
Holly started working at a coffee shop. After two days, she had earned a total of $42.50. After five days, she had earned a total of $106.25. Which equation models the amount of money Holly earned, y, after x days?
A. 85x-4y=0
B. 4x-85y=0
C. 85x+4y=0
D. 4x+85y=0

To find an equation that models Holly's earnings, we need to express the relationship between the number of days $ x $ and the total amount of money earned $ y $.

We know:
- After 2 days, Holly earned $42.50.
- After 5 days, Holly earned $106.25.

### Step 1: Determine the rate of earning
To find Holly's rate of earning per day, we calculate the difference in money earned over the difference in days.

\[
\text{Rate per day} = \frac{106.25 - 42.50}{5 - 2} = \frac{63.75}{3} = 21.25 \, \text{dollars per day}.
\]

### Step 2: Establish the linear equation
Since Holly earns $21.25 per day, the equation modeling her total earnings $ y $ after $ x $ days would be:

\[
y = 21.25x + b
\]

where $ b $ is the y-intercept, which represents her earnings at 0 days (likely $0 in this context). Thus, $ b = 0 $, and the equation simplifies to:

\[
y = 21.25x
\]

### Step 3: Verify the provided answer choices
We will manipulate the form of $ y = 21.25x $ to match one of the provided answer choices.

First, multiply both sides by 4 to eliminate decimals:

\[
4y = 4(21.25x) = 85x
\]

Rearrange to match one of the options:

\[
85x - 4y = 0
\]

This matches **option A: $ 85x - 4y = 0 $**.

### Final Answer:
The correct equation is **A. 85x - 4y = 0**.

Would you like more details or have any other questions? Here are 5 related questions to explore this concept further:

1. How do you derive a slope from two points in general?
2. What is the significance of the y-intercept in linear equations?
3. How can you convert a linear equation from slope-intercept form to standard form?
4. What is the difference between proportional and non-proportional linear relationships?
5. How do you graph a linear equation like $ y = 21.25x $?

**Tip:** Always start by identifying the slope when modeling real-world scenarios with linear equations. It represents the rate of change!

Holly started working at a coffee shop. After two days, she had earned a total of $42.50. After five days, she had earned a total of $106.25. Which equation models the amount of money Holly earned, y, after x days? A. 85x-4y=0 B. 4x-85y=0 C. 85x+4y=0 D. 4x+85y=0

This problem explores how to model Holly's earnings from working at a coffee shop using a linear equation. Learn how to calculate the rate of earnings per day and convert the equation into standard form. Suitable for students in Grades 8-10.

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