Finding The Equation Of A Perpendicular Line With The Same Y-Intercept
Hey everyone! Today, we're diving into a fun math problem: figuring out the equation of a line that's perpendicular to a given line and shares the same y-intercept. Sounds like a mouthful, right? But don't worry, we'll break it down step by step. Let's get started!
Understanding Perpendicular Lines and Y-Intercepts
Before we jump into the problem, let's quickly review what perpendicular lines and y-intercepts are. This foundational knowledge is key to solving our problem, guys. So, let's make sure we're all on the same page. We'll start by defining what we mean by perpendicular lines and then move onto understanding the y-intercept. Think of this as setting the stage for our main act – finding the equation of the perpendicular line!
Perpendicular Lines
Perpendicular lines are lines that intersect at a right angle (90 degrees). Imagine the corner of a square or a perfectly formed cross – that's what we're talking about. The crucial thing to remember about perpendicular lines is their slopes. If a line has a slope of m, a line perpendicular to it will have a slope of -1/m. This is often called the negative reciprocal of the slope. Why is this important? Well, the slope tells us how steep a line is and in what direction it's going. A line with a positive slope goes upwards as you move from left to right, while a line with a negative slope goes downwards. Perpendicular lines have slopes that are not only opposite in sign (one positive, one negative) but also reciprocals of each other. This ensures they meet at that perfect 90-degree angle. Think of it like this: one line is going up steeply, and the other is coming down just as steeply but in the opposite direction, creating that sharp, perpendicular intersection.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis. In other words, it's the y-coordinate of the point where the line intersects the vertical axis on a graph. This is a super handy point because it gives us a direct piece of information about the line's position on the coordinate plane. The y-intercept is usually represented as the 'b' value in the slope-intercept form of a linear equation, which is y = mx + b. Here, 'm' is the slope (which we just talked about!), and 'b' is the y-intercept. So, if you see an equation like y = 2x + 3, you immediately know that the line crosses the y-axis at the point (0, 3). The y-intercept is like the line's starting point on the vertical axis. It's the place where the line makes its grand entrance onto the coordinate plane. Understanding this concept is vital because, in our problem, we're looking for a line that shares this exact starting point with another line.
The Given Line: Analyzing y = rac{1}{5}x + 1
Okay, let's get down to the specifics of our problem. We're given the line y = (1/5)x + 1. Our mission, should we choose to accept it (and we do!), is to find the equation of a line that's perpendicular to this one and has the same y-intercept. To do this, we need to carefully analyze the given equation and extract the key pieces of information we need. This is like being a detective, guys, and gathering all the clues before we solve the mystery. We're looking for two main clues here: the slope of the given line and its y-intercept. Once we have these, we can use our knowledge of perpendicular lines and y-intercepts to find the equation of the line we're after.
Identifying the Slope
The first thing we need to do is identify the slope of the given line. Remember that the slope-intercept form of a linear equation is y = mx + b, where 'm' represents the slope. By comparing our given equation, y = (1/5)x + 1, to the slope-intercept form, we can clearly see that the coefficient of x is 1/5. This means that the slope of the given line is 1/5. So, m = 1/5. This tells us that for every 5 units we move to the right along the x-axis, the line goes up 1 unit along the y-axis. It's a relatively gentle upward slope. Now, remember our discussion about perpendicular lines? We know that the slope of a line perpendicular to this one will be the negative reciprocal of 1/5. This is a crucial piece of information that will help us find the slope of our new line.
Determining the Y-Intercept
Next, we need to determine the y-intercept of the given line. Again, looking at the slope-intercept form y = mx + b, 'b' represents the y-intercept. In our equation, y = (1/5)x + 1, the constant term is 1. This tells us that the y-intercept of the given line is 1. In other words, the line crosses the y-axis at the point (0, 1). This is the point that our new, perpendicular line will also have to pass through. We're not just looking for any line perpendicular to the given one; we're looking for one that shares this specific y-intercept. This common y-intercept is like the anchor point that ties our new line to the original one. It's a crucial constraint that helps us narrow down the possibilities and find the unique equation we're seeking.
Finding the Slope of the Perpendicular Line
Now that we know the slope of the given line is 1/5, we can find the slope of the line perpendicular to it. Remember the key concept: the slopes of perpendicular lines are negative reciprocals of each other. This is like a mathematical recipe, guys, where we have one ingredient (the original slope) and we need to transform it to get the ingredient we need for our new line. So, what's the negative reciprocal of 1/5? This is a crucial step in our problem-solving process. Getting this right is like laying the foundation for a building; if it's solid, everything else will fall into place. Let's take a closer look at how we find this negative reciprocal.
Calculating the Negative Reciprocal
To find the negative reciprocal of a number, we first flip the fraction (find the reciprocal) and then change its sign. For 1/5, the reciprocal is 5/1, which is simply 5. Then, we change the sign to negative, giving us -5. So, the slope of the line perpendicular to y = (1/5)x + 1 is -5. This means that our new line will be much steeper than the original line and will slope downwards as we move from left to right. The negative sign indicates the downward direction, and the larger absolute value (5 compared to 1/5) indicates the steeper slope. This slope of -5 is a critical piece of the puzzle. It tells us the rate at which our new line is changing vertically with respect to its horizontal change. With this in hand, along with the y-intercept, we're well on our way to finding the equation of the perpendicular line.
Constructing the Equation
We're in the home stretch now! We know the slope of the perpendicular line (-5) and the y-intercept (1). Now we can plug these values into the slope-intercept form of a linear equation, y = mx + b, to get the equation of the line. This is like the final brushstroke on a painting, guys, where we bring all the elements together to create the finished product. We've gathered all the necessary information, and now it's time to put it all together in a clear and concise equation. This equation will be the answer to our problem, the culmination of all our hard work and careful analysis.
Using Slope-Intercept Form
We have m = -5 (the slope) and b = 1 (the y-intercept). Substituting these values into y = mx + b, we get y = -5x + 1. And there you have it! This is the equation of the line that is perpendicular to y = (1/5)x + 1 and has the same y-intercept. It's a beautiful thing when all the pieces come together like this. We started with a problem, broke it down into smaller parts, solved each part, and then put the results together to arrive at our final answer. This equation, y = -5x + 1, represents a line that is not only perpendicular to the given line but also shares a crucial characteristic – the y-intercept. This makes it a unique solution to our problem.
Conclusion
So, we've successfully found the equation of the line perpendicular to y = (1/5)x + 1 with the same y-intercept, which is y = -5x + 1. Great job, everyone! Remember, the key is to understand the relationship between slopes of perpendicular lines and the significance of the y-intercept. This is like mastering a new skill, guys, whether it's riding a bike or solving a math problem. The more you practice and understand the underlying principles, the easier it becomes. We took a potentially complex problem and broke it down into manageable steps. We identified the key concepts, applied them systematically, and arrived at the solution. This is the essence of problem-solving in mathematics and in life. Keep practicing, keep exploring, and keep having fun with math!
