Finding Equations Of Perpendicular And Parallel Lines

In mathematics, understanding the relationships between lines is fundamental, especially when dealing with perpendicular and parallel lines. This article delves into the specifics of finding equations for lines that are either perpendicular or parallel to a given line, while also passing through a specific point. We'll take the line ${ y = \frac{7}{3}x - 1 }$ as our starting point and explore how to determine the equations of lines that meet these criteria. This comprehensive guide will not only provide you with the solutions but also equip you with a deep understanding of the underlying concepts.

Understanding the Basics

Before diving into the calculations, it's crucial to grasp the basic concepts of linear equations, slopes, and the relationships between parallel and perpendicular lines. Let's start by breaking down the key elements.

Slope-Intercept Form

The equation ${ y = \frac{7}{3}x - 1 }$ is in slope-intercept form, which is ${ y = mx + b }$, where:

• ${ m }$ represents the slope of the line.
• ${ b }$ represents the y-intercept (the point where the line crosses the y-axis).

In our given equation, ${ y = \frac{7}{3}x - 1 }$, the slope ${ m }$ is ${ \frac{7}{3} }$, and the y-intercept ${ b }$ is -1. The slope is a measure of the steepness of a line, indicating how much the line rises or falls for each unit of horizontal change. In this case, for every 3 units the line moves horizontally, it rises 7 units vertically. The y-intercept, -1, tells us that the line crosses the y-axis at the point (0, -1).

Parallel Lines

Parallel lines are lines in a plane that never intersect. A fundamental property of parallel lines is that they have the same slope. This means that if a line is parallel to ${ y = \frac{7}{3}x - 1 }$, it will also have a slope of ${ \frac{7}{3} }$. The only difference between parallel lines is their y-intercepts, which determine where they cross the y-axis. Understanding this concept is crucial for finding the equation of a line parallel to the given line.

Perpendicular Lines

Perpendicular lines, on the other hand, intersect at a right angle (90 degrees). The slopes of perpendicular lines have a unique relationship: they are negative reciprocals of each other. If a line has a slope of ${ m }$, a line perpendicular to it will have a slope of ${ -\frac{1}{m} }$. For our given line with a slope of ${ \frac{7}{3} }$, the slope of a line perpendicular to it will be ${ -\frac{1}{\frac{7}{3}} = -\frac{3}{7} }$. This negative reciprocal relationship ensures that the lines intersect at a right angle, a key characteristic of perpendicular lines.

Finding the Equation of a Perpendicular Line

To find the equation of a line perpendicular to ${ y = \frac{7}{3}x - 1 }$ and passing through the point (7, 6), we need to follow a systematic approach. This involves determining the slope of the perpendicular line and using the point-slope form to find the equation.

Step 1: Determine the Slope of the Perpendicular Line

As established earlier, the slope of a line perpendicular to ${ y = \frac{7}{3}x - 1 }$ is the negative reciprocal of ${ \frac{7}{3} }$, which is ${ -\frac{3}{7} }$. This slope, ${ -\frac{3}{7} }$, indicates that for every 7 units the line moves horizontally, it falls 3 units vertically. The negative sign signifies that the line has a downward slope, moving from left to right.

Step 2: Use the Point-Slope Form

The point-slope form of a linear equation is ${ y - y_1 = m(x - x_1) }$, where:

• ${ m }$ is the slope of the line.
• ${ (x_1, y_1) }$ is a point on the line.

We are given the point (7, 6), so ${ x_1 = 7 }$ and ${ y_1 = 6 }$. We also know the slope of the perpendicular line is ${ -\frac{3}{7} }$. Plugging these values into the point-slope form, we get:

${ y - 6 = -\frac{3}{7}(x - 7) }$

This equation represents the line that is perpendicular to the given line and passes through the point (7, 6). However, it is often preferable to express the equation in slope-intercept form or standard form.

Step 3: Convert to Slope-Intercept Form

To convert the equation to slope-intercept form (${ y = mx + b }$), we need to isolate ${ y }$ on one side of the equation. Start by distributing the slope ${ -\frac{3}{7} }$ across the terms inside the parentheses:

${ y - 6 = -\frac{3}{7}x + \frac{3}{7} \times 7 }$

${ y - 6 = -\frac{3}{7}x + 3 }$

Next, add 6 to both sides of the equation to isolate ${ y }$:

${ y = -\frac{3}{7}x + 3 + 6 }$

${ y = -\frac{3}{7}x + 9 }$

Thus, the equation of the line perpendicular to ${ y = \frac{7}{3}x - 1 }$ and passing through the point (7, 6) is ${ y = -\frac{3}{7}x + 9 }$. This equation is now in slope-intercept form, clearly showing the slope (${ -\frac{3}{7} }$) and the y-intercept (9).

Finding the Equation of a Parallel Line

Now, let's find the equation of a line parallel to ${ y = \frac{7}{3}x - 1 }$ and passing through the point (7, 6). The process is similar, but with a key difference: parallel lines have the same slope.

Step 1: Determine the Slope of the Parallel Line

Since parallel lines have the same slope, the line parallel to ${ y = \frac{7}{3}x - 1 }$ will also have a slope of ${ \frac{7}{3} }$. This means that for every 3 units the line moves horizontally, it rises 7 units vertically, just like the original line.

Step 2: Use the Point-Slope Form

Again, we use the point-slope form of a linear equation: ${ y - y_1 = m(x - x_1) }$. We have the slope ${ m = \frac{7}{3} }$ and the point (7, 6), so ${ x_1 = 7 }$ and ${ y_1 = 6 }$. Plugging these values into the point-slope form, we get:

${ y - 6 = \frac{7}{3}(x - 7) }$

This equation represents the line that is parallel to the given line and passes through the point (7, 6). To make it more readable and comparable, we convert it to slope-intercept form.

Step 3: Convert to Slope-Intercept Form

To convert the equation to slope-intercept form (${ y = mx + b }$), distribute the slope ${ \frac{7}{3} }$ across the terms inside the parentheses:

${ y - 6 = \frac{7}{3}x - \frac{7}{3} \times 7 }$

${ y - 6 = \frac{7}{3}x - \frac{49}{3} }$

Next, add 6 to both sides of the equation to isolate ${ y }$. Note that 6 can be written as ${ \frac{18}{3} }$ to have a common denominator with ${ \frac{49}{3} }$:

${ y = \frac{7}{3}x - \frac{49}{3} + \frac{18}{3} }$

${ y = \frac{7}{3}x - \frac{31}{3} }$

Thus, the equation of the line parallel to ${ y = \frac{7}{3}x - 1 }$ and passing through the point (7, 6) is ${ y = \frac{7}{3}x - \frac{31}{3} }$. This equation, in slope-intercept form, shows that the slope is the same as the original line (${ \frac{7}{3} }$), and the y-intercept is ${ -\frac{31}{3} }$.

Conclusion

In summary, finding the equations of lines that are either perpendicular or parallel to a given line involves understanding the relationship between their slopes and using the point-slope form to derive the equation. For a line perpendicular to ${ y = \frac{7}{3}x - 1 }$ and passing through (7, 6), the equation is ${ y = -\frac{3}{7}x + 9 }$. For a line parallel to ${ y = \frac{7}{3}x - 1 }$ and passing through (7, 6), the equation is ${ y = \frac{7}{3}x - \frac{31}{3} }$. By following these steps, you can confidently determine the equations of lines with specific properties, enhancing your understanding of linear equations and their applications in mathematics.

This comprehensive guide has walked you through the process step by step, ensuring that you not only arrive at the correct answers but also grasp the underlying principles. With this knowledge, you'll be well-equipped to tackle similar problems and deepen your understanding of linear equations and their geometric interpretations.