1. **State the problem:** We are given two points $(86,113)$ and $(176,239)$ and need to find the linear equation relating $R$ and $C$ in the form $R = mC + b$. 2. **Calculate the slope $m$:** The slope formula is $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ Substitute the points: $$m = \frac{239 - 113}{176 - 86} = \frac{126}{90} = 1.4$$ 3. **Find the intercept $b$:** Use the point-slope form $y = mx + b$ with one point, say $(86,113)$: $$113 = 1.4 \times 86 + b$$ Calculate: $$113 = 120.4 + b$$ Solve for $b$: $$b = 113 - 120.4 = -7.4$$ 4. **Write the linear equation:** $$R = 1.4C - 7.4$$ 5. **Solve for $C$ when $R = 100$:** Set $R=100$: $$100 = 1.4C - 7.4$$ Add $7.4$ to both sides: $$100 + 7.4 = 1.4C$$ $$107.4 = 1.4C$$ Divide both sides by $1.4$: $$C = \frac{107.4}{1.4}$$ Show cancellation: $$C = \frac{\cancel{107.4}}{\cancel{1.4}} = 76.71$$ **Final answer:** $$C = 76.71$$ This means when $R=100$, the value of $C$ is approximately $76.71$.