To solve this problem, let's break down the information step-by-step in detail:
1. Understand Eric's Earnings Equation:
- Eric's weekly earnings, [tex]\( y \)[/tex], are given by the equation [tex]\( y = 10x + 50 \)[/tex].
- In this equation, [tex]\( x \)[/tex] represents the number of items Eric sells in a week.
- The term [tex]\( 10x \)[/tex] represents Eric's commission, where he earns [tex]$10 per item sold.
- The constant term \( 50 \) represents Eric's fixed weekly salary.
2. Bailey's Earnings Parameters:
- Bailey earns a greater weekly salary than Eric but at the same commission rate.
- This means Bailey's earnings equation will have the same slope (commission rate) but a higher y-intercept (fixed weekly salary).
3. Formulating Bailey's Earnings Equation:
- Keeping the commission rate the same, Bailey's earnings equation can be written as \( y = 10x + b \), where \( b \) is Bailey's fixed weekly salary.
- Since Bailey earns a greater weekly salary than Eric, \( b \) must be greater than 50.
4. Example of Bailey's Earnings Equation:
- Let’s consider that Bailey's weekly salary is slightly higher than Eric's. For example, suppose Bailey's fixed weekly salary is $[/tex]60.
- Therefore, Bailey's earnings equation becomes [tex]\( y = 10x + 60 \)[/tex].
5. Graphical Representation:
- The graph of [tex]\( y = 10x + 60 \)[/tex] will have the same slope as the graph of [tex]\( y = 10x + 50 \)[/tex], but it will intersect the y-axis at [tex]\( y = 60 \)[/tex] rather than at [tex]\( y = 50 \)[/tex].
- This means Bailey's graph will be a straight line starting at a higher point on the y-axis (60 vs 50) but with the same angle of inclination as Eric's graph.
6. Example Calculation for Clarity:
- To illustrate, let’s calculate Bailey's earnings when Bailey sells 5 items ([tex]\( x = 5 \)[/tex]):
- Substitute [tex]\( x = 5 \)[/tex] into Bailey's equation: [tex]\( y = 10(5) + 60 \)[/tex].
- Simplifying, we find [tex]\( y = 50 + 60 = 110 \)[/tex].
- This means that if Bailey sells 5 items, the earnings will be [tex]\( \$110 \)[/tex].
In summary, Bailey's graph will be represented by the equation [tex]\( y = 10x + 60 \)[/tex] instead of Eric's [tex]\( y = 10x + 50 \)[/tex]. The critical difference is the higher y-intercept of 60 in Bailey's graph versus 50 in Eric's graph. This higher y-intercept reflects Bailey's greater weekly salary while maintaining the same commission rate per item sold.
Thus, the graph that represents Bailey's weekly earnings equation will be a straight line with the formula [tex]\( y = 10x + 60 \)[/tex], which starts at [tex]\( y = 60 \)[/tex] on the y-axis and rises at the same rate [tex]\( (10) \)[/tex] as Eric's graph.
