Parallel Lines: Equation Through A Point

Welcome to our deep dive into the fascinating world of coordinate geometry! Today, we're going to tackle a common yet crucial problem: finding the equation of a line parallel to a given line and passing through a specific point. This skill is fundamental in mathematics, serving as a building block for more complex concepts in algebra, calculus, and beyond. We'll break down the process step-by-step, ensuring you understand not just how to solve it, but why it works. Our specific challenge is to find an equation for the line parallel to $y=10x+9$ and passing through the point $(9,-4)$, and we'll present our final answer in the universally recognized slope-intercept form. So, grab your favorite thinking cap, and let's embark on this mathematical adventure together!

Understanding Parallel Lines and Slope-Intercept Form

Before we dive into solving our specific problem, let's refresh our understanding of the key concepts involved. Parallel lines are lines that run alongside each other and never intersect, no matter how far they are extended. In the realm of coordinate geometry, this geometric property translates into a very specific algebraic characteristic: parallel lines have the same slope. This is the golden rule we'll be relying on. The slope of a line tells us how steep it is and in which direction it's rising or falling. It's often represented by the letter 'm'.

Now, let's talk about slope-intercept form. This is a standard way of writing the equation of a straight line. It looks like this: $y = mx + b$. In this equation, '$m$' represents the slope of the line, and '$b$' represents the y-intercept – the point where the line crosses the y-axis. The beauty of this form is that it immediately tells you two critical pieces of information about the line: its steepness and where it crosses the vertical axis. Our goal is to find an equation that fits this $y = mx + b$ structure for our new line.

Our given line is $y = 10x + 9$. By simply looking at this equation, we can instantly identify its slope. Comparing it to the general $y = mx + b$ form, we see that the slope ($m$) of this given line is 10. Since we need to find a line parallel to this one, our new line must also have a slope of 10. This is the first and most important piece of information we've extracted. The second piece of information is the point our new line must pass through: $(9, -4)$. This point is like a fixed anchor for our line; it must go through this exact location on the coordinate plane. With these two pieces of information – the slope and a point it passes through – we have everything we need to construct our unique parallel line's equation.

Step-by-Step Solution: Finding the Parallel Line Equation

Let's get down to business and solve our problem: find an equation for the line parallel to $y=10x+9$ and passing through the point $(9,-4)$ in slope-intercept form. We've already established that since the line we're looking for is parallel to $y=10x+9$, its slope must also be 10. So, for our new line's equation, we know that $m=10$. Our equation now looks something like $y = 10x + b$. The only thing missing is the value of '$b$', the y-intercept.

This is where the point $(9, -4)$ comes into play. Since our line must pass through this point, the coordinates of this point must satisfy the equation. In other words, when $x=9$, the corresponding $y$ value must be $-4$. We can substitute these values into our partially formed equation: $y = 10x + b$. Substituting $x=9$ and $y=-4$, we get: $-4 = 10(9) + b$. Now, we just need to solve this equation for '$b$'.

First, calculate the product of 10 and 9: $10 imes 9 = 90$. So, the equation becomes: $-4 = 90 + b$. To isolate '$b$', we need to subtract 90 from both sides of the equation. Performing this subtraction, we get: $-4 - 90 = b$. This simplifies to $-94 = b$. So, the y-intercept of our parallel line is -94.

Now that we have both the slope ($m=10$) and the y-intercept ($b=-94$), we can write the complete equation of the line in slope-intercept form: $y = 10x - 94$. This is our final answer! We have successfully found the equation of a line that is parallel to the original line and passes through the specified point, all expressed in the convenient slope-intercept form. This process demonstrates how geometric properties translate directly into algebraic manipulations, making coordinate geometry a powerful tool for understanding spatial relationships.

Verifying Our Solution

It's always a good practice in mathematics to verify your solution. This ensures that you haven't made any calculation errors and that your final equation truly meets all the conditions of the problem. We found our equation to be $y = 10x - 94$. Let's check if it satisfies the two main requirements:

1. Is it parallel to $y=10x+9$? Two lines are parallel if they have the same slope. The slope of our new line, $y = 10x - 94$, is clearly 10. The slope of the original line, $y = 10x + 9$, is also 10. Since the slopes are identical, our line is indeed parallel to the given line. Success on the first condition!

2. Does it pass through the point $(9,-4)$? For the line to pass through this point, substituting $x=9$ into our equation should yield $y=-4$. Let's substitute $x=9$ into our equation $y = 10x - 94$: $y = 10(9) - 94$ $y = 90 - 94$ $y = -4$ The calculation shows that when $x=9$, $y$ is indeed $-4$. Success on the second condition!

Since both conditions are met, we can be confident that our equation $y = 10x - 94$ is the correct answer. This verification step is crucial for building confidence in your mathematical abilities and for catching any potential slip-ups. It transforms the process from just finding an answer to understanding and confirming it.

Practical Applications of Parallel Lines

While this problem might seem like a purely academic exercise, the concept of parallel lines and finding their equations has numerous practical applications in various fields. Understanding how to manipulate line equations is not just about solving textbook problems; it's about describing and interacting with the world around us.

In architecture and civil engineering, parallel lines are fundamental. Think about roads, bridges, railway tracks, or the walls of a building. These all rely on parallel structures for stability and functionality. When designing structures, engineers use coordinate geometry to ensure that elements are precisely parallel, preventing structural failures. For instance, ensuring that bridge supports are perfectly parallel to each other is critical for distributing weight evenly.

In computer graphics and design, parallel lines are everywhere. When creating 2D or 3D models, software relies heavily on geometric principles. Programmers use line equations to define shapes, create parallel edges for objects, and ensure consistent perspectives. This is essential for everything from video game development to architectural visualizations and graphic design.

Navigation systems, both on land and at sea, use principles of geometry. While not always explicitly dealing with $y=mx+b$ form, the underlying concepts of parallel paths and lines of bearing are derived from these fundamental geometric ideas. Ensuring parallel flight paths or parallel shipping lanes helps manage traffic and prevent collisions.

Even in everyday life, we encounter situations where parallel lines are important. Think about setting up a fence, aligning picture frames on a wall, or arranging furniture. While we might not be calculating slopes, the visual and functional importance of parallel elements is clear. The ability to understand and describe parallel relationships mathematically provides a powerful framework for problem-solving in these diverse areas. It highlights how abstract mathematical concepts have tangible, real-world impacts.

Conclusion

We've successfully navigated the process of finding the equation of a line parallel to a given line and passing through a specific point, culminating in our answer $y = 10x - 94$ in slope-intercept form. We learned that the key to parallel lines lies in their shared slope, and by using the given point, we could pinpoint the unique y-intercept needed to define our specific line. This exercise reinforces the power of coordinate geometry in translating geometric relationships into algebraic equations.

Remember, the slope of the original line ($y=10x+9$) is $m=10$. Since parallel lines have the same slope, our new line also has $m=10$. We then used the given point $(9,-4)$ to solve for the y-intercept ($b$), finding that $b=-94$. Combining these, we arrived at our final equation. This methodical approach, coupled with verification, ensures accuracy and understanding.

Keep practicing these types of problems, as they are foundational to many areas of mathematics and its applications. The more you work with lines, slopes, and intercepts, the more intuitive these concepts will become.

For further exploration into the fascinating world of linear equations and coordinate geometry, I recommend visiting these trusted resources:

• Khan Academy: Explore their extensive Linear Equations and Graphing** section for in-depth lessons and practice.
• Wolfram MathWorld: Delve into detailed mathematical explanations and definitions on their Line** page.