Finding Parallel Line Equation In Slope-Intercept Form

In this article, we'll explore how to determine the equation of a line that is parallel to a given line and passes through a specific point. We'll focus on expressing the equation in slope-intercept form, which is a fundamental concept in linear algebra. The given equation is $10x + 2y = -2$, and the point is $(0, 12)$. Our goal is to find the equation of the line that satisfies these conditions. This problem is a classic example of applying the principles of linear equations and coordinate geometry, which are essential in various fields such as physics, engineering, and computer graphics. Understanding how to solve such problems is crucial for anyone studying mathematics or related disciplines.

Understanding Slope-Intercept Form and Parallel Lines

To begin, it's crucial to understand the slope-intercept form of a linear equation, which is expressed as $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the y-intercept. The slope indicates the steepness and direction of the line, while the y-intercept is the point where the line crosses the y-axis. Parallel lines, a key concept in this problem, are lines that never intersect. A fundamental property of parallel lines is that they have the same slope. This means that if we find the slope of the given line, we automatically know the slope of any line parallel to it. To solve our problem effectively, we must first convert the given equation into slope-intercept form. This involves isolating $y$ on one side of the equation. Once we have the equation in slope-intercept form, we can easily identify the slope. Then, using the point-slope form, we can find the equation of the parallel line passing through the given point. This step-by-step approach ensures a clear and logical solution. Understanding these principles is not only essential for solving this particular problem but also for tackling a wide range of linear equation problems in mathematics and real-world applications. For instance, in physics, understanding the slope-intercept form can help in analyzing motion graphs, where the slope represents velocity and the y-intercept represents initial position. In engineering, this concept is used in designing structures and systems where linear relationships are prevalent. Therefore, mastering this concept is a significant step in mathematical proficiency.

Converting the Given Equation to Slope-Intercept Form

To effectively solve this problem, our first critical step is to convert the given equation, $10x + 2y = -2$, into the slope-intercept form, which is $y = mx + b$. This form allows us to easily identify the slope and y-intercept of the line. The process involves isolating $y$ on one side of the equation. We begin by subtracting $10x$ from both sides of the equation, which gives us $2y = -10x - 2$. Next, to completely isolate $y$, we divide both sides of the equation by 2. This results in $y = -5x - 1$. Now, the equation is in slope-intercept form. From this, we can clearly see that the slope, $m$, of the given line is $-5$, and the y-intercept, $b$, is $-1$. This conversion is a fundamental algebraic technique that is widely used in various mathematical contexts. It not only helps in visualizing the line on a coordinate plane but also in comparing different linear equations. Identifying the slope and y-intercept is crucial for understanding the behavior of the line, such as its steepness and where it crosses the y-axis. In practical applications, this conversion can be used to model linear relationships in various scenarios, such as cost analysis, where the slope represents the variable cost per unit and the y-intercept represents the fixed cost. Therefore, mastering the technique of converting equations to slope-intercept form is an invaluable skill in both mathematics and real-world problem-solving.

Determining the Slope of the Parallel Line

Knowing the slope of the given line is a critical piece of information, as it directly helps us find the slope of any line parallel to it. As we established earlier, parallel lines have the same slope. From the slope-intercept form of the given equation, $y = -5x - 1$, we identified the slope as $-5$. Therefore, any line parallel to the given line will also have a slope of $-5$. This principle is a cornerstone of coordinate geometry and is essential for solving a variety of problems related to parallel and perpendicular lines. Understanding this concept allows us to quickly determine the slope of a parallel line without needing to perform any additional calculations. This is particularly useful in scenarios where we need to find the equation of a line that satisfies specific geometric conditions, such as being parallel to a given line and passing through a particular point. In practical applications, this principle is used in fields such as architecture and engineering, where ensuring parallel lines is crucial for structural integrity and design aesthetics. For example, in building construction, parallel lines are essential for the alignment of walls and floors. In electrical engineering, parallel lines are used in the design of circuit boards to ensure efficient current flow. Thus, the concept of parallel lines having the same slope is not only a fundamental mathematical principle but also a practical tool in various real-world applications.

Using the Point-Slope Form to Find the Equation

With the slope of the parallel line determined to be $-5$, and given the point $(0, 12)$ through which the line must pass, we can now use the point-slope form to find the equation of the line. The point-slope form of a linear equation is given by $y - y_1 = m(x - x_1)$, where $m$ is the slope, and $(x_1, y_1)$ is a point on the line. In this case, $m = -5$ and $(x_1, y_1) = (0, 12)$. Substituting these values into the point-slope form, we get $y - 12 = -5(x - 0)$. Simplifying this equation, we have $y - 12 = -5x$. To convert this equation into slope-intercept form, we add 12 to both sides, which gives us $y = -5x + 12$. This is the equation of the line that is parallel to the given line and passes through the point $(0, 12)$. The point-slope form is a powerful tool for finding the equation of a line when we know a point on the line and its slope. It is particularly useful in situations where we don't have the y-intercept directly. The point-slope form provides a direct way to construct the equation, which can then be easily converted to other forms, such as slope-intercept or standard form. In various applications, such as navigation and mapping, the point-slope form is used to determine the path of an object or vehicle given its direction (slope) and a known location (point). Understanding and applying the point-slope form is therefore an essential skill in both mathematical and real-world problem-solving.

Final Equation in Slope-Intercept Form

After carefully applying the point-slope form and simplifying the equation, we have arrived at the final equation of the line in slope-intercept form: $y = -5x + 12$. This equation represents a line that is parallel to the given line, $10x + 2y = -2$, and passes through the point $(0, 12)$. The equation clearly shows that the slope of the line is $-5$, which is the same as the slope of the given line, confirming that the lines are indeed parallel. The y-intercept is $12$, which is the y-coordinate of the point through which the line passes. This final result demonstrates the power of combining the concepts of slope-intercept form, parallel lines, and the point-slope form to solve linear equation problems. The process we followed involved several key steps: converting the given equation to slope-intercept form, identifying the slope, using the point-slope form to construct the equation of the parallel line, and simplifying the equation to slope-intercept form. This systematic approach is crucial for solving similar problems accurately and efficiently. Understanding the relationship between different forms of linear equations and how to convert between them is a fundamental skill in algebra and has wide-ranging applications in various fields. For example, in economics, linear equations are used to model supply and demand curves, and understanding their slopes and intercepts is essential for market analysis. In statistics, linear regression models are used to analyze the relationship between variables, and the slope and intercept provide valuable insights into the nature of the relationship. Therefore, mastering these concepts is essential for success in both academic and professional settings.

Conclusion

In conclusion, we successfully found the equation of the line parallel to $10x + 2y = -2$ and passing through the point $(0, 12)$. The final equation in slope-intercept form is $y = -5x + 12$. This problem highlights the importance of understanding key concepts such as slope-intercept form, parallel lines, and the point-slope form. The systematic approach we used, involving converting to slope-intercept form, identifying the slope, applying the point-slope form, and simplifying, is a valuable problem-solving strategy that can be applied to a variety of linear equation problems. Mastering these concepts and techniques is crucial for success in algebra and related fields. The ability to manipulate linear equations and understand their properties is not only essential for academic pursuits but also for real-world applications in various fields such as engineering, physics, economics, and computer science. For example, in computer graphics, linear equations are used to model lines and planes, which are fundamental elements in creating 3D images and animations. In physics, linear equations are used to describe the motion of objects and the relationships between forces and acceleration. In economics, linear equations are used to model economic relationships and make predictions about market behavior. Therefore, a strong foundation in linear equations is a valuable asset for anyone pursuing a career in these fields. The skills and knowledge gained from solving problems like this one will serve as a solid foundation for more advanced mathematical concepts and real-world applications.