Slope Of A Perpendicular Line To 2x + 7y = 5
This article delves into the concept of perpendicular lines and their slopes, providing a step-by-step guide to solving the problem: What is the slope of a line that is perpendicular to the line whose equation is 2x + 7y = 5? We will explore the fundamental principles of linear equations, slope-intercept form, and the relationship between slopes of perpendicular lines. This comprehensive explanation will not only provide the answer but also equip you with the knowledge to tackle similar problems confidently.
Decoding Linear Equations and Slope
At its core, this problem requires a strong understanding of linear equations and the concept of slope. A linear equation, in its simplest form, represents a straight line on a coordinate plane. The slope of a line, often denoted by 'm', quantifies its steepness and direction. It tells us how much the line rises (or falls) for every unit increase in the horizontal direction. The slope is a crucial characteristic of a line, determining its orientation and how it interacts with other lines. To effectively address the question at hand, it's essential to first grasp how to extract the slope from a given linear equation. Several forms of linear equations exist, but the slope-intercept form is particularly useful for identifying the slope directly.
The slope-intercept form of a linear equation is expressed as y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept (the point where the line crosses the vertical y-axis). This form offers a clear and concise way to visualize the line's characteristics. By rearranging a given linear equation into slope-intercept form, we can easily pinpoint the value of the slope. This process typically involves isolating the 'y' variable on one side of the equation. Once we have the equation in slope-intercept form, the coefficient of the 'x' term directly reveals the slope of the line. For example, if we have the equation y = 3x + 2, we can immediately see that the slope of the line is 3. Understanding this fundamental concept is crucial for tackling problems involving perpendicular lines, as the relationship between their slopes is a key element in solving such problems. The ability to manipulate linear equations and extract the slope is a foundational skill in algebra and geometry, and it forms the basis for more advanced mathematical concepts.
Unveiling the Slope-Intercept Form
The slope-intercept form of a linear equation, y = mx + b, is an invaluable tool for understanding and manipulating lines. This form explicitly reveals the slope (m) and the y-intercept (b) of the line, making it easy to visualize the line's position and direction on the coordinate plane. The slope (m) indicates the line's steepness and whether it rises or falls as you move from left to right. A positive slope signifies an upward trend, while a negative slope indicates a downward trend. The larger the absolute value of the slope, the steeper the line. The y-intercept (b) represents the point where the line intersects the y-axis. This point has coordinates (0, b), and it serves as an anchor point for graphing the line. The slope-intercept form allows us to quickly sketch the line by plotting the y-intercept and then using the slope to find another point on the line. For example, if the slope is 2/3, we can start at the y-intercept and then move 3 units to the right and 2 units up to find another point. Connecting these two points gives us the line.
The beauty of the slope-intercept form lies in its simplicity and directness. It provides a clear and intuitive representation of a linear equation. By rearranging any linear equation into this form, we can immediately extract the slope and y-intercept, which are essential for various mathematical operations and applications. For instance, we can use the slope-intercept form to determine if two lines are parallel or perpendicular, to find the equation of a line given its slope and a point, or to solve systems of linear equations. Mastering the slope-intercept form is a fundamental step in understanding linear equations and their properties. It provides a solid foundation for more advanced topics in algebra and calculus. The ability to manipulate equations and convert them into slope-intercept form is a crucial skill for students and professionals alike, enabling them to analyze and solve a wide range of problems in mathematics, science, and engineering.
Perpendicular Lines and Their Slopes: The Inverse Relationship
Now, let's delve into the core concept of perpendicular lines and their slopes. Two lines are considered perpendicular if they intersect at a right angle (90 degrees). There's a fascinating relationship between the slopes of perpendicular lines: they are negative reciprocals of each other. This means that if one line has a slope of 'm', the slope of a line perpendicular to it will be '-1/m'. This inverse relationship is crucial for solving problems involving perpendicularity. To understand this relationship intuitively, consider a line with a positive slope. A line perpendicular to it must have a negative slope to slant in the opposite direction. Moreover, the reciprocal aspect ensures that the lines intersect at a right angle. For example, if a line has a slope of 2, a line perpendicular to it will have a slope of -1/2. Similarly, if a line has a slope of -3/4, a perpendicular line will have a slope of 4/3.
The negative reciprocal relationship between slopes of perpendicular lines is a fundamental principle in geometry and is widely used in various mathematical applications. It allows us to determine whether two lines are perpendicular, find the equation of a line perpendicular to a given line, or construct geometric figures with right angles. This concept is not only important in mathematics but also has practical applications in fields like engineering, architecture, and computer graphics, where precise angles and orientations are crucial. Understanding the inverse relationship between slopes of perpendicular lines enables us to solve a wide range of problems involving geometric shapes and spatial relationships. It's a powerful tool for analyzing and manipulating lines and angles, and it forms a cornerstone of geometric reasoning. Mastering this concept is essential for anyone working with geometric figures or spatial data, as it provides a fundamental understanding of how lines interact and intersect in space.
Solving the Problem: Finding the Perpendicular Slope
To answer the question, "What is the slope of a line that is perpendicular to the line whose equation is 2x + 7y = 5?", we'll follow a step-by-step approach. First, we need to transform the given equation, 2x + 7y = 5, into slope-intercept form (y = mx + b). This involves isolating 'y' on one side of the equation. Subtracting 2x from both sides, we get 7y = -2x + 5. Then, dividing both sides by 7, we obtain y = (-2/7)x + 5/7. Now, we can clearly see that the slope of the given line is -2/7. Remember, the slope is the coefficient of the 'x' term when the equation is in slope-intercept form.
Next, we need to find the slope of a line perpendicular to this one. As we discussed earlier, the slopes of perpendicular lines are negative reciprocals of each other. So, to find the perpendicular slope, we take the negative reciprocal of -2/7. The reciprocal of -2/7 is -7/2, and the negative of -7/2 is 7/2. Therefore, the slope of a line perpendicular to the line 2x + 7y = 5 is 7/2. This corresponds to option D in the given choices. This step-by-step process demonstrates how we can use our understanding of linear equations, slope-intercept form, and the relationship between slopes of perpendicular lines to solve this problem effectively. The key is to first identify the slope of the given line and then apply the negative reciprocal rule to find the perpendicular slope. By breaking down the problem into smaller, manageable steps, we can arrive at the correct solution with confidence.
Conclusion: Mastering Slopes and Perpendicularity
In conclusion, the slope of a line perpendicular to the line whose equation is 2x + 7y = 5 is 7/2. This problem highlights the importance of understanding the fundamental concepts of linear equations, slope-intercept form, and the relationship between slopes of perpendicular lines. By mastering these concepts, you can confidently tackle a wide range of problems involving lines and their properties. The ability to manipulate linear equations, extract the slope, and apply the negative reciprocal rule for perpendicular lines is a valuable skill in mathematics and various other fields. Remember, practice is key to solidifying your understanding. Work through similar problems, and you'll become more adept at recognizing and applying these concepts. This knowledge will serve as a strong foundation for more advanced mathematical studies and real-world applications. Understanding the relationship between perpendicular lines and their slopes is not just about solving specific problems; it's about developing a deeper understanding of geometric principles and spatial reasoning.
Therefore, the correct answer is D. 7/2.
Answer: D. 7/2
