Perpendicular Lines Equation Find The Perpendicular Line's Equation
Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of linear equations, specifically focusing on perpendicular lines and how to find their equations. We'll tackle a classic problem that involves converting an equation to slope-intercept form and then using that information to determine the equation of a perpendicular line passing through a given point. Buckle up, because we're about to embark on a mathematical adventure!
The Challenge Understanding Perpendicular Lines
Our journey begins with a line defined by the equation $2x + 12y = -1$. The mission, should we choose to accept it, is to find the equation of a new line. This new line must be perpendicular to our original line and also pass through the point $(0, 9)$. To conquer this challenge, we need to understand a few key concepts about lines and their equations. Firstly, let’s dive into slope-intercept form, a crucial tool in our mathematical arsenal. The slope-intercept form of a linear equation is expressed as $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the y-intercept. The slope, often denoted by $m$, quantifies the steepness and direction of a line. A positive slope indicates that the line rises as we move from left to right, while a negative slope indicates that the line falls. The larger the absolute value of the slope, the steeper the line. The y-intercept, denoted by $b$, is the point where the line crosses the y-axis. It's the value of $y$ when $x = 0$. Understanding these basics is crucial for manipulating linear equations and extracting valuable information. Next, we need to understand the relationship between the slopes of perpendicular lines. Two lines are perpendicular if they intersect at a right angle (90 degrees). There's a special connection between their slopes: the slopes of perpendicular lines are negative reciprocals of each other. This means that if one line has a slope of $m$, the slope of a line perpendicular to it will be $-\frac{1}{m}$. This negative reciprocal relationship is the cornerstone of solving our problem. Finally, we'll use the given point $(0, 9)$ and the slope we calculate to construct the equation of the perpendicular line. The point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, will be our handy tool for this step. Here, $(x_1, y_1)$ represents a point on the line and $m$ is the slope. By substituting the coordinates of our point and the calculated slope, we can easily derive the equation of the perpendicular line. So, with these fundamental concepts in our grasp, let's roll up our sleeves and solve this problem step by step!
Step 1 Transforming the Equation to Slope-Intercept Form
The first hurdle in our path is the given equation, $2x + 12y = -1$. It's not in the friendly slope-intercept form yet, so we need to transform it. Remember, our goal is to isolate $y$ on one side of the equation. To achieve this, we'll employ some algebraic maneuvering. First, let's subtract $2x$ from both sides of the equation. This maintains the balance of the equation and moves the $x$ term to the right side. The equation now looks like this: $12y = -2x - 1$. We're getting closer! Now, to completely isolate $y$, we need to divide both sides of the equation by 12. This will remove the coefficient of $y$ and give us the desired slope-intercept form. Performing the division, we get: $y = -\frac2}{12}x - \frac{1}{12}$. But wait, we can simplify the fraction $-\frac{2}{12}$. Both the numerator and denominator are divisible by 2, so let's reduce it to its simplest form. Dividing both by 2, we get $-\frac{1}{6}$. Our equation now looks much cleaner{6}x - \frac{1}{12}$. We've successfully transformed the equation into slope-intercept form! This is a significant victory because it allows us to easily identify the slope of the given line. By comparing our equation to the general slope-intercept form, $y = mx + b$, we can clearly see that the slope of the given line is $m = -\frac{1}{6}$. This slope is the key to unlocking the next step in our journey: finding the slope of the perpendicular line. So, with the slope of the original line in hand, let's move on to the next stage of our mathematical quest!
Step 2 Finding the Slope of the Perpendicular Line
Now that we've conquered the challenge of converting the original equation to slope-intercept form and identified its slope as $-\frac1}{6}$, we're ready to tackle the next crucial step{6}$. To find the negative reciprocal, we first flip the fraction, which gives us $-\frac{6}{1}$, or simply $-6$. Then, we change the sign. Since our original slope was negative, its negative reciprocal will be positive. So, changing the sign of $-6$ gives us $6$. Therefore, the slope of the line perpendicular to the given line is $6$. This is a critical piece of the puzzle! With the slope of the perpendicular line in hand, we're well on our way to finding its equation. We know the slope is $6$, and we also know that the line passes through the point $(0, 9)$. This is enough information to construct the equation using the point-slope form or by directly substituting into the slope-intercept form. The next step is where we put it all together and unveil the final equation. So, let's keep the momentum going and move on to the exciting conclusion of our mathematical journey!
Step 3 Constructing the Equation of the Perpendicular Line
We've arrived at the final, thrilling step! We know the slope of our perpendicular line is $6$, and we know it passes through the point $(0, 9)$. Now, we'll use this information to construct the equation of the line. There are two main approaches we can take here: using the point-slope form or directly substituting into the slope-intercept form. Let's start with the slope-intercept form, $y = mx + b$. We already know the slope, $m = 6$. The point $(0, 9)$ gives us the y-intercept directly! Remember, the y-intercept is the value of $y$ when $x = 0$. Since our point is $(0, 9)$, the y-intercept is $9$. So, $b = 9$. Now we simply substitute these values into the slope-intercept form: $y = 6x + 9$. And there you have it! We've found the equation of the perpendicular line. But just for fun, let's also use the point-slope form to verify our result. The point-slope form is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. We have $(x_1, y_1) = (0, 9)$ and $m = 6$. Substituting these values, we get: $y - 9 = 6(x - 0)$. Simplifying, we get: $y - 9 = 6x$. Adding 9 to both sides, we get: $y = 6x + 9$. Ta-da! We arrived at the same equation using both methods. This confirms that our solution is correct. The equation of the line perpendicular to the given line and passing through the point $(0, 9)$ is $y = 6x + 9$. We've successfully navigated the world of perpendicular lines and emerged victorious! This problem demonstrates the power of understanding fundamental concepts like slope-intercept form, the relationship between slopes of perpendicular lines, and the point-slope form. With these tools in your mathematical toolkit, you'll be well-equipped to tackle a wide range of linear equation challenges. So, go forth and conquer!
Final Answer
The equation of the line perpendicular to $2x + 12y = -1$ and passing through the point $(0, 9)$ is:
A. $y = 6x + 9$
