1. **Problem Statement:** Find the slope of a line and understand its meaning in different contexts. 2. **Formula for slope:** The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ This formula calculates the "rise" (change in $y$) over the "run" (change in $x$). 3. **Important rules:** - The slope measures the steepness of a line. - For proportional relationships, the slope equals the unit rate and constant of proportionality. - The slope between any two points on a straight line is always the same because the line is linear. 4. **Example 1: Theater Price** - Points: $(0,0)$ and $(6,90)$ - Calculate slope: $$m = \frac{90 - 0}{6 - 0} = \frac{90}{6} = 15$$ - Interpretation: The slope 15 means each ticket costs 15 units. 5. **Example 2: Cost of Grapes** - Points: $(0,0)$ and $(4,10)$ - Calculate slope: $$m = \frac{10 - 0}{4 - 0} = \frac{10}{4} = 2.5$$ - Interpretation: The price is 2.5 units per pound of grapes. 6. **Example 3: Model Airplane Scale** - Points: $(3,5)$ and $(6,10)$ - Calculate slope: $$m = \frac{10 - 5}{6 - 3} = \frac{5}{3} \approx 1.67$$ - Interpretation: Each centimeter on the model corresponds to approximately 1.67 feet in real size. **Final answers:** - Theater Price slope: 15 - Cost of Grapes slope: 2.5 - Model Airplane slope: $\frac{5}{3}$ or approximately 1.67 Understanding slope helps relate changes in one quantity to changes in another in real-world contexts.