Question

1. a movie theater offers a reward program that charges a yearly membership fee and a discounted rate per movie ticket. the total cost for a reward program member to see 5 movies is $40 and the total cost for 12 movies is $75. assume the relationship is linear. write the equation of the function in the form $y = mx + b$, where $x$ represents the number of movies and $y$ represents the total cost.

Explanation:

Step1: Identify two points

We have two points \((x_1, y_1)=(5, 40)\) and \((x_2, y_2)=(12, 75)\) from the problem.

Step2: Calculate the slope \(m\)

The formula for slope \(m\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substituting the values, we get \(m=\frac{75 - 40}{12 - 5}=\frac{35}{7} = 5\).

Step3: Find the y - intercept \(b\)

We use the point - slope form \(y - y_1=m(x - x_1)\) or substitute \(x = 5\), \(y = 40\) and \(m = 5\) into \(y=mx + b\). So, \(40=5\times5 + b\).
Simplify the right - hand side: \(40 = 25 + b\).
Subtract 25 from both sides: \(b=40 - 25=15\).

Step4: Write the equation

Now that we have \(m = 5\) and \(b = 15\), the equation of the line is \(y = 5x+15\).

Answer:

\(y = 5x + 15\)