4 - 3} = \frac{-6}{1} = -6$$ - Between $(4,-17)$ and $(5,-23)$: $$m = \frac{-23 - (-17)}{5 - 4} = \frac{-6}{1} = -6$$ 4. **Interpretation:** The slope is constant at $-6$ between all consecutive points, which is a key property of linear functions. 5. **Find the y-intercept $b$:** Using the point $(0,7)$ and the slope $m = -6$, substitute into the linear equation: $$7 = -6 \times 0 + b \implies b = 7$$ 6. **Write the linear equation:** $$y = -6x + 7$$ 7. **Verify the equation with another point:** For $x=2$, $$y = -6 \times 2 + 7 = -12 + 7 = -5$$ which matches the table. **Final answer:** The table satisfies the linear equation $$y = -6x + 7$$.