Equation Of A Line Parallel To Y=4x-10 And Passes Through (1,13)

In the fascinating realm of mathematics, parallel lines hold a special significance. These lines, forever side by side, never meeting, possess a unique relationship that translates beautifully into the world of equations. In this comprehensive exploration, we will embark on a journey to understand the concept of parallel lines and delve into the process of finding the equation of a line that mirrors the path of another, while gracefully passing through a specific point. This article will serve as your guiding star, illuminating the steps involved in unraveling this mathematical puzzle. We will dissect the underlying principles, ensuring that you not only grasp the solution but also develop a profound understanding of the geometrical and algebraic concepts at play. Mastering this skill is crucial for anyone venturing further into geometry, calculus, and beyond. So, let's embark on this enriching learning experience together, transforming what might seem like a daunting task into an enjoyable and rewarding endeavor.

Parallel lines, in their essence, are lines that stretch infinitely in the same direction, maintaining a constant distance from each other. This fundamental characteristic is mathematically expressed through their slopes. The slope of a line, often denoted as 'm' in the equation y = mx + b, represents the steepness of the line and its direction. Parallel lines share the same slope, embodying their unwavering alignment. This shared slope is the key that unlocks the mystery of finding the equation of a line parallel to another. Imagine two roads running side by side, never intersecting – that's the visual essence of parallel lines. To truly grasp this concept, it's essential to understand that the y-intercept (the point where the line crosses the y-axis) can differ for parallel lines; they simply march in the same direction. Think of it as two trains running on parallel tracks; they might start at different points, but their paths remain steadfastly aligned. This understanding forms the cornerstone of our exploration into parallel line equations. Grasping the concept of slope as the defining characteristic of parallelism is paramount. It's the compass that guides us through the sea of equations, leading us to the precise line we seek. So, let's keep this key principle at the forefront as we delve deeper into the process of finding these elusive equations.

The equation y = 4x - 10 serves as our starting point, our reference line in this mathematical quest. Dissecting this equation is paramount to understanding how to construct a parallel line. The equation is presented in slope-intercept form, a particularly insightful form that reveals the line's slope and y-intercept at a glance. In this form, 'y' is isolated on one side of the equation, and the other side is expressed as 'mx + b', where 'm' represents the slope and 'b' represents the y-intercept. In our equation, y = 4x - 10, the coefficient of 'x', which is 4, is the slope. This tells us the line's steepness – for every one unit we move along the x-axis, the line rises four units along the y-axis. The constant term, -10, is the y-intercept, indicating that the line crosses the y-axis at the point (0, -10). Understanding the slope is crucial because, as we've established, parallel lines share the same slope. Therefore, any line parallel to y = 4x - 10 will also have a slope of 4. The y-intercept, however, is free to vary, allowing us to create a multitude of parallel lines. This distinction is key – the slope dictates parallelism, while the y-intercept determines the line's specific position on the coordinate plane. So, with the slope firmly in our grasp, we can now proceed to incorporate the additional condition of the line passing through the point (1, 13).

The point (1, 13) acts as our anchor, the specific location through which our parallel line must pass. This point provides a crucial piece of information, allowing us to pinpoint a unique line from the infinite family of lines parallel to y = 4x - 10. Understanding coordinate pairs is fundamental here. The pair (1, 13) represents a specific location on the coordinate plane, where 1 is the x-coordinate (the horizontal position) and 13 is the y-coordinate (the vertical position). Our goal is to find the equation of a line that not only runs parallel to y = 4x - 10 but also gracefully traverses this specific point. To achieve this, we'll employ a technique known as the point-slope form. This form is particularly useful when we know a point on the line and its slope. It allows us to construct the equation of the line directly, without having to first determine the y-intercept. The point (1, 13) acts as our fixed point, the guiding star in our quest to define the equation of the parallel line. It constrains the possibilities, narrowing down the infinite number of parallel lines to the single one that satisfies this specific condition. This is where the beauty of mathematics shines – the ability to use precise information to arrive at a unique and definitive solution. So, with our anchor firmly in place, we are now ready to delve into the point-slope form and construct our desired equation.

The point-slope form serves as the bridge connecting the slope and the specific point to the equation of the line. This form is expressed as y - y1 = m(x - x1), where 'm' is the slope, and (x1, y1) is the given point. It's a powerful tool that allows us to build the equation of a line when we know its slope and a single point it passes through. In our case, we know the slope (m = 4, inherited from the parallel line) and the point (x1 = 1, y1 = 13). Substituting these values into the point-slope form, we get: y - 13 = 4(x - 1). This equation is a direct representation of the line we seek, a line parallel to y = 4x - 10 and passing through (1, 13). However, it's not yet in the familiar slope-intercept form (y = mx + b). To get there, we need to simplify and rearrange the equation. The point-slope form is a testament to the elegance of mathematical expression, encapsulating the relationship between slope, a point, and the line itself. It provides a clear and concise pathway to the equation we desire. Think of it as a blueprint, a step-by-step guide to constructing the line. By substituting our known values, we've laid the foundation for our equation. Now, we simply need to refine it, transforming it into the more commonly used slope-intercept form, which will reveal the line's y-intercept and solidify our understanding of its position on the coordinate plane.

Simplifying the equation y - 13 = 4(x - 1) into slope-intercept form (y = mx + b) is the final step in our quest. This transformation involves distributing the 4 on the right side of the equation and then isolating 'y' on the left side. First, we distribute the 4: y - 13 = 4x - 4. Next, we add 13 to both sides of the equation to isolate 'y': y = 4x - 4 + 13. Finally, we combine the constant terms: y = 4x + 9. This is our final answer – the equation of the line parallel to y = 4x - 10 and passing through the point (1, 13). The equation y = 4x + 9 reveals that the line has a slope of 4 (confirming its parallelism) and a y-intercept of 9 (meaning it crosses the y-axis at the point (0, 9)). This final step is a testament to the power of algebraic manipulation. By systematically applying the rules of algebra, we've transformed the equation from a less familiar form into the elegant and informative slope-intercept form. Think of it as polishing a gem, revealing its brilliance and clarity. The equation y = 4x + 9 is not just a solution; it's a complete description of the line, encapsulating its direction (slope) and position (y-intercept) on the coordinate plane. This marks the culmination of our journey, a satisfying conclusion to our mathematical exploration.

In conclusion, finding the equation of a line parallel to y = 4x - 10 and passing through the point (1, 13) has been a journey through the fundamental concepts of parallel lines, slopes, and equation forms. We started by understanding the essence of parallel lines – their shared slope. We then dissected the given equation, y = 4x - 10, identifying its slope as 4. The point (1, 13) served as our anchor, the specific location our parallel line must traverse. The point-slope form, y - y1 = m(x - x1), became our bridge, connecting the slope and the point to the equation of the line. Finally, we simplified the equation into slope-intercept form, y = 4x + 9, revealing the complete picture of our parallel line. This exercise underscores the power of mathematical reasoning and the elegance of algebraic manipulation. Mastering these skills opens doors to a deeper understanding of geometry, calculus, and countless other mathematical disciplines. It's not just about finding the answer; it's about understanding the process, the why behind the how. This is the true essence of mathematical learning, and it's a skill that will serve you well in any endeavor. So, embrace the challenge, delve into the details, and let the beauty of mathematics illuminate your path.