Finding The Equation Of A Parallel Line A Step-by-Step Guide
Understanding Parallel Lines and Their Equations
In the realm of mathematics, particularly in coordinate geometry, the concept of parallel lines is fundamental. Parallel lines, by definition, are lines that lie in the same plane and never intersect. A crucial property of parallel lines is that they have the same slope. Understanding this property is key to finding the equation of a line that is parallel to a given line and passes through a specific point.
The equation of a line is commonly expressed in the slope-intercept form, which is written as y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis). The slope, m, determines the steepness and direction of the line. A positive slope indicates that the line rises as you move from left to right, while a negative slope indicates that the line falls as you move from left to right. The magnitude of the slope represents the steepness of the line; a larger magnitude indicates a steeper line.
When two lines are parallel, their slopes are equal. This means that if we have a line with a slope of m, any line parallel to it will also have a slope of m. This is the cornerstone of solving problems involving parallel lines. The y-intercept, b, on the other hand, can be different for parallel lines, as it simply determines the vertical position of the line on the coordinate plane. Therefore, to find the equation of a line parallel to a given line, we first identify the slope of the given line and then use that slope for the new line. The only remaining task is to determine the y-intercept of the new line, which can be done using the given point that the line passes through.
Let's illustrate this with an example. Suppose we have a line with the equation y = 2x + 3. The slope of this line is 2. Any line parallel to this line will also have a slope of 2. The equation of a parallel line would then be in the form y = 2x + b, where b is the y-intercept that we need to find. If we are given a point that the parallel line passes through, such as (1, 5), we can substitute these coordinates into the equation to solve for b. In this case, we would have 5 = 2(1) + b, which simplifies to 5 = 2 + b. Solving for b, we get b = 3. Therefore, the equation of the line parallel to y = 2x + 3 and passing through the point (1, 5) is y = 2x + 3. This example demonstrates the fundamental process of finding the equation of a parallel line, which involves identifying the slope and then using a given point to determine the y-intercept.
Problem Statement: Finding the Parallel Line
Our goal is to determine the equation of a line that satisfies two specific conditions. First, it must be parallel to a given line. Second, it must pass through a particular point, which in this case is (12, -2). The challenge lies in using these two pieces of information to construct the equation of the desired line. We will leverage the fact that parallel lines share the same slope and employ the point-slope form of a linear equation to arrive at the solution.
The given options for the equation of the line are:
- A. y = -6/5 x + 10
- B. y = -6/5 x + 12
- C. y = -5/6 x - 10
- D. y = 5/6 x - 12
To solve this problem, we need to first identify the slope of the given line. Since the original question does not explicitly provide the equation of the given line, we must assume that one of the options presented is the line to which we want to find a parallel line. We will analyze each option and determine its slope. Remember, the slope of a line in slope-intercept form (y = mx + b) is represented by the coefficient m.
For option A, y = -6/5 x + 10, the slope is -6/5. This means any line parallel to this one will also have a slope of -6/5. Option B, y = -6/5 x + 12, also has a slope of -6/5, so it is parallel to option A. Option C, y = -5/6 x - 10, has a slope of -5/6, which is different from -6/5. Therefore, option C is not parallel to options A and B. Option D, y = 5/6 x - 12, has a slope of 5/6, which is also different from -6/5. Hence, option D is not parallel to options A and B either.
Since options A and B have the same slope, they are parallel to each other. To determine which of these two options is the correct answer, we need to check which one passes through the point (12, -2). We will substitute the coordinates of this point into the equations of options A and B. If the equation holds true after the substitution, then the line passes through the point. This process will help us narrow down the correct answer from the given options and identify the equation of the line that is parallel to the assumed given line and passes through the point (12, -2).
Step-by-Step Solution: Identifying the Correct Equation
To find the equation of the line parallel to the given line and passing through the point (12, -2), we will proceed step-by-step, building upon our understanding of slopes and parallel lines. First, let's assume option A, y = -6/5 x + 10, is the given line. This line has a slope of -6/5. Any line parallel to this line will also have a slope of -6/5. Now, we need to find the line with this slope that passes through the point (12, -2).
We can use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point. In this case, m = -6/5, x1 = 12, and y1 = -2. Substituting these values into the point-slope form, we get:
y - (-2) = -6/5 (x - 12)
Simplifying this equation:
y + 2 = -6/5 x + 72/5
Now, we need to convert this equation to slope-intercept form (y = mx + b). To do this, we subtract 2 from both sides of the equation:
y = -6/5 x + 72/5 - 2
To subtract 2, we need to express it as a fraction with a denominator of 5:
y = -6/5 x + 72/5 - 10/5
y = -6/5 x + 62/5
The y-intercept in this case is 62/5, which is equal to 12.4. This equation, y = -6/5 x + 62/5, represents the line parallel to y = -6/5 x + 10 and passing through the point (12, -2). However, this equation is not among the options provided. This indicates that option A might not be the correct "given line".
Let's now consider option B, y = -6/5 x + 12. This line also has a slope of -6/5, so it is parallel to option A. We will repeat the process using the point-slope form with the same slope, m = -6/5, and the point (12, -2):
y - (-2) = -6/5 (x - 12)
This is the same point-slope equation as before, which simplifies to:
y = -6/5 x + 62/5
Again, this result does not match option B. However, the fact that we obtained the same equation suggests that the error might be in the simplification process or that the correct answer is not explicitly listed among the options. Let's re-examine the options and our calculations to ensure accuracy. The key is to carefully substitute the given point into the potential equations and verify if the equation holds true.
Verification and Final Answer
To determine the correct answer, we need to check which of the given options satisfies two conditions: being parallel to the assumed given line (having the same slope) and passing through the point (12, -2). We've already established that options A and B have the same slope, -6/5, making them parallel. Let's substitute the point (12, -2) into options A and B to see which one holds true.
For option A, y = -6/5 x + 10:
-2 = -6/5 (12) + 10
-2 = -72/5 + 10
-2 = -72/5 + 50/5
-2 = -22/5
This is not true, so option A does not pass through the point (12, -2).
For option B, y = -6/5 x + 12:
-2 = -6/5 (12) + 12
-2 = -72/5 + 12
-2 = -72/5 + 60/5
-2 = -12/5
This is also not true, so option B does not pass through the point (12, -2).
It seems there might be an error in the provided options or in our initial assumption that the parallel line should be parallel to one of the given options. Let's revisit our calculations and the problem statement to ensure we haven't overlooked anything. We correctly identified the slopes and used the point-slope form, but the point (12, -2) does not satisfy either option A or B. This discrepancy suggests that we need to find the correct equation independently and then compare it to the options.
Let's use the point-slope form again, assuming the slope is -6/5 (from options A and B) and the point is (12, -2):
y - (-2) = -6/5 (x - 12)
y + 2 = -6/5 x + 72/5
y = -6/5 x + 72/5 - 2
y = -6/5 x + 72/5 - 10/5
y = -6/5 x + 62/5
This equation, y = -6/5 x + 62/5, is the correct equation of the line parallel to the lines with a slope of -6/5 and passing through the point (12, -2). Since none of the given options match this equation, there might be an error in the options provided.
Based on our calculations and verification, none of the given options is the correct equation of the line parallel to the assumed given line (with a slope of -6/5) and passing through the point (12, -2). The correct equation should be y = -6/5 x + 62/5.
