Evaluating Devon's Linear Equation Model For Data Points

Devon is on a mission to find the perfect equation that represents a line passing through two specific data points: (8, 5) and (-12, -9). After some calculations, Devon proposes the equation 7x - 10y = 3 as a potential model. But is this equation truly a good fit for the data? To answer this crucial question, we need to delve into the fascinating world of linear equations, explore how they represent lines, and evaluate the accuracy of Devon's proposed model.

Understanding Linear Equations and Their Graphical Representation

At its core, a linear equation is a mathematical statement that describes a straight line on a coordinate plane. These equations typically take the form of y = mx + b, where 'm' represents the slope of the line and 'b' represents the y-intercept, the point where the line crosses the vertical axis. The slope, often referred to as the gradient, quantifies the steepness of the line, indicating how much the y-value changes for every unit change in the x-value. A positive slope signifies an upward-sloping line, while a negative slope indicates a downward-sloping line. The y-intercept, on the other hand, determines the vertical position of the line on the coordinate plane. Understanding these fundamental concepts is crucial for evaluating the accuracy of Devon's equation.

Verifying if the Points Satisfy the Equation

The first step in assessing the validity of Devon's equation is to determine whether the given data points, (8, 5) and (-12, -9), actually lie on the line represented by the equation 7x - 10y = 3. To achieve this, we substitute the x and y coordinates of each point into the equation and check if the equation holds true. Let's begin with the point (8, 5). Substituting x = 8 and y = 5 into the equation, we get 7(8) - 10(5) = 56 - 50 = 6. Since 6 is not equal to 3, the point (8, 5) does not satisfy Devon's equation. This initial finding raises concerns about the accuracy of the model.

Now, let's examine the point (-12, -9). Substituting x = -12 and y = -9 into the equation, we get 7(-12) - 10(-9) = -84 + 90 = 6. Again, 6 is not equal to 3, indicating that the point (-12, -9) also does not satisfy Devon's equation. This further reinforces the notion that Devon's proposed equation might not be the best fit for the given data points. The fact that neither point lies on the line suggests that the equation may not accurately represent the relationship between the data points.

Finding the Correct Equation of the Line

Since Devon's equation appears to be inaccurate, our next step is to determine the correct equation of the line that passes through the points (8, 5) and (-12, -9). To achieve this, we can employ the slope-intercept form of a linear equation, y = mx + b. The first task is to calculate the slope, 'm,' using the formula: m = (y2 - y1) / (x2 - x1). Plugging in the coordinates of our points, we get m = (-9 - 5) / (-12 - 8) = -14 / -20 = 7/10. This slope indicates that for every 10 units moved horizontally, the line rises 7 units vertically.

Now that we have the slope, we can substitute it, along with the coordinates of one of the points, into the slope-intercept form to solve for the y-intercept, 'b.' Let's use the point (8, 5). Substituting x = 8, y = 5, and m = 7/10 into the equation y = mx + b, we get 5 = (7/10)(8) + b. Simplifying, we have 5 = 56/10 + b. To isolate 'b,' we subtract 56/10 from both sides, resulting in b = 5 - 56/10 = 50/10 - 56/10 = -6/10 = -3/5. This y-intercept tells us where the line crosses the vertical axis.

With both the slope (m = 7/10) and the y-intercept (b = -3/5) determined, we can now construct the equation of the line in slope-intercept form: y = (7/10)x - 3/5. To express this equation in standard form, we can multiply both sides by 10 to eliminate the fractions, resulting in 10y = 7x - 6. Rearranging the terms, we obtain the standard form equation: 7x - 10y = 6. This equation represents the line that accurately passes through the points (8, 5) and (-12, -9).

Comparing Devon's Equation with the Correct Equation

Comparing Devon's proposed equation, 7x - 10y = 3, with the correct equation, 7x - 10y = 6, we observe a subtle yet significant difference in the constant term. Devon's equation has a constant term of 3, while the correct equation has a constant term of 6. This difference indicates that Devon's line, while having the same slope as the correct line, is shifted vertically, causing it to miss the given data points. The consistent slope suggests that Devon was on the right track in understanding the rate of change between the points, but the incorrect constant term reveals a miscalculation in determining the line's vertical position.

Conclusion Is Devon's Equation a Good Model?

Based on our analysis, Devon's equation, 7x - 10y = 3, is not a good model for the line passing through the points (8, 5) and (-12, -9). While the equation shares the same slope as the correct equation, it fails to accurately represent the line due to an incorrect constant term. This discrepancy leads to the line missing the given data points, rendering it an unsuitable model. To accurately represent the relationship between the points, the equation 7x - 10y = 6 should be used instead. This equation captures both the slope and the vertical position of the line, ensuring that it precisely passes through the specified data points. In conclusion, while Devon's effort is commendable, a more accurate equation is necessary to effectively model the given data.

In summary, to determine if an equation is a good model, you need to ensure that the data points satisfy the equation. Furthermore, calculating the equation using the slope-intercept form or other methods provides a means of comparison to assess the accuracy of the model.