Peter's Perpendicular Line: Unpacking The Equation

Oct 14, 2025 by ADMIN 51 views

Hey guys! Let's dive into a cool math problem. We're going to figure out why Peter wrote the equation $1 = - rac{1}{2}(0) + b$ to find the equation of a line. This line is super special because it's perpendicular to another line, and it also has to go through a specific point. It's like a mathematical puzzle, and we're the detectives! Understanding this requires us to remember a few key concepts about lines, slopes, and how they relate to each other. Don't worry, it's not as scary as it sounds. I'll break it down step by step, so you can understand why Peter made those specific choices.

Understanding the Problem: Perpendicular Lines and Point-Slope Form

So, the heart of this problem revolves around two crucial ideas: perpendicular lines and the point-slope form of a linear equation. First, what does it mean for lines to be perpendicular? Simply put, it means they intersect at a right angle (90 degrees). Think of the corner of a square or a cross. The slopes of perpendicular lines have a special relationship: they are negative reciprocals of each other. That's a mouthful, but let's break it down. If one line has a slope of 2, the slope of a line perpendicular to it will be $- rac{1}{2}$ . We flip the fraction (2/1 becomes 1/2) and change the sign (positive becomes negative, and vice-versa). Got it? Nice!

Now, the point-slope form of a linear equation is a handy way to write the equation of a line when you know its slope and a point it passes through. The general form is: $y - y_1 = m(x - x_1)$ , where: $m$ represents the slope of the line, and $(x_1, y_1)$ represents the coordinates of a point on the line. This form is super convenient because it allows us to directly plug in the slope and the coordinates of the given point, and then solve for the equation of the line. Pretty neat, right?

Let's look at the given information: We have the line $y = 2x + 3$ . This is written in slope-intercept form ( $y = mx + b$ ), where $m$ (the slope) is 2, and $b$ (the y-intercept) is 3. We also know that our new line must be perpendicular to this one, and it must pass through the point $(0, 1)$ . We have all the ingredients; we just need to put them together.

To reiterate, Peter needs to find a line that intersects the line $y = 2x + 3$ at a right angle and goes through the point (0, 1). Therefore, we must first determine the slope of the new line. Since we know that the slope of the first line is 2, the slope of the new line must be its negative reciprocal, which is $-\frac{1}{2}$ . Next, we have a point (0, 1) that lies on the new line. So, Peter needs to use the information to find the equation of the new line.

Breaking Down Peter's Equation: $1 = - rac{1}{2}(0) + b$

Okay, now let's zoom in on Peter's equation: $1 = - rac{1}{2}(0) + b$ . This equation is a crucial step in finding the equation of the perpendicular line. Remember the point-slope form? While Peter didn't directly use the point-slope form to write this equation, the concept is still there. He's essentially using the slope-intercept form ( $y = mx + b$ ) to find the specific equation of the line. Let's see how it all fits together.

Peter already knows the slope ( $m$ ) of the perpendicular line is $- rac{1}{2}$ . He also knows that the line passes through the point $(0, 1)$ . This point is particularly important because it's the y-intercept of the line. The y-intercept is the point where the line crosses the y-axis (where $x = 0$ ).

By substituting the x and y values of the point $(0, 1)$ into the slope-intercept form, Peter is essentially solving for $b$ (the y-intercept). Here's how it breaks down: $y = mx + b$ . Replace $y$ with 1 (the y-coordinate of the point), $m$ with $- rac{1}{2}$ (the slope of the perpendicular line), and $x$ with 0 (the x-coordinate of the point): $1 = - rac{1}{2}(0) + b$ . When we simplify this, we will be able to determine the value of $b$ . By solving for $b$ , he is finding the point where the line crosses the y-axis. The value of $b$ is the y-coordinate where the line intersects with the y-axis. That's why this equation is so important. It allows us to pinpoint the exact equation of our line.

The equation $1 = - rac{1}{2}(0) + b$ is Peter's way of figuring out the y-intercept. By simplifying this equation, we can easily find that $b = 1$ . This means the line crosses the y-axis at the point $(0, 1)$ , which is also the given point. In fact, the only reason that Peter has to use this formula is because the point provided is at the y-intercept. If the point was not located at the y-intercept, we will have to apply the formula with the x-intercept and determine where the line intersects the x-axis.

Putting It All Together: The Equation of the Perpendicular Line

Now that we know the slope ( $m = - rac{1}{2}$ ) and the y-intercept ( $b = 1$ ), we can write the equation of the perpendicular line in slope-intercept form ( $y = mx + b$ ). Plugging in the values, we get $y = - rac{1}{2}x + 1$ . And there you have it, the equation of the line that is perpendicular to $y = 2x + 3$ and passes through the point $(0, 1)$ ! We've successfully cracked the code and solved the mathematical puzzle. Congratulations! You see, it's not so difficult once you break it down into smaller, manageable steps.

To sum it up: Peter's equation, $1 = - rac{1}{2}(0) + b$ , was a clever way to find the y-intercept of the perpendicular line. He used the given point and the calculated slope to solve for $b$ , enabling him to write the complete equation of the line. Remember, the equation is a tool to help find a specific line by using the properties of perpendicular lines and the slope-intercept form.

So, next time you encounter a similar problem, remember the steps: Find the slope of the perpendicular line, use the point to find the y-intercept, and then write the equation. You got this, guys!

Why is the Point (0, 1) Significant?

The point (0, 1) is actually super important in this specific problem, because it makes the calculation for the y-intercept ( $b$ ) incredibly easy. As mentioned earlier, the point (0, 1) is the y-intercept. When $x = 0$ , the equation simplifies to $y = b$ . This means that the y-coordinate of the point is the same as the y-intercept. In this particular case, it makes the whole problem so much easier to solve! If the point wasn't on the y-axis, we would have to do a few more steps. Namely, we would have to use the slope and the point to determine the equation of the line. However, because the point lies on the y-axis, Peter can immediately determine the value of $b$ and write the whole equation without much work.

It's a shortcut, guys! Imagine if the point had been (2, 0). We'd still use the slope-intercept form, but we'd have to substitute the x and y values and then solve for $b$ : $0 = - rac{1}{2}(2) + b$ . Then we'd simplify, and find that $0 = -1 + b$ , which means $b = 1$ . Although the y-intercept is still 1, the process would be a little longer. This problem, though, is easy because the point is at the y-intercept.

Deeper Dive: The Math Behind the Scenes

Let's go a little deeper to see why this all works. The slope-intercept form ( $y = mx + b$ ) is actually derived from the point-slope form we talked about earlier. By rearranging the point-slope form, you can derive the slope-intercept form. Both are just different ways of representing the same linear relationship. The point-slope form is great when you have a point and the slope. Slope-intercept form is great when you want to easily see the slope and the y-intercept. They are both tools that help you understand and manipulate linear equations.

Also, the idea of negative reciprocals is important. This property guarantees that the two lines will be perpendicular. The negative reciprocal concept ensures that the lines will meet at a right angle. It's rooted in the fundamental geometry of how slopes and angles relate to each other. If you were to graph the two lines, you'd visually see the 90-degree angle created at the intersection. Pretty cool, right?

Key Takeaways

Perpendicular Lines: They have slopes that are negative reciprocals. If you know the slope of one line, you can easily find the slope of a line perpendicular to it. The product of the slopes of perpendicular lines is always -1. This is a quick way to verify your work. This property is crucial for solving problems related to perpendicular lines. It is the very basis of this problem. You must understand this to begin.
Point-Slope Form and Slope-Intercept Form: These are tools for finding and writing the equations of lines. They provide flexibility in solving problems based on what you know (a point and a slope, or a slope and a y-intercept).
The Y-Intercept: The point where a line crosses the y-axis. It is represented by the value of $b$ in the slope-intercept form ( $y = mx + b$ ). Knowing the y-intercept simplifies the process of finding the equation of a line, especially when given the y-intercept point.
Peter's Equation: $1 = - rac{1}{2}(0) + b$ was a clever way to determine the y-intercept ( $b$ ) of the perpendicular line, given that the point $(0, 1)$ was on the line. It simplified the process and helped Peter find the line's equation quickly and efficiently.

Conclusion

So, in essence, Peter wrote the equation $1 = - rac{1}{2}(0) + b$ to find the y-intercept of the line, which, along with the calculated slope, allowed him to complete the equation of the perpendicular line. It's a clear demonstration of how mathematical concepts connect to solve problems. By understanding the relationship between slopes, perpendicularity, and the slope-intercept form, you can tackle similar problems with confidence. Keep practicing, and soon you'll be writing equations like Peter too! Keep up the great work, guys!