Aexus Answers Article

Eric's Earnings: $y=10x+50$

Dec 10, 2025 by Alex Johnson 28 views

Eric's Earnings: $y=10x+50$

Eric's Earnings: $y=10x+50$

Let's dive into the world of weekly earnings and commissions, specifically looking at Eric's situation. Eric's weekly earnings are modeled by the equation $y = 10x + 50$ . This equation tells us a few important things. The variable ' $y$ ' represents the total amount of money Eric earns in a week. The variable ' $x$ ' represents the number of items that Eric sells in that week. The number '50' in the equation is his fixed weekly salary, meaning no matter how many items he sells, he is guaranteed to earn at least $50. This is his base pay, the foundation of his weekly income. The number '10' is his commission rate per item. This means for every single item Eric sells, he earns an additional $10 on top of his base salary. So, if Eric sells 0 items, he earns $y = 10(0) + 50 = 50$ . If he sells 1 item, he earns $y = 10(1) + 50 = 60$ . If he sells 10 items, he earns $y = 10(10) + 50 = 150$ . It's a straightforward linear relationship, where his total earnings increase directly with the number of items sold. Understanding this equation is key to comparing his earnings with others, like Bailey, who has a slightly different earning structure.

Now, let's consider Bailey. We are told that Bailey earns a greater weekly salary than Eric. This is a crucial piece of information. If Eric's fixed weekly salary is $50, then Bailey's fixed weekly salary must be more than $50. Let's say Bailey's weekly salary is $S_B$ . Then, $S_B > 50$ . Bailey also has the same commission rate as Eric. Eric's commission rate is $10 per item. Therefore, Bailey also earns $10 for every item she sells. This means the coefficient of '$ x

(the number of items sold) in Bailey's earning equation will also be 10. So, if we were to write an equation for Bailey's weekly earnings, it would look something like

y = 10x + S_B

, where

S_B

is her weekly salary and

S_B > 50

. The core difference between their earning structures lies solely in their base weekly salary. Both are incentivized to sell items at the same rate, but Bailey starts her week with a higher guaranteed income. This difference in their fixed weekly salary will be visually represented when we compare their earnings graphically.

When we think about comparing Eric and Bailey's earnings, we often turn to graphs to visualize these relationships. Eric's equation, $y = 10x + 50$ , represents a straight line on a coordinate plane. The ' $y$ -intercept' of this line is 50, which corresponds to his base salary when he sells zero items. The ' $slope$ ' of the line is 10, indicating that for every one unit increase in ' $x$ ' (selling one more item), ' $y$ ' (his earnings) increases by 10. Bailey's equation would also be a straight line with a slope of 10, but her ' $y$ -intercept' would be a value greater than 50. This means Bailey's line would be parallel to Eric's line, but shifted upwards. The fact that their commission rates are the same means their lines have the same steepness or slope. However, because Bailey's base salary is higher, her line will start at a higher point on the ' $y$ -axis. This graphical representation makes it immediately clear who earns more at any given number of sales, and highlights the impact of a higher base salary in a commission-based earning structure. It's a powerful way to understand how different components of a salary package affect overall income.

Let's consider the implications of these differences. If Eric sells 5 items, his earnings are $y = 10(5) + 50 = 100$ . If Bailey sells 5 items, and let's assume her salary is $60 (which is greater than Eric's 50), her earnings would be $y = 10(5) + 60 = 110$ . In this scenario, Bailey earns $10 more than Eric. Now, what if Eric sells 20 items? His earnings would be $y = 10(20) + 50 = 250$ . If Bailey, with her $60 base salary, also sells 20 items, she would earn $y = 10(20) + 60 = 260$ . Again, she earns $10 more. Notice that the difference in their earnings remains constant at $10 ($110 - $100 = $10 and $260 - $250 = $10). This is because the commission rate is the same. The only factor driving the difference in their total earnings is the difference in their **fixed weekly salaries**. The question asks which graph could represent this situation. We are looking for a graph that shows two parallel lines. One line (representing Eric) must have a '$ y $-intercept' of 50. The other line (representing Bailey) must have a '$ y$-intercept' greater than 50. Both lines should have the same positive slope, indicating a commission of $10 per item sold. Understanding these graphical properties is essential for solving problems that involve comparing linear relationships, especially in contexts like salary and commission structures. It allows us to see the impact of changes in the fixed component versus the variable component of income.

Furthermore, the concept of break-even points and point of intersection becomes relevant when comparing different earning models. In this specific case, because Bailey's base salary is higher and their commission rates are the same, their graphs (lines) will never intersect. Bailey will always earn more than Eric, regardless of the number of items sold. If their commission rates were different, then their lines might intersect at a certain number of sales, indicating a point where one person overtakes the other in earnings. However, with identical commission rates and a higher base salary for Bailey, the lines are parallel and separated by the difference in their base salaries. This scenario highlights how even small differences in fixed components can lead to consistent differences in outcomes over time, especially when the variable components are scaled identically. When analyzing these types of problems, it's crucial to pay close attention to both the slope and the y-intercept of the linear equations involved, as they encapsulate the core dynamics of the situation being modeled. The visual representation provided by a graph is invaluable for quickly grasping these relationships and making informed comparisons.

In summary, when tasked with identifying the correct graph comparing Eric's earnings ( $y = 10x + 50$ ) and Bailey's earnings (where her weekly salary is greater than $50 but her commission rate is the same as Eric's), you should look for two parallel lines. Eric's line will have a '$ y $-intercept' of 50, and Bailey's line will have a '$ y$-intercept' greater than 50. Both lines will exhibit the same upward trend (positive slope) due to the equal commission rates. This visual cue will help you distinguish the correct representation of their weekly earnings.

For more on linear equations and their graphical representations, you can explore resources on ****Khan Academy.