Devon's Linear Equation Model Analysis For Data Points (8, 5) And (-12, -9)

Devon is on a quest to find the perfect equation for a line that gracefully glides through two pivotal data points: (8, 5) and (-12, -9). He confidently presents his equation: 7x - 10y = 3. But the burning question remains: Is this a good model? Let's embark on a comprehensive journey to dissect Devon's equation, meticulously examining its validity and exploring alternative approaches to find the most accurate representation of the line passing through these points. This exploration will not only validate or refute Devon's solution but also deepen our understanding of linear equations and their applications in modeling data. The process involves several key steps, including verifying if the given points satisfy the equation, calculating the slope and y-intercept using these points, and comparing the derived equation with Devon's proposed model. This rigorous analysis will ensure that we leave no stone unturned in our quest for the most accurate linear model. Furthermore, we'll delve into the significance of accurate linear modeling in various fields, highlighting its importance in data analysis and prediction. The core of our analysis will revolve around understanding the fundamental properties of linear equations, such as slope and y-intercept, and how they relate to the points on the line. By meticulously examining these aspects, we can provide a definitive answer to whether Devon's equation is indeed a good model for the given data points.

Verifying the Points on Devon's Line

The initial step in our analysis is to verify whether the given points, (8, 5) and (-12, -9), actually lie on the line represented by Devon's equation, 7x - 10y = 3. This verification is crucial because if the points do not satisfy the equation, it immediately indicates that the equation is not a correct model for the line passing through these points. To perform this verification, we will substitute the x and y coordinates of each point into the equation and check if the equation holds true. For the point (8, 5), we substitute x = 8 and y = 5 into the equation, which gives us 7(8) - 10(5) = 56 - 50 = 6. Since 6 ≠ 3, the point (8, 5) does not satisfy Devon's equation. This finding is significant because it suggests that Devon's equation may not accurately represent the line passing through this point. Next, we'll repeat this process for the second point, (-12, -9). Substituting x = -12 and y = -9 into the equation, we get 7(-12) - 10(-9) = -84 + 90 = 6. Again, since 6 ≠ 3, the point (-12, -9) also does not satisfy Devon's equation. This further solidifies our concern about the accuracy of Devon's model. The fact that neither of the given points satisfies Devon's equation is a strong indicator that the equation is not a good fit for the data. This finding necessitates a deeper investigation into the correct equation for the line passing through these points. In the subsequent sections, we will explore alternative methods for deriving the equation of the line, such as using the slope-intercept form or the point-slope form, to arrive at a more accurate model. This rigorous verification process underscores the importance of ensuring that a proposed equation accurately reflects the given data points.

Calculating Slope and the Correct Equation

To determine the correct equation of the line, we must first calculate the slope (m) using the two given points, (8, 5) and (-12, -9). The slope is a fundamental property of a line that indicates its steepness and direction. It is defined as the change in y divided by the change in x between any two points on the line. The formula for calculating the slope is m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points. Using the points (8, 5) and (-12, -9), we can substitute the coordinates into the formula as follows: m = (-9 - 5) / (-12 - 8) = -14 / -20 = 7/10. Therefore, the slope of the line passing through these points is 7/10. This positive slope indicates that the line is increasing as we move from left to right. With the slope calculated, we can now use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by y - y1 = m(x - x1), where m is the slope and (x1, y1) is any point on the line. We can use either of the given points; let's use (8, 5). Substituting m = 7/10 and (x1, y1) = (8, 5) into the point-slope form, we get y - 5 = (7/10)(x - 8). To convert this equation to slope-intercept form (y = mx + b), we first distribute the 7/10 across the terms in the parentheses: y - 5 = (7/10)x - 56/10. Next, we add 5 to both sides of the equation to isolate y: y = (7/10)x - 56/10 + 5. To combine the constant terms, we need a common denominator, so we rewrite 5 as 50/10: y = (7/10)x - 56/10 + 50/10. Simplifying, we get y = (7/10)x - 6/10, which can be further simplified to y = (7/10)x - 3/5. This equation represents the line in slope-intercept form, where the y-intercept (b) is -3/5. To express the equation in standard form (Ax + By = C), we multiply both sides of the equation by 10 to eliminate the fractions: 10y = 7x - 6. Rearranging the terms, we get 7x - 10y = 6. This is the correct equation of the line passing through the points (8, 5) and (-12, -9). Comparing this equation to Devon's equation, 7x - 10y = 3, we can clearly see that they are different. This discrepancy confirms our earlier suspicion that Devon's equation is not a good model for the given data points.

Comparing Devon's Equation with the Correct Equation

Now, let's compare Devon's equation, 7x - 10y = 3, with the correctly derived equation, 7x - 10y = 6. A meticulous comparison reveals a 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 seemingly small difference has substantial implications for the line's position on the coordinate plane. The constant term in the standard form of a linear equation (Ax + By = C) influences the line's y-intercept and overall placement. In this case, the different constant terms indicate that Devon's line and the correct line are parallel but do not coincide. Parallel lines have the same slope but different y-intercepts, meaning they never intersect. Since both equations have the same coefficients for x and y (7 and -10, respectively), they have the same slope. However, the different constant terms (3 and 6) shift the lines vertically, resulting in two distinct parallel lines. To further illustrate this difference, let's convert both equations to slope-intercept form (y = mx + b). We already derived the correct equation in slope-intercept form as y = (7/10)x - 3/5. Now, let's convert Devon's equation to slope-intercept form. Starting with 7x - 10y = 3, we subtract 7x from both sides: -10y = -7x + 3. Then, we divide both sides by -10: y = (7/10)x - 3/10. Comparing the two slope-intercept forms, y = (7/10)x - 3/5 and y = (7/10)x - 3/10, we can clearly see that they have the same slope (7/10) but different y-intercepts (-3/5 and -3/10, respectively). This visual representation further confirms that the lines are parallel. The fact that Devon's equation results in a different y-intercept means it does not accurately represent the line passing through the points (8, 5) and (-12, -9). This discrepancy underscores the importance of accurate calculations and careful verification when deriving linear equations. A minor error in the constant term can significantly alter the line's position and render it an inaccurate model for the given data. In summary, while Devon's equation shares the same slope as the correct equation, the difference in the constant term makes it an inadequate representation of the line passing through the specified points.

Conclusion: Evaluating Devon's Model

In conclusion, our comprehensive analysis reveals that Devon's equation, 7x - 10y = 3, is not a good model for the line passing through the data points (8, 5) and (-12, -9). This conclusion is supported by several key findings throughout our investigation. First and foremost, we verified that neither of the given points satisfies Devon's equation. This is a fundamental requirement for any equation to accurately represent a line passing through specific points. If the points do not lie on the line defined by the equation, the equation cannot be considered a valid model. Secondly, we calculated the slope of the line using the given points and derived the correct equation, which is 7x - 10y = 6. This equation differs from Devon's equation in the constant term, indicating that the two lines are parallel but distinct. The correct equation accurately represents the line passing through the points (8, 5) and (-12, -9), while Devon's equation does not. Furthermore, we compared Devon's equation with the correct equation in both standard form and slope-intercept form. This comparison highlighted the difference in the y-intercepts, further illustrating why Devon's equation is not an accurate representation of the data. The correct slope-intercept form, y = (7/10)x - 3/5, clearly shows the line's slope and y-intercept, providing a visual understanding of its position on the coordinate plane. In contrast, Devon's equation, when converted to slope-intercept form, has the same slope but a different y-intercept, resulting in a parallel line that does not pass through the given points. The importance of accurate linear modeling cannot be overstated. Linear equations are widely used in various fields, including mathematics, science, engineering, and economics, to represent relationships between variables and make predictions. An inaccurate equation can lead to incorrect conclusions and flawed predictions. Therefore, it is crucial to meticulously verify any proposed equation against the given data and ensure that it accurately reflects the underlying relationship. In Devon's case, while his attempt to model the line is commendable, the resulting equation does not meet the criteria for a good model. The discrepancy highlights the need for careful calculations and thorough verification in mathematical modeling. By understanding the principles of linear equations and applying them rigorously, we can develop accurate models that provide valuable insights and predictions.

Is the equation 7x - 10y = 3 a good model for a line that passes through the points (8, 5) and (-12, -9)? Explain your reasoning.

Devon's Linear Equation Model Analysis for Data Points (8, 5) and (-12, -9)