Question

i can write equations and represent linear relationships. 8.51

1. elyse has a gift card to a local movie theater. the graph shows the amount of money

remaining on her gift card based on the number of movies she has seen.
a. write an equation to represent the situation.
b. what is the y - intercept, and what does it mean in the
context of the situation?
c. what is the slope, and what does it mean in the context
of the situation?

Explanation:

Step1: Find two points on the line

From the graph, two points are \((0,50)\) and \((1,42)\).

Step2: Calculate the slope \(m\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Substituting \((x_1,y_1)=(0,50)\) and \((x_2,y_2)=(1,42)\), we get \(m=\frac{42 - 50}{1-0}=\frac{-8}{1}=-8\).

Step3: Use the slope - intercept form \(y = mx + b\)

Since \(b = 50\) (the \(y\) - value when \(x = 0\)) and \(m=-8\), the equation is \(y=-8x + 50\).

Step4: Determine the \(y\) - intercept

The \(y\) - intercept is \(b = 50\). In the context, it means the initial amount of money on the gift card (when \(x = 0\) movies are seen).

Step5: Define the slope

The slope \(m=-8\). In the context, it means the amount of money the gift - card balance decreases per movie seen.

Answer:

a. The equation is \(y=-8x + 50\).
b. The \(y\) - intercept is \(50\). It means the initial amount of money on the gift card.
c. The slope is \(-8\). It means the gift - card balance decreases by \(8\) dollars per movie seen.