Equation Of A Parallel Line In Slope-Intercept Form
In the realm of coordinate geometry, understanding the relationships between lines is fundamental. Among these relationships, parallelism stands out as a key concept. Parallel lines, by definition, never intersect, and this characteristic is intrinsically linked to their slopes. To truly grasp the equation of a parallel line, especially when expressed in slope-intercept form, we must delve into the core principles of linear equations and their graphical representations. This article serves as a comprehensive guide, walking you through the process of determining the equation of a line that is parallel to a given line and passes through a specific point.
The slope-intercept form, a cornerstone of linear equations, provides a clear and concise way to represent a line. It is expressed as y = mx + b, where 'm' denotes the slope of the line and 'b' represents the y-intercept, the point where the line intersects the y-axis. The slope, often referred to as the 'rise over run,' quantifies the steepness and direction of the line. A positive slope indicates an upward slant from left to right, while a negative slope signifies a downward slant. The y-intercept, on the other hand, pinpoints the location where the line crosses the vertical axis. Understanding these components is crucial for manipulating and interpreting linear equations.
Parallel lines share a unique characteristic: they possess the same slope. This shared slope is the cornerstone of their non-intersecting nature. If two lines have identical slopes, they will maintain a consistent distance from each other, extending infinitely without ever meeting. This principle is paramount when determining the equation of a line parallel to a given one. Once you've identified the slope of the given line, you've essentially unlocked the slope of any line parallel to it. This understanding forms the bedrock of our problem-solving approach.
Problem Statement: A Step-by-Step Solution
Let's tackle the given problem head-on. We are presented with a line whose equation is 10x + 2y = -2. Our mission is to find the equation of a line that is parallel to this one and gracefully passes through the point (0, 12). To achieve this, we'll embark on a step-by-step journey, leveraging the power of the slope-intercept form and the principle of parallel slopes.
Step 1: Unveiling the Slope
Our first step is to unveil the slope of the given line. Currently, the equation 10x + 2y = -2 is not in the coveted slope-intercept form. To transform it, we need to isolate 'y' on one side of the equation. Let's embark on this algebraic transformation:
- Subtract 10x from both sides: This maneuver leaves us with 2y = -10x - 2.
- Divide both sides by 2: This crucial division isolates 'y', giving us y = -5x - 1.
Behold! We've successfully transformed the equation into slope-intercept form. By direct observation, we can now identify the slope of the given line as -5. This value, the coefficient of 'x', is the key to unlocking the slope of any parallel line.
Step 2: Embracing the Parallel Slope
The principle of parallel lines dictates that our target line, the one we seek to define, must possess the same slope as the given line. Therefore, the slope of our parallel line is also -5. We now have a crucial piece of the puzzle, the 'm' in our y = mx + b equation.
Step 3: The Y-Intercept Revelation
We're not just seeking any line with a slope of -5; we need the one that gracefully passes through the point (0, 12). This point holds the key to determining the y-intercept, 'b', in our equation. Recall that the y-intercept is the point where the line intersects the y-axis. The point (0, 12), with its x-coordinate of 0, lies precisely on the y-axis. Therefore, the y-coordinate, 12, directly reveals the y-intercept.
Step 4: Crafting the Equation
With the slope (m = -5) and the y-intercept (b = 12) in our grasp, we can now construct the equation of the parallel line in its full glory. Substituting these values into the slope-intercept form y = mx + b, we arrive at:
y = -5x + 12
This equation, a testament to our step-by-step approach, represents the line that is parallel to 10x + 2y = -2 and elegantly passes through the point (0, 12).
Verifying Our Solution: A Double-Check
In the pursuit of accuracy, it's always prudent to verify our solution. Let's ensure that our derived equation, y = -5x + 12, indeed satisfies the given conditions.
Parallelism Check
We've already established that the slope of our derived line is -5, which matches the slope of the given line. This confirms the parallelism criterion.
Point Verification
To verify that our line passes through (0, 12), we substitute these coordinates into our equation:
12 = -5(0) + 12
Simplifying, we get 12 = 12, a resounding confirmation that the point lies on our line.
Conclusion: Mastering Parallel Lines
In this comprehensive exploration, we've successfully navigated the realm of parallel lines and slope-intercept form. We've meticulously determined the equation of a line parallel to a given one, ensuring it gracefully passes through a specified point. The key takeaways from this journey are:
- Parallel lines share the same slope, a fundamental principle.
- The slope-intercept form (y = mx + b) provides a powerful framework for representing and manipulating linear equations.
- The y-intercept is directly revealed when the x-coordinate is 0.
- Verification is crucial to ensure the accuracy of our solutions.
By mastering these concepts, you'll be well-equipped to tackle a wide range of problems involving parallel lines, linear equations, and coordinate geometry. The ability to navigate these concepts is not just a mathematical skill; it's a powerful tool for understanding and interpreting the world around us, where linear relationships abound.
This problem serves as a microcosm of the broader applications of linear equations in various fields, from physics and engineering to economics and computer science. The ability to find equations of parallel lines is a stepping stone to more complex concepts, such as systems of equations, transformations, and optimization problems. By solidifying your understanding of these fundamental principles, you're laying a strong foundation for future mathematical endeavors.
So, embrace the power of the slope-intercept form, remember the principle of parallel slopes, and continue your exploration of the fascinating world of linear equations. The journey of mathematical discovery is a rewarding one, filled with challenges and triumphs, and each step forward brings you closer to a deeper understanding of the universe.
Original Question
A given line has the equation $10x + 2y = -2$. What is the equation, in slope-intercept form, of the line that is parallel to the given line and passes through the point $(0, 12)$?
Rewritten Question
Determine the slope-intercept form equation of a line that is parallel to the line defined by $10x + 2y = -2$ and intersects the point $(0, 12)$.
