Equation Of Line Perpendicular To Given Line: A Step-by-Step Guide

Dec 3, 2025 by Alex Johnson 67 views

Equation of a Line Perpendicular to a Given Line: A Step-by-Step Guide

Have you ever found yourself staring at a graph, wondering how to find the equation of a line that's not just any line, but one that's perfectly perpendicular to another? It might seem daunting at first, but don't worry! This guide will walk you through the process, step by step, with a friendly and conversational tone. We'll tackle the challenge of finding the equation of a line that passes through the point (-1, 4) and is perpendicular to a given line. Let's dive in and make this concept crystal clear!

Understanding Perpendicular Lines

Before we jump into the equation itself, let's first understand what it means for lines to be perpendicular. In simple terms, perpendicular lines are lines that intersect at a right angle (90 degrees). Think of the corner of a square or a perfectly formed cross. The key to understanding perpendicular lines lies in their slopes.

The slope of a line is a measure of its steepness and direction. It tells us how much the line rises (or falls) for every unit it runs horizontally. We often represent slope with the letter 'm'. Now, here's the crucial part: if two lines are perpendicular, their slopes have a special relationship. The slope of one line is the negative reciprocal of the slope of the other. Let's break that down:

Reciprocal: To find the reciprocal of a number, you simply flip it. For example, the reciprocal of 2 is 1/2, and the reciprocal of 3/4 is 4/3.
Negative Reciprocal: This means you first find the reciprocal and then change its sign. So, the negative reciprocal of 2 would be -1/2, and the negative reciprocal of -3/4 would be 4/3.

This relationship is absolutely fundamental to solving our problem. Once we know the slope of the given line, we can easily find the slope of the perpendicular line by taking the negative reciprocal. This knowledge is our starting point, and from here, we will find the equation of the line we are looking for.

Identifying the Slope of the Given Line

Now that we understand the relationship between perpendicular lines and their slopes, let's figure out how to find the slope of our given line. Often, the equation of a line is presented in slope-intercept form, which looks like this:

y = mx + b

Where:

y is the vertical coordinate
x is the horizontal coordinate
m is the slope of the line
b is the y-intercept (the point where the line crosses the y-axis)

If the equation of the given line is already in slope-intercept form, identifying the slope is a breeze! It's simply the coefficient of x (the number multiplying x). For instance, if the equation is y = 3x + 2, the slope is 3. However, sometimes the equation might be in a different form, like standard form (Ax + By = C). In that case, you'll need to rearrange the equation to slope-intercept form by solving for y. Let's say you have the equation 2x + y = 5. To convert it, you'd subtract 2x from both sides to get y = -2x + 5. Now it's clear that the slope is -2.

Even if you're given two points on the line instead of an equation, you can still find the slope. The formula for calculating slope using two points (x1, y1) and (x2, y2) is:

m = (y2 - y1) / (x2 - x1)

This formula represents the change in y (the rise) divided by the change in x (the run). Once you have the slope of the given line, you're one step closer to finding the equation of the perpendicular line.

Remember, the key is to isolate 'm', either by recognizing it in the slope-intercept form or by calculating it using the two-point formula. This value is the cornerstone of our next step.

Calculating the Slope of the Perpendicular Line

With the slope of the given line in hand, we can now easily determine the slope of the line perpendicular to it. Remember our golden rule: the slopes of perpendicular lines are negative reciprocals of each other. This means we need to flip the fraction (find the reciprocal) and change the sign.

Let's say the slope of our given line (m1) is 2/3. To find the slope of the perpendicular line (m2), we first find the reciprocal of 2/3, which is 3/2. Then, we change the sign, making it -3/2. So, if m1 = 2/3, then m2 = -3/2. If the slope of the given line is a whole number, like 5, think of it as 5/1. The reciprocal would be 1/5, and the negative reciprocal would be -1/5. If the slope is already negative, like -4, the negative reciprocal would be 1/4.

This simple yet powerful transformation gives us the slope we need for our new line. This negative reciprocal slope is what ensures our new line intersects the original at a perfect 90-degree angle.

This step is crucial, so double-check your calculations! A small mistake here will throw off the entire equation. With the correct perpendicular slope, we're ready to move on to the next step: using the point-slope form to craft the equation of our desired line.

Using the Point-Slope Form

Now that we have the slope of the perpendicular line, we need to find its equation. This is where the point-slope form comes in handy. The point-slope form is a way to write the equation of a line when you know a point on the line and its slope. The formula looks like this:

y - y1 = m(x - x1)

Where:

y and x are the variables representing the coordinates of any point on the line
(x1, y1) is a specific point on the line (in our case, (-1, 4))
m is the slope of the line (the negative reciprocal we calculated earlier)

This form is incredibly useful because it directly incorporates the information we have: the slope (m) and a point ((-1, 4)) that the line passes through. To use the point-slope form, simply substitute the known values into the formula. In our example, x1 = -1, y1 = 4, and let's say we've calculated the perpendicular slope m to be -2 (for the sake of example). Plugging these values in, we get:

y - 4 = -2(x - (-1))

Notice how we've carefully substituted each value into its correct place in the formula. The x - (-1) becomes x + 1. This equation is now in point-slope form. It accurately represents the line that passes through the point (-1, 4) and has a slope of -2.

The beauty of the point-slope form is its directness. It allows us to build the equation of the line using the most essential pieces of information: a point and the slope. From here, we can easily transform it into other forms if needed.

Converting to Slope-Intercept Form (Optional)

While the point-slope form is a perfectly valid way to express the equation of a line, sometimes it's helpful to convert it to slope-intercept form (y = mx + b). This form makes it easy to see the slope and y-intercept of the line at a glance. To convert from point-slope form to slope-intercept form, simply simplify the equation and solve for y.

Let's continue with our example from the previous section. We had the equation in point-slope form:

y - 4 = -2(x + 1)

First, distribute the -2 on the right side of the equation:

y - 4 = -2x - 2

Next, isolate y by adding 4 to both sides:

y = -2x - 2 + 4

Finally, simplify:

y = -2x + 2

Now the equation is in slope-intercept form. We can clearly see that the slope (m) is -2 and the y-intercept (b) is 2. Converting to slope-intercept form can be particularly useful if you need to graph the line or compare it to other lines. It provides a clear and concise representation of the line's characteristics.

However, remember that converting to slope-intercept form is optional. The point-slope form is already a complete and accurate representation of the line. The choice of which form to use often depends on the specific problem or the desired format for the answer.

Putting It All Together: A Worked Example

Let's solidify our understanding with a complete example. Suppose we are asked to find the equation of the line that passes through the point (-1, 4) and is perpendicular to the line y = (1/3)x - 2. Let's walk through each step:

Identify the slope of the given line: The given line is in slope-intercept form, y = (1/3)x - 2. The slope (m1) is the coefficient of x, which is 1/3.
Calculate the slope of the perpendicular line: The slope of the perpendicular line (m2) is the negative reciprocal of 1/3. The reciprocal of 1/3 is 3/1, or simply 3. The negative reciprocal is -3. So, m2 = -3.
Use the point-slope form: We know the slope (m2 = -3) and a point on the line ((-1, 4)). Plug these values into the point-slope form: y - y1 = m(x - x1). This gives us y - 4 = -3(x - (-1)), which simplifies to y - 4 = -3(x + 1).
Convert to slope-intercept form (optional): Distribute the -3: y - 4 = -3x - 3. Add 4 to both sides: y = -3x + 1.

Therefore, the equation of the line that passes through (-1, 4) and is perpendicular to the line y = (1/3)x - 2 is y = -3x + 1 (in slope-intercept form) or y - 4 = -3(x + 1) (in point-slope form). This example demonstrates how each step fits together to solve the problem. By following these steps methodically, you can confidently tackle any similar problem.

Conclusion

Finding the equation of a line perpendicular to a given line might have seemed like a complex task at first, but hopefully, this step-by-step guide has made the process clear and manageable. Remember the key concepts: perpendicular lines have slopes that are negative reciprocals of each other, and the point-slope form is a powerful tool for constructing the equation of a line. By understanding these principles and practicing the steps, you'll be able to confidently solve these types of problems.

Don't hesitate to revisit this guide and work through the example again if needed. With practice, finding equations of perpendicular lines will become second nature. Keep exploring and expanding your mathematical skills! For further learning and resources on linear equations, you can visit Khan Academy's Linear Equations section.