Identifying The Table Representing Ava's Line Slope

In mathematics, understanding the concept of slope is crucial for analyzing linear relationships. The slope of a line, often denoted as 'm', quantifies the steepness and direction of the line. It represents the change in the vertical direction (y-axis) for every unit change in the horizontal direction (x-axis). Ava has written the expression $\frac{4-2}{3-1}$ to determine the slope of a line. Our task is to identify which table of values might represent Ava's line. This involves calculating the slope from the given expression and then comparing it with the slopes derived from the provided tables. Understanding slope is fundamental not only in algebra but also in various real-world applications, such as determining the steepness of a road or the rate of change in a business context. Before we dive into analyzing the tables, let's first dissect Ava's expression and calculate the slope she has determined.

Decoding Ava's Expression for Slope

Ava's expression, $\frac{4-2}{3-1}$, directly relates to the formula for calculating the slope of a line given two points. The slope, often denoted by 'm', is calculated using the formula: m = $\frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are two distinct points on the line. Let's break down Ava's expression in this context. The numerator, 4 - 2, represents the difference in the y-coordinates of two points on the line. The denominator, 3 - 1, represents the difference in the corresponding x-coordinates of the same two points. Evaluating the expression, we get: $\frac{4-2}{3-1}$ = $\frac{2}{2}$ = 1. This calculation tells us that the slope of the line Ava is considering is 1. This means that for every one unit increase in the x-coordinate, the y-coordinate also increases by one unit. Now that we have determined the slope, we can proceed to analyze the given tables to identify the one that represents a line with a slope of 1. This involves applying the slope formula to the coordinates provided in each table and comparing the result with our calculated slope.

Analyzing the Tables to Find Ava's Line

Now, let's examine each table to determine which one represents a line with a slope of 1, as calculated from Ava's expression. We will apply the slope formula (m = $\frac{y_2 - y_1}{x_2 - x_1}$) to each table, using the given x and y values as coordinates. Our goal is to find the table where the calculated slope matches the slope derived from Ava's expression. This process involves careful substitution and arithmetic to ensure accuracy. For each table, we will select two points, substitute their coordinates into the formula, and simplify the expression to find the slope. This methodical approach will allow us to confidently identify the table that represents Ava's line. After calculating the slope for each table, we will compare the results and identify the one that matches our target slope of 1. This comparison is the final step in determining which table represents Ava's line.

Table 1:

Using the points (4, 2) and (3, 1), the slope is calculated as follows: m = $\frac{1 - 2}{3 - 4}$ = $\frac{-1}{-1}$ = 1

Table 2:

Using the points (3, 4) and (-1, -2), the slope is calculated as follows: m = $\frac{-2 - 4}{-1 - 3}$ = $\frac{-6}{-4}$ = $\frac{3}{2}

Table 3:

Using the points (1, 2) and (3, 4), the slope is calculated as follows: m = $\frac{4 - 2}{3 - 1}$ = $\frac{2}{2}$ = 1

Identifying the Correct Table and Its Implications

After calculating the slopes for each table, we can now identify the one that represents Ava's line. We found that Table 1 has a slope of 1, Table 2 has a slope of $\frac{3}{2}$, and Table 3 has a slope of 1. Since Ava's expression yielded a slope of 1, both Table 1 and Table 3 could potentially represent Ava's line. However, the context of the problem often requires us to consider additional factors or information to make a definitive choice. In this case, without further context, both Table 1 and Table 3 are valid representations of Ava's line. This exercise highlights the importance of accurately calculating slope and interpreting its meaning in the context of linear equations. The slope not only tells us about the steepness of the line but also its direction – whether it's increasing or decreasing. Understanding these concepts is crucial for solving a wide range of mathematical problems and real-world applications. The ability to calculate and interpret slope is a fundamental skill in algebra and beyond.

In conclusion, the tables that might represent Ava's line are:

• Table 1:

  x	y
  4	2
  3	1

• Table 3:

  x	y
  1	2
  3	4