Finding Equations Of Perpendicular Lines A Step By Step Guide

To determine the equations that represent the line perpendicular to the given line $5x - 2y = -6$ and passing through the point $(5, -4)$, we need to follow a step-by-step approach. This involves finding the slope of the given line, determining the slope of the perpendicular line, and then using the point-slope form to construct the equation of the new line. Finally, we can compare our result with the provided options to identify the correct equations. Understanding these concepts is crucial in coordinate geometry, and this article aims to provide a comprehensive explanation.

1. Understanding Perpendicular Lines and Their Slopes

When delving into coordinate geometry, understanding the concept of perpendicular lines and their slopes is paramount. Perpendicular lines are lines that intersect at a right angle (90 degrees). The relationship between their slopes is a fundamental aspect of this concept. If we have two lines, line 1 and line 2, with slopes $m_1$ and $m_2$ respectively, and these lines are perpendicular, then the product of their slopes is -1. Mathematically, this can be expressed as:

$$m_1 * m_2 = -1$$

This relationship is crucial because it allows us to find the slope of a line perpendicular to another line if we know the slope of the original line. For instance, if a line has a slope of $m$, then any line perpendicular to it will have a slope of $-\frac{1}{m}$. This is often referred to as the negative reciprocal.

To further illustrate this concept, consider a line with a slope of 2. The slope of any line perpendicular to this line would be $-\frac{1}{2}$. Similarly, if a line has a slope of $-\frac{3}{4}$, the slope of a perpendicular line would be $\frac{4}{3}$. This negative reciprocal relationship ensures that the lines intersect at a right angle.

Understanding this principle is not only vital for solving problems involving perpendicular lines but also for various applications in geometry and calculus. Recognizing and applying this relationship allows for the efficient determination of line equations and geometric properties.

2. Finding the Slope of the Given Line: $5x - 2y = -6$

To embark on the journey of finding the equation of a line perpendicular to the given line and passing through a specific point, the initial crucial step involves determining the slope of the given line. The equation provided is $5x - 2y = -6$. To find its slope, we need to transform this equation into the slope-intercept form, which is $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept.

Starting with the given equation:

$$5x - 2y = -6$$

We want to isolate $y$ on one side of the equation. First, subtract $5x$ from both sides:

$$-2y = -5x - 6$$

Next, divide both sides by $-2$ to solve for $y$:

$$y = \frac{-5x - 6}{-2}$$

$$y = \frac{5}{2}x + 3$$

Now, the equation is in the slope-intercept form, $y = mx + b$. By comparing this with our transformed equation, we can clearly see that the slope, $m$, of the given line is $\frac{5}{2}$. This slope is a critical piece of information because it allows us to determine the slope of any line perpendicular to the given line.

The process of converting the equation to slope-intercept form is a fundamental technique in algebra and coordinate geometry. It enables us to easily identify the slope and y-intercept, which are essential for further analysis and calculations. In this case, finding the slope of the given line is the stepping stone to finding the equation of the perpendicular line.

3. Determining the Slope of the Perpendicular Line

Now that we've successfully identified the slope of the given line, which is $\frac{5}{2}$, the next pivotal step is to ascertain the slope of the line that is perpendicular to it. As discussed earlier, the slopes of perpendicular lines have a unique relationship: they 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 $-\frac{1}{m}$.

In our scenario, the given line has a slope of $\frac{5}{2}$. To find the slope of the perpendicular line, we need to calculate the negative reciprocal of $\frac{5}{2}$. This involves two operations: first, we find the reciprocal, and second, we change the sign.

The reciprocal of $\frac{5}{2}$ is $\frac{2}{5}$. Now, we change the sign to make it negative. Therefore, the slope of the line perpendicular to the given line is $-\frac{2}{5}$. This value is crucial because it will be used in the equation of the new line we are trying to find.

Understanding and applying the concept of negative reciprocals is fundamental in coordinate geometry. It allows us to quickly determine the relationship between lines that intersect at right angles. In this case, knowing the slope of the perpendicular line is essential for constructing its equation, which will pass through the specified point $(5, -4)$.

4. Using the Point-Slope Form to Find the Equation

With the slope of the perpendicular line now determined as $-\frac{2}{5}$, and the point through which it passes given as $(5, -4)$, we can now construct the equation of this line. The most convenient form for this purpose is the point-slope form of a linear equation. The point-slope form is expressed as:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line. This form is particularly useful when we know a point on the line and its slope, which is exactly our situation.

In our case, we have the point $(5, -4)$, so $x_1 = 5$ and $y_1 = -4$. We also have the slope of the perpendicular line, m = -rac{2}{5}. Plugging these values into the point-slope form, we get:

y - (-4) = -rac{2}{5}(x - 5)

Simplifying this equation, we have:

y + 4 = -rac{2}{5}(x - 5)

This equation represents the line that is perpendicular to the given line $5x - 2y = -6$ and passes through the point $(5, -4)$. This form is one of the options provided in the question, and it clearly demonstrates how the point-slope form can be directly used to construct the equation of a line given its slope and a point on it.

The point-slope form is a versatile tool in linear algebra and coordinate geometry, allowing for the easy determination of a line's equation. In this scenario, it has enabled us to create the equation of the perpendicular line, which we can now compare with other forms to identify the correct options.

5. Converting to Slope-Intercept and Standard Forms

While the point-slope form, y + 4 = -rac{2}{5}(x - 5), is a valid representation of the line perpendicular to $5x - 2y = -6$ and passing through $(5, -4)$, it is beneficial to convert it into other common forms such as the slope-intercept form ($y = mx + b$) and the standard form ($Ax + By = C$). This allows us to easily compare our result with different equation formats and identify equivalent expressions.

Converting to Slope-Intercept Form:

Starting from the point-slope form:

y + 4 = -rac{2}{5}(x - 5)

First, distribute the $-\frac{2}{5}$ on the right side:

y + 4 = -rac{2}{5}x + 2

Next, subtract 4 from both sides to isolate $y$:

y = -rac{2}{5}x + 2 - 4

y = -rac{2}{5}x - 2

This is the slope-intercept form of the equation. We can see that the slope is $-\frac{2}{5}$ and the y-intercept is $-2$.

Converting to Standard Form:

Starting from the slope-intercept form:

y = -rac{2}{5}x - 2

First, add $\frac{2}{5}x$ to both sides:

$$\frac{2}{5}x + y = -2$$

To eliminate the fraction, multiply the entire equation by 5:

$$5(\frac{2}{5}x + y) = 5(-2)$$

$$2x + 5y = -10$$

This is the standard form of the equation. The standard form is particularly useful for quickly identifying coefficients and constants, which can be helpful in various algebraic manipulations and comparisons.

By converting the equation into both slope-intercept and standard forms, we gain a more comprehensive understanding of the line's properties and can readily compare it with other equations presented in different formats. In the context of the original question, this step is crucial for selecting the correct options.

6. Identifying the Correct Options

After determining the equation of the line perpendicular to $5x - 2y = -6$ and passing through the point $(5, -4)$, we found the equations in point-slope form, slope-intercept form, and standard form. These are:

• Point-slope form: y + 4 = -rac{2}{5}(x - 5)
• Slope-intercept form: y = -rac{2}{5}x - 2
• Standard form: $2x + 5y = -10$

Now, we need to compare these equations with the options provided in the question to identify the correct ones. The options are:

• y = -rac{2}{5}x - 2
• $2x + 5y = -10$
• $2x - 5y = -10$
• y + 4 = -rac{2}{5}(x - 5)
• $y - 4 = ...$ (This option is incomplete and cannot be evaluated)

By direct comparison, we can see that the following options match our derived equations:

• y = -rac{2}{5}x - 2 (Slope-intercept form)
• $2x + 5y = -10$ (Standard form)
• y + 4 = -rac{2}{5}(x - 5) (Point-slope form)

The option $2x - 5y = -10$ does not match our standard form equation, and the incomplete option $y - 4 = ...$ cannot be determined without further information.

Therefore, the three equations that represent the line perpendicular to the given line and passing through the point $(5, -4)$ are:

• y = -rac{2}{5}x - 2
• $2x + 5y = -10$
• y + 4 = -rac{2}{5}(x - 5)

This step-by-step comparison ensures that we accurately identify the equations that satisfy the given conditions, highlighting the importance of converting equations into different forms for effective analysis.

7. Conclusion: Mastering Perpendicular Lines and Equation Forms

In conclusion, finding the equation of a line perpendicular to a given line and passing through a specific point involves several key steps. First, we must determine the slope of the given line. Then, we use the concept of negative reciprocals to find the slope of the perpendicular line. Next, the point-slope form is utilized to construct the equation of the new line. Finally, converting this equation into slope-intercept and standard forms allows for easy comparison with various options and a deeper understanding of the line's properties.

This process not only reinforces our understanding of perpendicular lines and their slopes but also enhances our ability to manipulate linear equations in different forms. Mastering these concepts is crucial for success in coordinate geometry and related mathematical fields. By systematically working through each step, we can confidently solve problems involving perpendicular lines and ensure accurate results.

Understanding these principles provides a solid foundation for tackling more complex problems in geometry and calculus, where the relationships between lines and their equations play a vital role. This comprehensive approach ensures that we can confidently address similar challenges in the future.

Which equations represent the line that is perpendicular to the line $5x - 2y = -6$ and passes through the point $(5, -4)$? In mathematics, especially in coordinate geometry, understanding the relationships between lines is crucial. This article breaks down how to find the equation of a line that is perpendicular to a given line and passes through a specific point. This involves understanding the concept of slopes, the relationship between perpendicular lines, and the various forms of linear equations.

Repair Input Keyword

Find the equations representing a line perpendicular to the line $5x - 2y = -6$ that passes through the point $(5, -4)$. Select three options.