Question: 9. A caterer charges $120 to cater a party for 15 people and $200 for 25 people. Assume that the cost, $y$, is a linear function of the number of $x$ people. Write an equation in slope-intercept form for this function. What does the slope represent? How much would a party for 40 people cost?
1. **State the problem:** We need to find the linear function $y = mx + b$ that models the cost $y$ based on the number of people $x$. Given points are $(15,120)$ and $(25,200)$.
2. **Find the slope $m$:** Use the formula $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{200 - 120}{25 - 15} = \frac{80}{10} = 8.$$ The slope $m = 8$ means the cost increases by 8 units per additional person.
3. **Find the y-intercept $b$:** Use one point, say $(15,120)$, and the equation $y = mx + b$:
$$120 = 8 \times 15 + b$$
$$120 = 120 + b$$
$$b = 120 - 120 = 0.$$ So, $b = 0$.
4. **Write the equation:**
$$y = 8x + 0$$
or simply
$$y = 8x.$$ This means the cost is $8$ times the number of people.
5. **Interpret the slope:** The slope $8$ represents the cost per person.
6. **Calculate cost for 40 people:**
$$y = 8 \times 40 = 320.$$ So, a party for 40 people costs $320$.