Finding The Equation Of A Perpendicular Line Slope Intercept Form Guide
In the realm of coordinate geometry, the relationship between lines, especially perpendicular lines, is a fundamental concept. Understanding how to determine the equation of a line that is perpendicular to a given line and passes through a specific point is a crucial skill. This article delves into the process of finding the equation of a line perpendicular to a given line, expressed in slope-intercept form, and containing a specified point. We will use the example of line LM with the equation 5x - y = -4 and the point (-3, 2) to illustrate the steps involved.
Understanding the Basics: Slope-Intercept Form and Perpendicular Lines
Before diving into the problem, let's revisit some key concepts. The slope-intercept form of a linear equation is expressed as y = mx + b, where 'm' represents the slope of the line and 'b' represents the y-intercept. The slope indicates the steepness and direction of the line, while the y-intercept is the point where the line crosses the y-axis.
Perpendicular lines are lines that intersect at a right angle (90 degrees). A crucial property of perpendicular lines is that their slopes are negative reciprocals of each other. This means that if a line has a slope of 'm', a line perpendicular to it will have a slope of '-1/m'. This negative reciprocal relationship is the cornerstone of finding the equation of a perpendicular line.
Transforming the Given Equation into Slope-Intercept Form
The given equation of line LM is 5x - y = -4. To work with slopes and intercepts effectively, we need to convert this equation into slope-intercept form (y = mx + b). This involves isolating 'y' on one side of the equation. Let's walk through the steps:
- Subtract 5x from both sides: -y = -5x - 4
- Multiply both sides by -1: y = 5x + 4
Now we have the equation in slope-intercept form. We can clearly see that the slope of line LM (m₁) is 5 and the y-intercept is 4. This transformation is a critical first step as it allows us to easily identify the slope of the given line, which we'll use to determine the slope of the perpendicular line. This conversion to slope-intercept form not only reveals the slope but also provides a clear visual representation of the line's position on the coordinate plane. Understanding this process is fundamental to solving various problems related to linear equations and their graphical representations. Remember, mastering the manipulation of equations into different forms is a key skill in algebra and geometry.
Determining the Slope of the Perpendicular Line
Now that we know the slope of line LM (m₁) is 5, we can find the slope of a line perpendicular to it. As mentioned earlier, the slopes of perpendicular lines are negative reciprocals. Therefore, the slope of the perpendicular line (m₂) is -1/m₁. Substituting the value of m₁, we get:
m₂ = -1/5
This calculation is a critical step in finding the equation of the perpendicular line. The negative reciprocal relationship is a fundamental geometric concept that dictates the orientation of perpendicular lines. Understanding this relationship allows us to directly calculate the slope of any line perpendicular to a given line, provided we know the slope of the original line. This concept extends beyond simple line equations; it's a key element in various geometric proofs and constructions. The ability to quickly determine the slope of a perpendicular line is a valuable tool in solving problems related to geometric shapes, transformations, and spatial reasoning. Remember, the negative reciprocal is not just a mathematical trick; it's a reflection of the geometric relationship between lines that intersect at right angles.
Utilizing the Point-Slope Form
We now have the slope of the perpendicular line (-1/5) and a point it passes through (-3, 2). This is where the point-slope form of a linear equation comes in handy. The point-slope form is expressed as:
y - y₁ = m(x - x₁)
where (x₁, y₁) is a point on the line and 'm' is the slope. This form is particularly useful when we have a point and a slope but want to find the equation in slope-intercept form. Let's plug in our values:
y - 2 = (-1/5)(x - (-3))
This equation represents the line we are looking for, but it is not yet in slope-intercept form. The point-slope form provides a direct way to construct the equation of a line given a point and a slope. It is a powerful tool because it bypasses the need to calculate the y-intercept directly. Instead, it leverages the known point and slope to define the line's path. This form is not only useful for finding the equation of a line but also for understanding the relationship between a line's slope, a point on the line, and its overall equation. Mastering the point-slope form is essential for a comprehensive understanding of linear equations and their applications in various fields, including physics, engineering, and computer graphics.
Converting to Slope-Intercept Form
To get the equation in slope-intercept form (y = mx + b), we need to simplify the equation we obtained from the point-slope form. Let's distribute and isolate 'y':
y - 2 = (-1/5)(x + 3) y - 2 = (-1/5)x - 3/5
Now, add 2 to both sides:
y = (-1/5)x - 3/5 + 2
To combine the constants, we need a common denominator:
y = (-1/5)x - 3/5 + 10/5
Finally, we get:
y = (-1/5)x + 7/5
This is the equation of the line perpendicular to line LM and passing through the point (-3, 2), expressed in slope-intercept form. This final step demonstrates the process of transforming an equation from one form to another, highlighting the flexibility of linear equations. The conversion from point-slope form to slope-intercept form is a common algebraic manipulation that allows us to clearly identify the slope and y-intercept of the line. This conversion also makes it easier to compare the equation with other linear equations and analyze their relationships. Understanding how to manipulate equations into different forms is a crucial skill in mathematics, enabling us to solve problems efficiently and gain a deeper understanding of the underlying concepts.
Conclusion
In this article, we successfully determined the equation of a line perpendicular to a given line and passing through a specific point. We started with the equation of line LM (5x - y = -4) and the point (-3, 2). By converting the equation to slope-intercept form, we identified the slope of line LM. We then used the negative reciprocal relationship to find the slope of the perpendicular line. Applying the point-slope form and simplifying, we arrived at the equation of the perpendicular line in slope-intercept form: y = (-1/5)x + 7/5. This process highlights the importance of understanding fundamental concepts like slope-intercept form, perpendicular lines, and the point-slope form. It also emphasizes the power of algebraic manipulation in solving geometric problems. This step-by-step approach can be applied to a wide range of problems involving linear equations and perpendicular lines, making it a valuable skill for anyone studying mathematics or related fields. Mastering these concepts not only enhances problem-solving abilities but also fosters a deeper appreciation for the elegance and interconnectedness of mathematical ideas.
