Find The Equation Of A Line Perpendicular To The Line $x + Y = 5$ That Passes Through The Point $(2, 1)$.

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In mathematics, determining the equation of a line that is perpendicular to a given line and passes through a specific point is a fundamental concept in coordinate geometry. This concept has wide-ranging applications in various fields, including physics, engineering, and computer graphics. This article delves into the process of finding the equation of a line perpendicular to a given line and passing through a given point. We will explore the underlying principles, step-by-step methods, and practical examples to solidify your understanding of this important mathematical concept.

Understanding Perpendicular Lines

Before we dive into the process of finding the equation, it is crucial to understand the properties of perpendicular lines. Two lines are said to be perpendicular if they intersect at a right angle (90 degrees). A key characteristic of perpendicular lines is the relationship between their slopes. The slope of a line, often denoted by m, represents the steepness and direction of the line. For two lines to be perpendicular, the product of their slopes must be -1. In other words, if one line has a slope of m₁, then the slope of a line perpendicular to it, m₂, must satisfy the condition m₁ * m₂* = -1. This relationship forms the foundation for determining the equation of a perpendicular line.

Step-by-Step Method

To find the equation of a line perpendicular to a given line and passing through a specific point, we can follow a step-by-step method:

1. Determine the slope of the given line: The first step involves identifying the slope of the given line. The equation of a line can be expressed in various forms, including slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept. If the equation is not in slope-intercept form, you can rearrange it to this form to easily identify the slope. For example, if the given line is expressed in the form Ax + By = C, you can solve for y to obtain the slope-intercept form.

2. Calculate the slope of the perpendicular line: Once you have the slope of the given line (m₁), you can calculate the slope of the perpendicular line (m₂) using the relationship m₁ * m₂* = -1. This formula implies that the slope of the perpendicular line is the negative reciprocal of the slope of the given line. To find the negative reciprocal, simply flip the fraction and change its sign. For example, if the slope of the given line is 2/3, the slope of the perpendicular line would be -3/2.

3. Use the point-slope form of a line: The point-slope form of a line is a convenient way to express the equation of a line when you know a point on the line and its slope. The point-slope form is given by y - y₁ = m( x - x₁), where (x₁, y₁) is the given point and m is the slope. You can substitute the coordinates of the given point and the slope of the perpendicular line (calculated in step 2) into this equation. This will give you the equation of the perpendicular line in point-slope form.

4. Convert to slope-intercept form (optional): If desired, you can convert the equation from point-slope form to slope-intercept form (y = mx + b). This is done by simplifying the equation and solving for y. The slope-intercept form provides a clear representation of the slope and y-intercept of the line, which can be useful for graphing and further analysis.

Example

Let's illustrate this method with an example. Suppose we want to find the equation of a line perpendicular to the line x + y = 5 and passing through the point (2, 1).

1. Determine the slope of the given line: First, we need to rewrite the equation x + y = 5 in slope-intercept form. Subtracting x from both sides, we get y = -x + 5. Comparing this to y = mx + b, we see that the slope of the given line (m₁) is -1.

2. Calculate the slope of the perpendicular line: Next, we calculate the slope of the perpendicular line (m₂) using the relationship m₁ * m₂* = -1. Substituting m₁ = -1, we get -1 * m₂ = -1. Solving for m₂, we find that m₂ = 1. Therefore, the slope of the perpendicular line is 1.

3. Use the point-slope form of a line: Now, we use the point-slope form of a line, y - y₁ = m( x - x₁), with the point (2, 1) and the slope m₂ = 1. Substituting these values, we get y - 1 = 1*( x - 2). This is the equation of the perpendicular line in point-slope form.

4. Convert to slope-intercept form (optional): To convert to slope-intercept form, we simplify the equation y - 1 = 1*( x - 2). Distributing the 1 on the right side, we get y - 1 = x - 2. Adding 1 to both sides, we obtain y = x - 1. This is the equation of the perpendicular line in slope-intercept form.

Common Mistakes to Avoid

While the process of finding the equation of a perpendicular line is relatively straightforward, there are a few common mistakes to watch out for:

  • Incorrectly calculating the slope of the perpendicular line: Remember that the slope of the perpendicular line is the negative reciprocal of the original line's slope. Failing to both flip the fraction and change the sign is a common error.
  • Using the wrong point-slope form: Double-check that you are substituting the correct values for (x₁, y₁) and m into the point-slope equation. Ensure you are using the coordinates of the given point and the slope of the perpendicular line.
  • Algebraic errors: Be careful when simplifying and rearranging the equations. Errors in algebraic manipulation can lead to an incorrect final equation.

Applications in Real-World Scenarios

The concept of perpendicular lines and their equations has numerous applications in real-world scenarios. Here are a few examples:

  • Architecture and Construction: Architects and engineers use perpendicular lines extensively in building design and construction. Ensuring walls are perpendicular to the floor or ceiling is crucial for structural integrity and stability. The equations of these lines help in precise planning and execution.
  • Navigation: In navigation, perpendicular lines are used to determine directions and courses. For example, in air traffic control, understanding perpendicular paths is vital for avoiding collisions and maintaining safe distances between aircraft.
  • Computer Graphics: In computer graphics, perpendicular lines are used for various tasks such as creating 3D models, rendering images, and implementing lighting effects. The precise manipulation of lines and surfaces is essential for realistic visualizations.
  • Physics: In physics, perpendicular lines are fundamental in understanding concepts such as forces, motion, and fields. For example, the normal force acting on an object is perpendicular to the surface it rests on, and the components of a force are often resolved into perpendicular directions for analysis.

Conclusion

Finding the equation of a line perpendicular to a given line and passing through a specific point is a fundamental concept in coordinate geometry with broad applications. By understanding the relationship between the slopes of perpendicular lines, using the point-slope form, and avoiding common errors, you can confidently tackle these problems. The ability to work with perpendicular lines is not only essential in mathematics but also valuable in various fields where geometric principles play a critical role. Whether you are designing a building, navigating a ship, or creating a computer animation, the principles discussed in this article provide a solid foundation for understanding and applying the concept of perpendicular lines.

Now, let's solidify your understanding with a practical example. Consider the line defined by the equation $x + y = 5$. Our goal is to determine the equation of a line that is perpendicular to this line and passes through the point $(2, 1)$. This is a classic problem that combines the concepts of slope, perpendicularity, and line equations. By working through this example step-by-step, we'll reinforce the methodology and clarify any remaining questions. This will not only help in solving similar problems but also in appreciating the versatility of these geometric principles in various mathematical and real-world contexts.

Step 1: Rewrite the Equation in Slope-Intercept Form

The first crucial step in finding the equation of a perpendicular line involves transforming the given equation into a more convenient form. Specifically, we want to rewrite the equation $x + y = 5$ into the slope-intercept form, which is expressed as $y = mx + b$. Here, $m$ represents the slope of the line, and $b$ represents the y-intercept. This form is particularly useful because it allows us to easily identify the slope, which is essential for determining the slope of the perpendicular line.

To convert $x + y = 5$ into slope-intercept form, we need to isolate $y$ on one side of the equation. This can be achieved by subtracting $x$ from both sides of the equation. Let's perform this operation step by step:

x+y=5x + y = 5

Subtract $x$ from both sides:

x+y−x=5−xx + y - x = 5 - x

This simplifies to:

y=−x+5y = -x + 5

Now, the equation is in the slope-intercept form $y = mx + b$. By comparing this equation with the general form, we can clearly see that the coefficient of $x$ is $-1$. Therefore, the slope ($m$) of the given line is $-1$. This value is crucial for the next step, where we'll determine the slope of the line perpendicular to this one. Understanding this transformation is key to solving the problem efficiently and accurately. The slope-intercept form not only simplifies the identification of the slope but also provides a clear picture of how the line behaves on a graph, making it an invaluable tool in coordinate geometry.

Step 2: Determine the Slope of the Perpendicular Line

Once we've identified the slope of the given line, the next critical step is to find the slope of the line that is perpendicular to it. Recall the fundamental property of perpendicular lines: the product of their slopes is always $-1$. This relationship is the key to finding the slope of the perpendicular line. If we denote the slope of the given line as $m_1$ and the slope of the perpendicular line as $m_2$, then the relationship can be expressed mathematically as:

m1⋅m2=−1m_1 \cdot m_2 = -1

In the previous step, we determined that the slope of the given line, $x + y = 5$, is $m_1 = -1$. Now, we can use this information to find $m_2$, the slope of the perpendicular line. Substitute $m_1 = -1$ into the equation:

(−1)⋅m2=−1(-1) \cdot m_2 = -1

To solve for $m_2$, we can divide both sides of the equation by $-1$:

(−1)⋅m2−1=−1−1\frac{(-1) \cdot m_2}{-1} = \frac{-1}{-1}

This simplifies to:

m2=1m_2 = 1

Therefore, the slope of the line perpendicular to $x + y = 5$ is $1$. This value is crucial for constructing the equation of the perpendicular line. It tells us the rate at which the y-coordinate changes with respect to the x-coordinate for the perpendicular line. A positive slope indicates that the line rises as you move from left to right, and in this case, a slope of 1 means that for every unit increase in $x$, $y$ also increases by one unit. Understanding this relationship between slopes is essential for grasping the geometry of perpendicular lines and their equations.

Step 3: Use the Point-Slope Form to Write the Equation

With the slope of the perpendicular line determined, the next step is to construct the equation of the line. We know that the line must pass through the point $(2, 1)$. Given a point and a slope, the most convenient way to write the equation of a line is to use the point-slope form. The point-slope form of a linear equation is expressed as:

y−y1=m(x−x1)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.

In our case, we know that the line passes through the point $(2, 1)$, so we can identify $x_1 = 2$ and $y_1 = 1$. We also determined in the previous step that the slope of the perpendicular line is $m = 1$. Now, we can substitute these values into the point-slope form:

y−1=1(x−2)y - 1 = 1(x - 2)

This equation represents the line that is perpendicular to $x + y = 5$ and passes through the point $(2, 1)$. The equation is in a form that directly incorporates the slope and a specific point on the line, making it a powerful tool for defining linear relationships. The point-slope form is particularly useful in scenarios where you have a point and a slope and need to quickly write the equation of the line. It provides a clear and concise representation of the line's properties, allowing for easy manipulation and analysis. In the next step, we can further simplify this equation into other forms, such as slope-intercept form, if desired.

Step 4: Simplify the Equation (Optional)

While the point-slope form we derived in the previous step, $y - 1 = 1(x - 2)$, is a valid representation of the equation of the line, it's often useful to simplify it into other forms, such as the slope-intercept form or the standard form. This simplification can make it easier to compare the equation with other lines, graph the line, or perform further analysis. Let's simplify the equation into slope-intercept form, which is expressed as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

To simplify $y - 1 = 1(x - 2)$, we need to isolate $y$ on one side of the equation. First, we can distribute the $1$ on the right side:

y−1=x−2y - 1 = x - 2

Next, we can add $1$ to both sides of the equation to isolate $y$:

y−1+1=x−2+1y - 1 + 1 = x - 2 + 1

This simplifies to:

y=x−1y = x - 1

Now, the equation is in slope-intercept form, $y = mx + b$. We can see that the slope is $m = 1$ and the y-intercept is $b = -1$. This form provides a clear picture of the line's characteristics: it has a positive slope of 1, meaning it rises as you move from left to right, and it intersects the y-axis at the point $(0, -1)$. Simplifying the equation into slope-intercept form not only provides a clear understanding of the line's properties but also facilitates graphing the line and comparing it with other linear equations. This final step completes the process of finding the equation of a line perpendicular to a given line and passing through a specific point.

Conclusion

In summary, we have successfully found the equation of a line perpendicular to $x + y = 5$ and passing through the point $(2, 1)$. By systematically following the steps—rewriting the equation in slope-intercept form, determining the slope of the perpendicular line, using the point-slope form, and simplifying the equation—we arrived at the solution: $y = x - 1$. This equation represents a line that is perpendicular to the given line and satisfies the condition of passing through the specified point. This exercise demonstrates the power and versatility of coordinate geometry in solving problems involving lines and their relationships. Understanding these concepts is crucial for various applications in mathematics, science, and engineering. The ability to find the equation of a perpendicular line is a fundamental skill that opens the door to more complex geometric problems and real-world applications.