Finding The Equation Of A Line Parallel To 2x + Y - 10 = 0

Finding the equation of a line that satisfies specific conditions is a fundamental problem in coordinate geometry. This article delves into a common scenario: determining the equation of a line that passes through a given point and is parallel to another given line. We will explore the underlying principles, the step-by-step process, and illustrate the solution with a detailed example. Understanding these concepts is crucial for various mathematical applications, including linear algebra, calculus, and physics.

Understanding Parallel Lines and Their Equations

To effectively find the equation of a line parallel to another, we must first grasp the concept of parallel lines and how their equations relate. Parallel lines are lines in the same plane that never intersect. A crucial property of parallel lines is that they have the same slope. The slope of a line, often denoted by m, represents its steepness and direction. It is defined as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line.

The equation of a line can be expressed in several forms, but the most relevant for this problem are the slope-intercept form and the point-slope form. The slope-intercept form is given by y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). The point-slope form is given by y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. Understanding these forms is essential for manipulating and finding the equation of a line.

The given line in the problem is in the general form, 2x + y - 10 = 0. To find the slope of this line, we need to rewrite it in the slope-intercept form. This involves isolating y on one side of the equation. Subtracting 2x and adding 10 to both sides, we get y = -2x + 10. From this equation, we can clearly see that the slope of the given line is -2. Since parallel lines have the same slope, the line we are trying to find will also have a slope of -2. This is a critical piece of information that we will use in the subsequent steps. Recognizing and utilizing the relationship between the slopes of parallel lines is a fundamental aspect of solving this type of problem. The ability to convert between different forms of linear equations is also crucial, as it allows us to extract the necessary information, such as the slope, and apply it to find the equation of a parallel line.

Step-by-Step Process to Find the Equation

To find the equation of the line that passes through the point (5, -7) and is parallel to the line 2x + y - 10 = 0, we will follow a step-by-step process. This systematic approach will ensure that we arrive at the correct solution efficiently. This process involves finding the slope of the given line, using this slope for the parallel line, and then applying the point-slope form to determine the equation of the new line. Each step is crucial and builds upon the previous one, leading us to the final equation.

Step 1: Find the slope of the given line. As discussed earlier, we rewrite the given equation 2x + y - 10 = 0 in slope-intercept form, which is y = -2x + 10. From this form, it is evident that the slope of the given line is m = -2. This slope is the foundation for finding the equation of the parallel line. Identifying the slope correctly is essential, as it dictates the steepness and direction of the line we are trying to find.

Step 2: Use the same slope for the parallel line. Since parallel lines have the same slope, the slope of the line we are looking for is also m = -2. This is a direct application of the property of parallel lines. Knowing the slope allows us to use either the slope-intercept form or the point-slope form to find the equation. In this case, the point-slope form is particularly useful since we have a point that the line passes through.

Step 3: Use the point-slope form to find the equation. The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. We are given the point (5, -7) and have determined the slope m = -2. Substituting these values into the point-slope form, we get: y - (-7) = -2(x - 5). This equation represents the line we are looking for, but it is often simplified to a more standard form.

Step 4: Simplify the equation to slope-intercept or general form. To simplify the equation, we first distribute the -2 on the right side: y + 7 = -2x + 10. Next, we isolate y by subtracting 7 from both sides: y = -2x + 3. This is the slope-intercept form of the equation. We can also rewrite it in the general form by adding 2x to both sides and subtracting 3 from both sides, resulting in 2x + y - 3 = 0. Both forms are valid representations of the same line. The choice of which form to use often depends on the specific context or requirements of the problem. This step-by-step process provides a clear and organized method for finding the equation of a line parallel to a given line, ensuring accuracy and understanding.

Detailed Example: Finding the Equation

To solidify the understanding of the process, let's work through the given example in detail. We aim to find the equation of the line that passes through the point (5, -7) and is parallel to the line 2x + y - 10 = 0. This example will demonstrate the application of the steps outlined in the previous section and provide a concrete illustration of the solution.

Step 1: Find the slope of the given line.

The given line is 2x + y - 10 = 0. To find its slope, we rewrite it in slope-intercept form (y = mx + b). Subtracting 2x and adding 10 to both sides, we get:

y = -2x + 10

From this equation, we can clearly see that the slope of the given line is m = -2. This slope is the foundation for finding the equation of the parallel line. Identifying the slope correctly is essential, as it dictates the steepness and direction of the line we are trying to find. The ability to manipulate equations and convert them into different forms is a fundamental skill in algebra, and it is crucial for solving problems like this one.

Step 2: Use the same slope for the parallel line.

Since parallel lines have the same slope, the line we are looking for also has a slope of m = -2. This is a direct application of the property of parallel lines. Knowing the slope allows us to use either the slope-intercept form or the point-slope form to find the equation. In this case, the point-slope form is particularly useful since we have a point that the line passes through. The concept of parallel lines sharing the same slope is a cornerstone of coordinate geometry and is essential for understanding the relationships between lines.

Step 3: Use the point-slope form to find the equation.

The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. We are given the point (5, -7) and have determined the slope m = -2. Substituting these values into the point-slope form, we get:

y - (-7) = -2(x - 5)

This equation represents the line we are looking for, but it is often simplified to a more standard form. The point-slope form is a powerful tool for finding the equation of a line when a point and the slope are known. It provides a direct way to construct the equation without having to solve for the y-intercept first.

Step 4: Simplify the equation to slope-intercept or general form.

To simplify the equation, we first distribute the -2 on the right side:

y + 7 = -2x + 10

Next, we isolate y by subtracting 7 from both sides:

y = -2x + 3

This is the slope-intercept form of the equation. We can also rewrite it in the general form by adding 2x to both sides and subtracting 3 from both sides, resulting in:

2x + y - 3 = 0

Both forms are valid representations of the same line. The choice of which form to use often depends on the specific context or requirements of the problem. Simplifying the equation is a crucial step in presenting the solution in a clear and concise manner. It also allows for easier comparison and manipulation of the equation in future calculations or applications. Through this detailed example, we have demonstrated the process of finding the equation of a line parallel to a given line, providing a comprehensive understanding of the steps involved and the underlying principles.

Alternative Methods and Considerations

While the step-by-step method outlined above is effective, there are alternative approaches to finding the equation of a line parallel to a given line. Additionally, there are certain considerations that can further refine our understanding and problem-solving skills. Exploring these alternative methods and considerations can enhance our flexibility and efficiency in tackling similar problems.

One alternative method involves directly using the slope-intercept form (y = mx + b) after finding the slope. We know the slope of the parallel line is m = -2, so the equation will be of the form y = -2x + b. To find the y-intercept (b), we substitute the coordinates of the given point (5, -7) into the equation:

-7 = -2(5) + b

-7 = -10 + b

b = 3

Thus, the equation of the line is y = -2x + 3, which is consistent with our previous result. This method offers a more direct approach to finding the y-intercept and can be useful when the slope-intercept form is preferred.

Another consideration is the possibility of the given line being in a special form, such as a horizontal or vertical line. A horizontal line has the equation y = c, where c is a constant, and its slope is 0. A line parallel to a horizontal line will also be horizontal and have the same y-value if it passes through a given point. A vertical line has the equation x = c, where c is a constant, and its slope is undefined. A line parallel to a vertical line will also be vertical and have the same x-value if it passes through a given point. Recognizing these special cases can simplify the process of finding the equation of a parallel line.

Furthermore, it is important to verify the solution by ensuring that the resulting line indeed passes through the given point and is parallel to the original line. This can be done by substituting the coordinates of the point into the equation and checking if the equation holds true. Additionally, we can compare the slopes of the two lines to confirm that they are the same. Verification is a crucial step in problem-solving, as it helps to identify and correct any errors.

In summary, while the step-by-step method is a reliable approach, alternative methods and considerations can provide additional insights and efficiencies. Understanding these nuances allows for a more comprehensive and adaptable approach to finding the equation of a line parallel to a given line. This flexibility is essential for tackling a wide range of problems in coordinate geometry and related fields.

Conclusion

In conclusion, finding the equation of a line that passes through a given point and is parallel to another line is a fundamental problem in coordinate geometry. This article has provided a comprehensive guide to solving this type of problem, covering the underlying principles, the step-by-step process, a detailed example, and alternative methods and considerations. By understanding the relationship between parallel lines and their slopes, and by applying the point-slope form or slope-intercept form of a linear equation, we can effectively determine the equation of the desired line.

The step-by-step process involves first finding the slope of the given line, then using the same slope for the parallel line, and finally applying the point-slope form to find the equation. Simplifying the equation to slope-intercept or general form provides a clear and concise representation of the solution. The detailed example illustrates the application of these steps, providing a concrete understanding of the process.

Alternative methods, such as directly using the slope-intercept form, offer additional approaches to solving the problem. Considerations such as special cases involving horizontal and vertical lines further enhance our problem-solving skills. Verification of the solution is crucial to ensure accuracy and to identify any potential errors.

The ability to find the equation of a line parallel to a given line is a valuable skill in various mathematical applications, including linear algebra, calculus, and physics. It reinforces our understanding of linear equations, slopes, and intercepts, and it provides a foundation for more advanced concepts. By mastering this skill, we can confidently tackle a wide range of problems in coordinate geometry and related fields. This article serves as a valuable resource for students, educators, and anyone interested in deepening their understanding of linear equations and parallel lines. The principles and techniques discussed here can be applied to various scenarios, making it a versatile and essential tool in mathematical problem-solving.