Equation Of A Parallel Line Passing Through A Point
Finding the equation of a line that is parallel to a given line and passes through a specific point is a fundamental concept in linear algebra and coordinate geometry. This article provides a step-by-step guide to solving this type of problem, focusing on the given example: finding the equation of the line parallel to y = -7x + 2 and containing the point (-5, 32). By understanding the underlying principles and applying the appropriate techniques, you'll be well-equipped to tackle similar challenges.
Understanding Parallel Lines and Their Equations
When dealing with parallel lines, the most important concept to grasp is that they have the same slope. The slope of a line determines its steepness and direction. In the slope-intercept form of a linear equation, y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept, the coefficient of 'x' is the slope. Therefore, if two lines are parallel, their 'm' values will be equal. Grasping this concept is the cornerstone to solving problems related to parallel lines.
In our given problem, the line y = -7x + 2 is in slope-intercept form. By directly observing the equation, we can identify that the slope of this line is -7. Any line parallel to this line will also have a slope of -7. This is a crucial piece of information that we will use to determine the equation of the parallel line we are seeking. The y-intercept, represented by 'b' in the slope-intercept form, can be different for parallel lines, as it simply shifts the line up or down on the coordinate plane without changing its inclination. The fact that the y-intercept can vary while the slope remains constant is what allows for an infinite number of lines to be parallel to each other. Understanding this distinction between slope and y-intercept is essential for accurately determining the equation of a parallel line that satisfies specific conditions.
Step-by-Step Solution: Finding the Equation
To find the equation of the line parallel to y = -7x + 2 and containing the point (-5, 32), we will follow these steps:
Step 1: Identify the Slope of the Given Line
The given line is in the slope-intercept form, y = mx + b, where 'm' is the slope and 'b' is the y-intercept. Comparing y = -7x + 2 with the general form, we can clearly see that the slope, 'm', is -7. As established earlier, parallel lines have the same slope, so the line we are trying to find will also have a slope of -7.
Step 2: Use the Point-Slope Form
The point-slope form of a linear equation is a powerful tool for finding the equation of a line when you know its slope and a point it passes through. The point-slope form is given by: **y - y₁ = m(x - x₁) **, where 'm' is the slope and **(x₁, y₁) ** is the given point. In this case, we know the slope m = -7 and the point (-5, 32), so x₁ = -5 and y₁ = 32. Substituting these values into the point-slope form, we get: y - 32 = -7(x - (-5)). This equation represents the line we are looking for, but it is not yet in slope-intercept form. The point-slope form is a versatile tool because it directly incorporates the given information – the slope and a point – making it a convenient starting point for determining the line's equation.
Step 3: Simplify and Convert to Slope-Intercept Form
To convert the equation from point-slope form to slope-intercept form (y = mx + b), we need to simplify and isolate 'y'. First, distribute the -7 on the right side of the equation: y - 32 = -7(x + 5) becomes y - 32 = -7x - 35. Next, add 32 to both sides of the equation to isolate 'y': y - 32 + 32 = -7x - 35 + 32. This simplifies to y = -7x - 3. Now the equation is in slope-intercept form, and we can clearly see the slope (-7) and the y-intercept (-3). This final form is useful for graphing the line and for comparing it to other linear equations.
Step 4: Verification (Optional)
To verify that the equation y = -7x - 3 is correct, we can substitute the given point (-5, 32) into the equation. If the equation holds true, then the point lies on the line. Substituting x = -5 into the equation, we get: y = -7(-5) - 3 = 35 - 3 = 32. Since the calculated y-value matches the given y-value, we can confidently say that the point (-5, 32) lies on the line y = -7x - 3. This verification step provides an additional layer of assurance that our solution is accurate.
Final Answer
Therefore, the equation of the line that is parallel to y = -7x + 2 and contains the point (-5, 32) is y = -7x - 3. This line has the same slope as the given line but a different y-intercept, ensuring that it is parallel and passes through the specified point.
Key Concepts and Takeaways
- Parallel Lines: Parallel lines have the same slope. This is the most critical concept for solving these types of problems.
- Slope-Intercept Form: y = mx + b, where 'm' is the slope and 'b' is the y-intercept.
- Point-Slope Form: **y - y₁ = m(x - x₁) **, where 'm' is the slope and **(x₁, y₁) ** is a point on the line.
- Step-by-Step Approach: Identifying the slope, using the point-slope form, simplifying to slope-intercept form, and verifying the solution are crucial steps.
By mastering these concepts and following the step-by-step approach, you can confidently solve problems involving parallel lines and their equations. Remember that understanding the fundamental principles of linear equations and coordinate geometry is key to success in mathematics.
Additional Practice Problems
To further solidify your understanding, try solving these additional practice problems:
- Find the equation of the line parallel to y = 2x - 5 and passing through the point (1, -3).
- Determine the equation of the line parallel to y = -x + 8 and containing the point (4, 0).
- What is the equation of the line parallel to y = (1/2)x + 1 and going through the point (-2, 5)?
Working through these problems will help you internalize the process and develop your problem-solving skills. Remember to focus on identifying the slope, applying the point-slope form, and simplifying to slope-intercept form. The more you practice, the more comfortable and confident you will become in solving these types of problems.
Conclusion
Finding the equation of a line parallel to a given line and passing through a specific point is a valuable skill in mathematics. By understanding the relationship between parallel lines and their slopes, and by applying the point-slope form, you can confidently solve these problems. This article has provided a comprehensive guide, including a step-by-step solution and key concepts. With practice and a solid understanding of the underlying principles, you can master this fundamental concept in linear algebra and coordinate geometry.
