Finding The Equation Of A Perpendicular Line And Graphing It

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Hey guys! Today, we're diving into the exciting world of linear equations and perpendicular lines. We're going to tackle a common problem in mathematics: finding the equation of a line that's perpendicular to a given line and then, just for fun, we'll graph it! So, buckle up and let's get started!

Understanding the Basics: Slopes and Perpendicularity

Before we jump into solving the problem, let's quickly refresh our understanding of slopes and what it means for lines to be perpendicular. The slope of a line, often denoted by 'm', tells us how steep the line is and whether it's going upwards or downwards as we move from left to right. A line with a positive slope goes upwards, while a line with a negative slope goes downwards. A horizontal line has a slope of 0, and a vertical line has an undefined slope.

Now, what about perpendicular lines? Two lines are perpendicular if they intersect at a right angle (90 degrees). The key relationship between the slopes of perpendicular lines is this: the product of their slopes is -1. In other words, if one line has a slope of 'm', the slope of a line perpendicular to it will be '-1/m'. This is also known as the negative reciprocal of the slope. Understanding this relationship is crucial for solving our problem.

For instance, let's say we have a line with a slope of 2. To find the slope of a line perpendicular to it, we take the negative reciprocal of 2, which is -1/2. Similarly, if a line has a slope of -3, the slope of a perpendicular line would be 1/3. See how we flipped the fraction and changed the sign? That's the magic of negative reciprocals!

So, remember this: perpendicular lines have slopes that are negative reciprocals of each other. This concept is the foundation for finding the equation of a perpendicular line.

The Problem: Finding the Perpendicular Line

Okay, let's get to the heart of the matter. Our problem is to find the equation of a line that is perpendicular to the line given by the equation y = 5 - x. The first step is to identify the slope of the given line. To do this, we need to rewrite the equation in the slope-intercept form, which is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. In our case, we can rewrite y = 5 - x as y = -x + 5. Now it's clear that the slope of the given line is -1 (the coefficient of x).

Next, we need to find the slope of a line perpendicular to this line. As we discussed earlier, we do this by taking the negative reciprocal of the slope. The negative reciprocal of -1 is -1/(-1), which simplifies to 1. So, any line perpendicular to y = 5 - x will have a slope of 1. That's a big step forward!

Now that we know the slope of the perpendicular line, we need to find its equation. We'll use the slope-intercept form again, y = mx + b. We know 'm' is 1, so our equation looks like y = 1x + b, or simply y = x + b. The only thing left to find is 'b', the y-intercept.

Here's where things get interesting. There are infinitely many lines that are perpendicular to y = 5 - x, each with a different y-intercept. So, we can choose any value for 'b' and get a valid perpendicular line. For simplicity, let's choose b = 0. This gives us the equation y = x, which is a line that passes through the origin (0,0) with a slope of 1. Of course, we could have chosen any other value for 'b', such as 1, -2, or even pi, and we would have gotten a different perpendicular line. The important thing is that the slope is 1.

So, one possible equation for a line perpendicular to y = 5 - x is y = x. But remember, this is just one solution. There are countless others!

Graphing the Lines: Visualizing Perpendicularity

Now, let's bring these equations to life by graphing them! Graphing is a fantastic way to visualize the relationship between the original line and the perpendicular line.

First, let's graph y = 5 - x. We already know the slope is -1 and the y-intercept is 5. To graph it, we can start by plotting the y-intercept at the point (0, 5). Then, using the slope of -1, we can move one unit to the right and one unit down to find another point on the line (1, 4). Connecting these two points gives us the graph of y = 5 - x. You'll notice it's a line that slopes downwards from left to right.

Next, let's graph y = x, our perpendicular line. This line has a slope of 1 and a y-intercept of 0. We can plot the y-intercept at the origin (0, 0). Then, using the slope of 1, we can move one unit to the right and one unit up to find another point on the line (1, 1). Connecting these points gives us the graph of y = x. This line slopes upwards from left to right.

Now, the moment of truth! If you've graphed these lines correctly, you'll see that they intersect at a right angle. This visual confirmation reinforces the concept of perpendicularity. You can clearly see that the two lines form a perfect 90-degree angle where they meet.

Graphing not only helps us visualize the solution but also acts as a check for our work. If the lines don't appear to be perpendicular on the graph, it's a sign that we might have made a mistake in our calculations.

Key Takeaways and Further Exploration

Let's recap what we've learned today. We successfully found the equation of a line perpendicular to a given line and graphed both lines. We reinforced the concept of slopes and how they relate to perpendicular lines. Remember, the key is that perpendicular lines have slopes that are negative reciprocals of each other.

We also saw that there isn't just one perpendicular line; there are infinitely many! Each perpendicular line has the same slope (the negative reciprocal of the original slope) but a different y-intercept. This gives us a whole family of lines that are perpendicular to the original line.

To further explore this topic, you can try the following:

  • Practice with different equations: Try finding the equations of lines perpendicular to other given lines. Vary the slopes and y-intercepts to challenge yourself.
  • Explore different values for 'b': Choose different values for the y-intercept 'b' in the equation y = x + b and graph the resulting lines. See how they all remain perpendicular to the original line y = 5 - x.
  • Investigate parallel lines: Parallel lines have the same slope. How would you find the equation of a line parallel to a given line?
  • Real-world applications: Think about where perpendicular lines are used in the real world, such as in architecture, construction, and navigation.

By practicing and exploring these concepts, you'll solidify your understanding of linear equations and perpendicular lines. Keep up the great work, and remember, math can be fun!

Conclusion

So, there you have it! We've successfully navigated the world of perpendicular lines, found an equation for one, and even graphed it. Remember, the key takeaway is the relationship between the slopes of perpendicular lines. By understanding this concept, you can confidently tackle any problem involving perpendicularity. Keep practicing, and you'll become a pro in no time! Happy graphing, guys!