1. **State the problem:** We want to find the linear equation $y = mx + b$ that best fits the data points $(1,71), (2,91), (3,109), (4,125), (5,142), (6,157)$ using linear regression. 2. **Formula for linear regression slope and intercept:** $$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$b = \frac{\sum y - m \sum x}{n}$$ where $n$ is the number of points. 3. **Calculate sums:** $$\sum x = 1+2+3+4+5+6 = 21$$ $$\sum y = 71+91+109+125+142+157 = 695$$ $$\sum x^2 = 1^2+2^2+3^2+4^2+5^2+6^2 = 91$$ $$\sum xy = 1\times71 + 2\times91 + 3\times109 + 4\times125 + 5\times142 + 6\times157 = 2707$$ 4. **Calculate slope $m$:** $$m = \frac{6 \times 2707 - 21 \times 695}{6 \times 91 - 21^2} = \frac{16242 - 14595}{546 - 441} = \frac{1647}{105}$$ Show cancellation: $$m = \frac{\cancel{1647}}{\cancel{105}} = 15.69$$ 5. **Calculate intercept $b$:** $$b = \frac{695 - 15.69 \times 21}{6} = \frac{695 - 329.49}{6} = \frac{365.51}{6}$$ Show cancellation: $$b = \frac{\cancel{365.51}}{\cancel{6}} = 60.92$$ 6. **Final equation:** $$y = 15.69x + 60.92$$ This is the linear equation that best fits the data using linear regression, rounded to two decimal places.