Question 1166455: Eric earns a weekly salary and a commission on each item that he sells. The equation y = 10x + 50 represents the amount of money that Eric earns weekly. Bailey earns a greater weekly salary than Eric but the same commission rate. Which graph could represent the amount of money that Bailey earns weekly, y, based on the number of items sold, x?
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! This is a comparison problem based on the properties of linear equations, specifically the slope and the y-intercept.
The correct graph representing Bailey's weekly earnings will be a line that is **parallel to Eric's graph** but **shifted vertically higher** on the coordinate plane .
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## 📈 Analysis of the Earnings Equations
Both Eric's and Bailey's earnings can be modeled by a linear equation in the slope-intercept form: $y = mx + b$.
* $y$: Total weekly earnings.
* $x$: Number of items sold.
* $m$: **Commission rate** (slope).
* $b$: **Weekly salary** (y-intercept).
### 1. Eric's Earnings
Eric's equation is: $y = 10x + 50$
* **Slope ($m_{Eric}$):** $10$. This is his commission rate per item sold.
* **Y-intercept ($b_{Eric}$):** $50$. This is his weekly salary.
### 2. Bailey's Earnings
The prompt gives two conditions for Bailey's equation:
* **Same commission rate:** This means Bailey's slope ($m_{Bailey}$) must be the same as Eric's slope ($m_{Eric}$).
$$m_{Bailey} = 10$$
* In graphical terms, the two lines must be **parallel**.
* **Greater weekly salary:** This means Bailey's y-intercept ($b_{Bailey}$) must be greater than Eric's y-intercept ($b_{Eric} = 50$).
$$b_{Bailey} > 50$$
* In graphical terms, Bailey's line must cross the y-axis at a **higher point** than Eric's line.
## 🖼️ Conclusion for Bailey's Graph
The graph that represents Bailey's earnings must satisfy these two conditions:
1. **It must be a line with the same steepness (slope of 10) as Eric's line.**
2. **It must be a line that starts at a higher y-intercept (a point above 50 on the y-axis).**
Therefore, the graph that represents Bailey's earnings is a **parallel line that lies entirely above Eric's line**.
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