Determining Perpendicular Lines In The Xy-Plane
To determine which of the given equations represents a line perpendicular to the line represented by the equation , we need to understand the relationship between the slopes of perpendicular lines. In this comprehensive guide, we will explore the concept of perpendicular lines, how to find the slope of a line, and apply this knowledge to solve the given problem. This will involve transforming the given equation into slope-intercept form, identifying its slope, finding the negative reciprocal of the slope, and then checking which of the provided options has a slope that matches this negative reciprocal. Understanding these steps is crucial not only for this particular problem but also for various other problems involving linear equations and geometry in the coordinate plane.
Understanding Perpendicular Lines
In coordinate geometry, perpendicular lines are lines that intersect at a right angle (90 degrees). A fundamental property of perpendicular lines is that their slopes are negative reciprocals of each other. This means if one line has a slope m, a line perpendicular to it will have a slope of -1/m. This relationship is crucial for identifying perpendicular lines when given their equations. To effectively work with this property, it's essential to be able to quickly determine the slope of a line from its equation and understand how to manipulate equations to find the slope. This involves converting equations to different forms, such as the slope-intercept form, which we will explore in detail. Furthermore, recognizing and applying this property is a key skill in various mathematical contexts, including geometry, calculus, and linear algebra.
Finding the Slope of the Given Line
To find the slope of the given line, , we need to rewrite the equation in slope-intercept form, which is , where m represents the slope and b represents the y-intercept. This form allows us to easily identify the slope by looking at the coefficient of x. Let’s start by isolating y in the given equation. First, we add to both sides of the equation: $4y = 3x + 5$. Next, we divide both sides by 4 to solve for y: $y = \frac{3}{4}x + \frac{5}{4}$. Now, the equation is in slope-intercept form. By comparing this equation with , we can see that the slope of the given line is $m = \frac{3}{4}$. This process of converting a linear equation to slope-intercept form is a fundamental skill in algebra and is frequently used in problems involving linear equations and their graphical representations. Mastering this technique enables quick identification of the slope and y-intercept, which are crucial for sketching lines and solving related problems.
Determining the Slope of the Perpendicular Line
Now that we have found the slope of the given line to be $\frac{3}{4}$, we can determine the slope of a line perpendicular to it. As mentioned earlier, the slopes of perpendicular lines are negative reciprocals of each other. Therefore, to find the slope of the perpendicular line, we need to take the negative reciprocal of $\frac{3}{4}$. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$, and the negative reciprocal is $-\frac{4}{3}$. Thus, any line perpendicular to the given line must have a slope of $-\frac{4}{3}$. Understanding how to find the negative reciprocal is a crucial concept in coordinate geometry, as it directly relates to the perpendicularity of lines. This concept is not only important for solving problems in a textbook setting but also has practical applications in fields like engineering and computer graphics, where the relationships between lines and angles are critical.
Checking the Options
Now, we need to examine the given options to determine which equation represents a line with a slope of $-\frac{4}{3}$. We will do this by converting each equation into slope-intercept form () and identifying the slope. This involves algebraic manipulation similar to what we did with the original equation. Let's go through each option:
Option A: $3x + 6y = 5$
To convert this to slope-intercept form, we first subtract from both sides: $6y = -3x + 5$. Then, we divide both sides by 6: $y = -\frac3}{6}x + \frac{5}{6}$. Simplifying the fraction, we get{2}x + \frac{5}{6}$. The slope of this line is $-\frac{1}{2}$, which is not $-\frac{4}{3}$, so option A is incorrect.
Option B: $3x + 8y = 2$
Subtract from both sides: $8y = -3x + 2$. Divide both sides by 8: $y = -\frac{3}{8}x + \frac{2}{8}$. The slope of this line is $-\frac{3}{8}$, which is also not $-\frac{4}{3}$, so option B is incorrect.
Option C: $4x + 3y = 5$
Subtract from both sides: $3y = -4x + 5$. Divide both sides by 3: $y = -\frac{4}{3}x + \frac{5}{3}$. The slope of this line is $-\frac{4}{3}$, which matches the slope we are looking for. Therefore, option C is the correct answer.
Option D: $4x + 6y = 5$
Subtract from both sides: $6y = -4x + 5$. Divide both sides by 6: $y = -\frac4}{6}x + \frac{5}{6}$. Simplifying the fraction, we get{3}x + \frac{5}{6}$. The slope of this line is $-\frac{2}{3}$, which does not match $-\frac{4}{3}$, so option D is incorrect.
Conclusion
After analyzing each option, we found that the equation $4x + 3y = 5$ (Option C) represents a line perpendicular to the given line . This is because the slope of the line represented by Option C is $-\frac{4}{3}$, which is the negative reciprocal of the slope of the given line. Understanding how to find slopes and negative reciprocals is essential for solving problems involving perpendicular lines. This skill is not only important for academic success in mathematics but also has practical applications in various fields that require geometric and spatial reasoning.
By systematically converting each equation to slope-intercept form and comparing the slopes, we were able to accurately identify the correct option. This method highlights the importance of understanding the fundamental properties of linear equations and their graphical representations. The ability to manipulate equations and extract key information, such as the slope, is a critical skill in algebra and is frequently used in higher-level mathematics and related disciplines. Furthermore, this problem reinforces the connection between algebraic representations and geometric concepts, which is a core theme in coordinate geometry.
