1. **State the problem:** We are given a table of values for $x$ and $y$ and need to find the linear equation $y = mx + b$ that fits the data. 2. **Identify the points:** The table gives points $(31, 27)$, $(47, 43)$, $(63, 59)$, and $(79, 75)$. 3. **Find the slope $m$:** Use the formula for slope between two points: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Using points $(31, 27)$ and $(47, 43)$: $$m = \frac{43 - 27}{47 - 31} = \frac{16}{16} = 1$$ 4. **Verify slope consistency:** Check slope between $(47, 43)$ and $(63, 59)$: $$m = \frac{59 - 43}{63 - 47} = \frac{16}{16} = 1$$ Slope is consistent, so the function is linear with slope $m=1$. 5. **Find the y-intercept $b$:** Use the equation $y = mx + b$ and one point, for example $(31, 27)$: $$27 = 1 \times 31 + b$$ $$27 = 31 + b$$ Subtract 31 from both sides: $$27 - 31 = \cancel{31} + b - \cancel{31}$$ $$-4 = b$$ 6. **Write the final equation:** $$y = 1x - 4$$ Or simply: $$y = x - 4$$ **Answer:** The linear equation that fits the table is $y = x - 4$.