Understanding how to determine the equation of a line is a fundamental skill in algebra and geometry. In this article, we will focus on finding the equation of line CD, which passes through the points (0, 2) and (4, 6). This task involves analyzing the slope-intercept form of the line’s equation, as well as justifying our calculations using the provided key points. By breaking down the process step by step, we can arrive at a clear and accurate representation of line CD.
Analyzing the Slope-Intercept Form for Line CD’s Equation
The slope-intercept form of a linear equation is expressed as ( y = mx + b ), where ( m ) represents the slope and ( b ) is the y-intercept. To derive the equation of line CD, it is imperative first to calculate the slope between the two points. The slope ( m ) can be found using the formula ( m = frac{y_2 – y_1}{x_2 – x_1} ). By substituting our coordinates, we find that ( m = frac{6 – 2}{4 – 0} = frac{4}{4} = 1 ). This indicates that for every unit increase in ( x ), ( y ) increases by the same amount, demonstrating a direct linear relationship.
Next, we identify the y-intercept ( b ), which represents the value of ( y ) when ( x = 0 ). Since one of our key points is (0, 2), we can directly deduce that when ( x = 0 ), ( y ) is indeed 2. Hence, the value of ( b ) is 2. With both the slope and y-intercept determined, we can now write the equation of line CD in slope-intercept form as ( y = 1x + 2 ) or simply ( y = x + 2 ).
This equation signifies that line CD not only passes through the specified points but also conveys important information about its behavior. The slope of 1 indicates that the line rises at a 45-degree angle, affirming a consistent growth in ( y ) with respect to ( x ). This understanding is crucial, especially in fields such as physics and economics, where linear equations often model real-world phenomena.
Justifying the Calculation of Line CD Through Key Points
To further substantiate our findings, we must validate our calculations using the key points provided—(0, 2) and (4, 6). By substituting these coordinates back into our derived equation ( y = x + 2 ), we can confirm the line’s accuracy. For the first point (0, 2), substituting ( x = 0 ) yields ( y = 0 + 2 = 2 ), which matches the point exactly. Next, for the second point (4, 6), substituting ( x = 4 ) produces ( y = 4 + 2 = 6 ). Both points satisfy the equation, thereby reinforcing our conclusion that the equation ( y = x + 2 ) accurately describes line CD.
Moreover, the choice of points is significant as they are not arbitrary; they were intentionally selected to represent the line’s trajectory. By using the endpoints of the segment defined by line CD, we ensure that our calculations encapsulate the entire linear relationship. This method of verification is a standard practice in mathematics, serving as a check that our derived equation is consistent with the geometric representation of the line.
Lastly, understanding the importance of these calculations is essential for applications across various disciplines. In geometry, verifying equations through graphical representation allows students and professionals alike to visualize relationships, while in more complex applications, such as data analysis or predictive modeling, accurately derived linear equations can significantly influence decision-making processes. Thus, the justification of our calculations holds profound relevance beyond a mere academic exercise.
In conclusion, determining the equation of line CD through the points (0, 2) and (4, 6) exemplifies not only a fundamental algebraic skill but also highlights the importance of thorough analysis and verification in mathematical problem-solving. By employing the slope-intercept form, we derived the equation ( y = x + 2 ) and justified our calculations through careful validation of the key points. Such rigorous methodology ensures accuracy and enhances our understanding of linear relationships, thereby equipping us with the necessary tools to tackle more complex mathematical challenges in the future.
