Finding The Equation Of A Perpendicular Line With A Given X-Intercept
When working with linear equations, understanding the relationships between lines, particularly perpendicularity and intercepts, is crucial. This article will guide you through the process of finding the equation of a line that is perpendicular to a given line and possesses a specific x-intercept. We will delve into the underlying concepts, step-by-step procedures, and illustrative examples to solidify your understanding.
Understanding Perpendicular Lines
In the realm of coordinate geometry, perpendicular lines hold a special significance. Two lines are considered perpendicular if they intersect at a right angle (90 degrees). The slopes of perpendicular lines exhibit a unique relationship: 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 property forms the cornerstone of determining the equation of a perpendicular line.
To truly grasp the concept of perpendicular lines, it's essential to first understand the slope-intercept form of a linear equation, which is expressed as y = mx + b, where m represents the slope and b represents the y-intercept. The slope, often denoted by m, quantifies the steepness and direction of a line. It indicates how much the y-value changes for every unit change in the x-value. A positive slope signifies an upward slant, while a negative slope indicates a downward slant. The y-intercept, represented by b, is the point where the line intersects the y-axis. It is the value of y when x is equal to zero.
Now, let's delve deeper into the relationship between the slopes of perpendicular lines. Consider a line with a slope of 2. A line perpendicular to this would have a slope of -1/2. Similarly, if a line has a slope of -3/4, a line perpendicular to it would have a slope of 4/3. This inverse relationship is crucial for solving problems involving perpendicular lines. When visualizing perpendicular lines, imagine two lines crossing at a perfect 90-degree angle. The slopes of these lines will always be negative reciprocals of each other. Understanding this fundamental principle allows us to manipulate linear equations and solve for unknowns, such as the equation of a perpendicular line with a specific x-intercept.
Determining the Slope of the Perpendicular Line
Our primary goal is to find the equation of a line perpendicular to a given line. To achieve this, the first critical step involves determining the slope of the perpendicular line. As we discussed earlier, the slopes of perpendicular lines are negative reciprocals of each other. This means we need to identify the slope of the given line and then calculate its negative reciprocal. Let's consider the given line, represented by the equation y = -3/4x + 8. From this equation, we can readily identify the slope as -3/4. The slope is the coefficient of the x term in the slope-intercept form (y = mx + b).
Now, to find the slope of the line perpendicular to this given line, we need to take the negative reciprocal of -3/4. The negative reciprocal is obtained by flipping the fraction and changing its sign. So, we flip -3/4 to get -4/3, and then we change the sign from negative to positive, resulting in 4/3. Therefore, the slope of the line perpendicular to y = -3/4x + 8 is 4/3. This value is crucial as it will be the m value in the slope-intercept form of our new perpendicular line's equation.
Understanding this process is fundamental to solving a wide range of problems involving perpendicular lines. By correctly identifying the slope of the given line and calculating its negative reciprocal, we lay the groundwork for determining the equation of the perpendicular line. This slope will guide us in constructing the equation and ensuring that the new line is indeed perpendicular to the original. Furthermore, this concept extends beyond simple linear equations and finds applications in more complex mathematical problems and real-world scenarios, such as determining optimal angles in construction or navigation.
Incorporating the X-Intercept
Now that we've successfully determined the slope of the perpendicular line, the next crucial piece of information is the x-intercept. The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. Knowing the x-intercept provides us with a specific point (x, 0) that lies on the line. This point is invaluable in determining the y-intercept (b) in the slope-intercept form of the equation (y = mx + b).
In our problem, we are given that the x-intercept is 6. This means that the line passes through the point (6, 0). We can use this point, along with the slope we calculated earlier (4/3), to find the y-intercept. We substitute the x and y values of the point (6, 0) and the slope m = 4/3 into the slope-intercept equation: y = mx + b. This gives us 0 = (4/3)(6) + b. Now, we can solve for b.
First, we multiply (4/3) by 6, which equals 8. So, our equation becomes 0 = 8 + b. To isolate b, we subtract 8 from both sides of the equation, resulting in b = -8. This tells us that the y-intercept of our perpendicular line is -8. The y-intercept is where the line intersects the y-axis, and in this case, it's at the point (0, -8). With both the slope (m = 4/3) and the y-intercept (b = -8) determined, we are now fully equipped to write the complete equation of the perpendicular line.
The ability to incorporate the x-intercept into the equation-solving process is a powerful tool in linear algebra. It allows us to pinpoint a specific line that satisfies both the perpendicularity condition and the x-intercept requirement. This skill is not only valuable in academic settings but also in practical applications where determining the precise path of a line is essential, such as in construction, engineering, and computer graphics.
Constructing the Equation
With the slope and y-intercept in hand, we can now construct the equation of the perpendicular line. Recall the slope-intercept form of a linear equation: y = mx + b. We have already determined that the slope (m) of the perpendicular line is 4/3, and the y-intercept (b) is -8. Substituting these values into the slope-intercept form, we get the equation of the perpendicular line: y = (4/3)x - 8. This equation represents a line that is perpendicular to the given line y = -3/4x + 8 and has an x-intercept of 6.
This final step demonstrates the power of combining our knowledge of slopes, intercepts, and the slope-intercept form. By systematically determining the slope of the perpendicular line and using the x-intercept to find the y-intercept, we were able to construct the unique equation that satisfies both conditions. The equation y = (4/3)x - 8 is the definitive answer to our problem. It not only tells us the relationship between x and y on this line but also visually represents a line that crosses the x-axis at 6 and forms a right angle with the line y = -3/4x + 8.
The process of constructing the equation from the slope and y-intercept is a fundamental skill in linear algebra and coordinate geometry. It allows us to translate geometric properties into algebraic expressions, providing a powerful tool for analyzing and manipulating linear relationships. This skill is essential for various applications, including graphing lines, solving systems of equations, and modeling real-world phenomena that can be represented linearly.
Final Answer
In conclusion, the equation of the line that is perpendicular to the given line y = -3/4x + 8 and has an x-intercept of 6 is y = (4/3)x - 8. We arrived at this solution by systematically determining the slope of the perpendicular line, using the x-intercept to find the y-intercept, and then substituting these values into the slope-intercept form of a linear equation.
This problem highlights the importance of understanding the relationships between slopes, intercepts, and the equations of lines. By mastering these concepts, you can confidently tackle a wide range of problems in coordinate geometry and linear algebra. Remember, the key to success lies in breaking down the problem into smaller, manageable steps and applying the appropriate principles at each stage. With practice and a solid understanding of the fundamentals, you can confidently navigate the world of linear equations and their applications.
This process of finding the equation of a perpendicular line with a specific x-intercept is not just an academic exercise; it has practical applications in various fields. For example, in construction, ensuring that walls are perpendicular to the floor is crucial for structural integrity. In navigation, understanding the relationships between lines and angles is essential for plotting courses and avoiding obstacles. And in computer graphics, linear equations are used to draw lines and shapes on the screen. Therefore, the skills you've learned in this article are not only valuable for solving math problems but also for understanding and interacting with the world around you.
