Comparing Slopes: Marty's Equation Vs. Ethan's Table

Let's dive into the world of linear functions and slopes! We've got two folks here, Marty and Ethan, each representing a linear function in their own way. Marty's got an equation, and Ethan's rocking a table of values. The big question? Which one of them has the steeper slope? That is, which function's graph is climbing faster? So, let's find out how to compare Marty's slope with Ethan's slope, figuring out who's on the steeper climb.

Understanding Slope: The Key to Linear Functions

Before we jump into the nitty-gritty, let's make sure we're all on the same page about slope. Think of slope as the rate of change in a line. It tells us how much the 'y' value changes for every one-unit change in the 'x' value. A steeper slope means the line is rising or falling more quickly. Mathematically, we often represent slope with the letter 'm', and we calculate it using the formula: m = (change in y) / (change in x), often written as Δy/Δx. Understanding this fundamental concept is key to comparing Marty's and Ethan's functions effectively and accurately. We need to break down how each of them presents their function and then translate that into a slope value that we can compare directly.

Marty's Equation: Unveiling the Slope

Marty's giving us his function as an equation: v + 3 = rac{1}{3}(x + 9). Now, to easily spot the slope, we want to get this equation into the slope-intercept form, which looks like $y = mx + b$. Here, 'm' is our slope, and 'b' is the y-intercept (where the line crosses the y-axis). Let's do some algebraic maneuvering to transform Marty's equation. First, we'll replace 'v' with 'y' to keep our variables consistent. So, we have y + 3 = rac{1}{3}(x + 9). Next, we distribute the $\frac{1}{3}$ on the right side: $y + 3 = \frac{1}{3}x + 3$. Finally, we subtract 3 from both sides to isolate 'y': $y = \frac{1}{3}x$. Aha! Now it's in slope-intercept form. We can clearly see that Marty's slope is $\frac{1}{3}$. Remember, the slope is the coefficient in front of 'x' when the equation is in $y = mx + b$ form. This is a crucial step in comparing the two functions, as it provides a clear numerical value for Marty's rate of change. By converting the equation to slope-intercept form, we make it much easier to compare with Ethan's function, which is presented in a different format.

Ethan's Table: Finding Slope from Points

Ethan's showing off his function with a table of values. This table gives us specific points (x, y) that lie on the line. To find the slope from a table, we need to pick any two points and use our slope formula: m = Δy/Δx. Let's grab the first two points from Ethan's table: (-4, 9.2) and (-2, 9.6). Now, we calculate the change in y (Δy) and the change in x (Δx). Δy is 9.6 - 9.2 = 0.4, and Δx is -2 - (-4) = 2. So, Ethan's slope (m) is 0.4 / 2 = 0.2. We could have chosen any two points from the table, and we would have gotten the same slope because it's a linear function, meaning the rate of change is constant. This method demonstrates the flexibility of working with linear functions, as we can determine the slope from an equation, a table, or even a graph, as long as we understand the underlying principles.

The Slope Showdown: Marty vs. Ethan

Alright, we've crunched the numbers and we've got our slopes! Marty's slope is $\frac{1}{3}$ (which is approximately 0.33), and Ethan's slope is 0.2. Now it's time for the showdown! Which slope is bigger? Comparing 0.33 and 0.2, we can clearly see that 0.33 is greater than 0.2. Therefore, Marty's function has the larger slope. This means that for every unit increase in 'x', Marty's 'y' value increases by a larger amount than Ethan's 'y' value. In practical terms, Marty's line is climbing more steeply than Ethan's line. Understanding this comparison allows us to visualize the differences in the behavior of the two functions and to appreciate how slope affects the overall graph of a linear equation. This kind of comparison is a fundamental skill in algebra and is essential for understanding more complex mathematical concepts.

The Verdict: Whose Function Climbs Higher?

So, there you have it, guys! The final verdict is in: Marty's function has the larger slope. This corresponds to option A in the original question, which stated Marty's slope is $\frac{2}{3}$. Wait a minute! Did we make a mistake? Our calculation showed Marty's slope is $\frac{1}{3}$, not $\frac{2}{3}$. It seems there might be an error in the provided option. Our analysis, based on correctly converting Marty's equation and calculating Ethan's slope, definitively shows Marty's slope as $\frac{1}{3}$, which is still greater than Ethan's 0.2. This highlights the importance of carefully working through the steps and not simply relying on provided answers. Even if an option seems plausible, it's crucial to verify the results through independent calculation and analysis. In this case, we've not only determined the correct relationship between the slopes but also identified a potential error in the given information, showcasing the power of critical thinking in mathematics.

Why Does Slope Matter? Real-World Applications

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