Finding Equations Of Perpendicular Lines Passing Through A Point

In this article, we will tackle a common problem in coordinate geometry: finding the equation of a line perpendicular to a given line and passing through a specific point. This is a fundamental concept in mathematics, often encountered in algebra and calculus. We will dissect the problem step by step, ensuring a clear understanding of the underlying principles.

The problem we're addressing is as follows: Given the line $y = \frac{-1}{3}x + 1$, which of the following equations represents a line perpendicular to it and passes through the point (2, 7)?

This problem tests our understanding of several key concepts:

• Slope-intercept form of a line: The given equation is in slope-intercept form ($y = mx + b$), where m represents the slope and b represents the y-intercept.
• Perpendicular lines: We need to know the relationship between the slopes of perpendicular lines.
• Point-slope form of a line: This form helps us construct the equation of a line when we know a point on the line and its slope.
• Conversion to standard form: The answer choices are given in standard form ($Ax + By = C$), so we need to be able to convert between different forms of linear equations.

Let's embark on a detailed solution to this problem, ensuring we cover all the necessary concepts and steps.

Identifying the Slope of the Given Line

The first step in solving this problem is to identify the slope of the given line. The equation $y = \frac{-1}{3}x + 1$ is in slope-intercept form, which is generally expressed as $y = mx + b$. In this form, m represents the slope of the line, and b represents the y-intercept.

By comparing the given equation with the slope-intercept form, we can directly identify the slope. In our case, $m = \frac{-1}{3}$. This means that for every 3 units we move to the right along the x-axis, the line moves 1 unit down along the y-axis. The negative sign indicates that the line slopes downwards from left to right.

Understanding the slope of the given line is crucial because it allows us to determine the slope of any line perpendicular to it. The concept of perpendicular slopes is a cornerstone of coordinate geometry and is essential for solving a wide range of problems. In the next section, we will delve into the relationship between the slopes of perpendicular lines and how we can use this relationship to find the slope of the line we are looking for.

Determining the Slope of a Perpendicular Line

Now that we've identified the slope of the given line as $-\frac{1}{3}$, our next crucial step is to determine the slope of a line perpendicular to it. The relationship between the slopes of two perpendicular lines is a fundamental concept in coordinate geometry. Two lines are perpendicular if and only if the product of their slopes is -1. In other words, if a line has a slope of m, then a line perpendicular to it will have a slope of $-\frac{1}{m}$, which is the negative reciprocal of m.

This relationship stems from the fact that perpendicular lines intersect at a right angle (90 degrees). The negative reciprocal ensures that the lines meet at this angle. Visualizing this can be helpful: if one line has a positive slope (rising from left to right), a perpendicular line must have a negative slope (falling from left to right), and vice versa. The reciprocal part of the relationship accounts for the steepness of the lines; a steeper line will have a less steep perpendicular line.

In our case, the slope of the given line is $-\frac{1}{3}$. To find the slope of a line perpendicular to it, we take the negative reciprocal:

$$m_{\perp} = -\frac{1}{(-\frac{1}{3})} = 3$$

So, the slope of any line perpendicular to the given line is 3. This means that for every 1 unit we move to the right along the x-axis, the perpendicular line moves 3 units up along the y-axis. Knowing this slope is a key step towards finding the equation of the perpendicular line that passes through the point (2, 7). In the next section, we'll utilize the point-slope form to construct the equation of this line.

Using the Point-Slope Form to Find the Equation

With the slope of the perpendicular line determined to be 3, and the point (2, 7) through which it passes, we can now construct the equation of the line. The most convenient form for this purpose is the point-slope form of a linear equation. The point-slope form is given by:

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

where (x₁, y₁) is a known point on the line, and m is the slope of the line. This form is particularly useful when you have a point and a slope and want to find the equation of the line. It directly incorporates the given information, making the process straightforward.

In our case, we have m = 3 and (x₁, y₁) = (2, 7). Substituting these values into the point-slope form, we get:

$$y - 7 = 3(x - 2)$$

This equation represents the line we are looking for, but it's not yet in the form of the answer choices, which are in standard form. To match the answer choices, we need to simplify this equation and convert it to standard form. The next step involves distributing the 3 on the right side of the equation and then rearranging the terms to get the equation in the desired format. This conversion is a crucial algebraic manipulation that allows us to compare our result with the given options and select the correct answer.

Converting to Standard Form

Having obtained the equation in point-slope form, $y - 7 = 3(x - 2)$, our next task is to convert it to standard form, which is generally expressed as $Ax + By = C$, where A, B, and C are constants. This form is useful for various purposes, including easily identifying intercepts and comparing equations.

To convert our equation, we first distribute the 3 on the right side:

$$y - 7 = 3x - 6$$

Next, we want to rearrange the terms so that the x and y terms are on the same side of the equation and the constant term is on the other side. To do this, we can subtract 3x from both sides:

$$-3x + y - 7 = -6$$

Then, we add 7 to both sides to isolate the constant term:

$$-3x + y = 1$$

Now, the equation is in standard form. We can compare this equation with the given answer choices to identify the correct option. In this case, the equation $-3x + y = 1$ matches one of the provided options. This confirms that we have successfully found the equation of the line perpendicular to the given line and passing through the specified point.

Comparing with Answer Choices and Selecting the Correct Answer

Now that we have derived the equation $-3x + y = 1$, we need to compare it with the given answer choices to select the correct one. The answer choices were:

A. $3x - y = 1$ B. $-3x + y = -1$ C. $-3x - y = 1$ D. $3x - y = -1$

By direct comparison, we can see that our derived equation, $-3x + y = 1$, matches option B.

Therefore, the correct answer is B. This concludes our step-by-step solution to the problem. We started by identifying the slope of the given line, then determined the slope of the perpendicular line, used the point-slope form to construct the equation, converted it to standard form, and finally, compared it with the answer choices to select the correct option.

Conclusion

In summary, we have successfully found the equation of a line perpendicular to the line $y = \frac{-1}{3}x + 1$ and passing through the point (2, 7). The correct equation is $-3x + y = 1$, which corresponds to option B. This problem highlights the importance of understanding the following key concepts:

• The slope-intercept form of a line and how to identify the slope.
• The relationship between the slopes of perpendicular lines.
• The point-slope form of a line and its utility in constructing equations.
• The standard form of a line and how to convert between different forms.

By mastering these concepts, you can confidently tackle similar problems in coordinate geometry. Remember to break down the problem into smaller, manageable steps, and utilize the appropriate formulas and techniques. Practice is key to solidifying your understanding and improving your problem-solving skills in mathematics.

This problem serves as a great example of how different concepts in coordinate geometry are interconnected. The ability to seamlessly move between these concepts is crucial for success in more advanced mathematical topics. Keep practicing and exploring, and you'll find that these concepts become second nature.

What is the equation of the line that is perpendicular to the line defined by $y=\frac{-1}{3} x+1$ and passes through the point (2,7)?