Finding Equations Of Parallel Lines A Step-by-Step Guide

In mathematics, parallel lines are lines in a plane that never intersect. A fundamental property of parallel lines is that they have the same slope. This concept is crucial when dealing with linear equations and coordinate geometry. When we talk about finding the equation of a line parallel to another, we're essentially looking for a line that shares the same steepness or inclination but has a different y-intercept, which determines where the line crosses the y-axis. The equation of a line is commonly represented in slope-intercept form, which is ${ y = mx + b }$, where ${ m }$ represents the slope and ${ b }$ represents the y-intercept. The slope, ${ m }$, is the measure of how much the line rises or falls for each unit of horizontal change, and it is the key characteristic we focus on when dealing with parallel lines. For two lines to be parallel, their slopes must be identical. If we're given a line and asked to find the equation of another line that's parallel to it, the first thing we identify is the slope of the given line. Then, we use this slope for our new line. The challenge then becomes finding the correct y-intercept for the new line, which is determined by a specific point that the line must pass through. This point provides the necessary information to solve for the y-intercept, ensuring that the new line not only has the correct slope but also passes through the designated location on the coordinate plane. Understanding this principle allows us to accurately determine the equation of any line parallel to a given one, making it a fundamental skill in algebra and geometry. The slope-intercept form, ${ y = mx + b }$, not only simplifies the identification of parallel lines but also facilitates the construction of new equations based on known slopes and points. This form allows for a straightforward substitution of values to find the missing y-intercept, making it an indispensable tool in solving problems related to parallel lines.

Problem Statement: Finding a Parallel Line

In this particular problem, we're tasked with finding the equation of a line that is parallel to a given line, which is represented by the equation ${ y = \frac{3}{4}x + 7 }$. Additionally, we have the condition that the line we're looking for must pass through the point (-12, 36). This problem combines the concept of parallel lines with the practical application of finding a specific linear equation. To solve this, we must first recognize that the slope of the line we're trying to find will be the same as the slope of the given line. The given line, ${ y = \frac{3}{4}x + 7 }$, is already in slope-intercept form, which makes it easy to identify the slope. By comparing the given equation to the general form ${ y = mx + b }$, we can see that the slope, ${ m }$, is ${ \frac{3}{4} }$. This means that the line parallel to it will also have a slope of ${ \frac{3}{4} }$. The next step is to use the point-slope form or the slope-intercept form to find the equation of the new line. Since we know the slope and a point that the line passes through, we can substitute these values into the slope-intercept form, ${ y = mx + b }$, and solve for the y-intercept, ${ b }$. This will give us the complete equation of the line. This problem illustrates a common type of question in algebra where understanding the properties of parallel lines is essential. By carefully applying the principles of slope and y-intercept, we can find the unique equation that satisfies both conditions: being parallel to the given line and passing through the specified point. The ability to solve such problems demonstrates a solid understanding of linear equations and their geometric interpretations. Furthermore, the process involves algebraic manipulation and problem-solving skills that are broadly applicable in mathematics and related fields. The key to success in this type of problem is recognizing the significance of the slope in parallel lines and using the given point to determine the y-intercept, thus defining the specific line that meets the criteria.

Step-by-Step Solution

To solve this problem, let's break it down step by step. First, we identify the slope of the given line, ${ y = \frac{3}{4}x + 7 }$. As discussed, the slope is the coefficient of ${ x }$, which in this case is ${ \frac{3}{4} }$. Since parallel lines have the same slope, the line we're looking for will also have a slope of ${ \frac{3}{4} }$. Now, we know that our line has the form ${ y = \frac{3}{4}x + b }$, where ${ b }$ is the y-intercept that we need to find. To find ${ b }$, we use the fact that the line passes through the point (-12, 36). This means that when ${ x = -12 }$, ${ y = 36 }$. We substitute these values into our equation: ${ 36 = \frac{3}{4}(-12) + b }$. Next, we simplify the equation. ${ \frac{3}{4} }$ times -12 is -9, so the equation becomes ${ 36 = -9 + b }$. To solve for ${ b }$, we add 9 to both sides of the equation: ${ 36 + 9 = b }$, which gives us ${ b = 45 }$. Now that we have the slope ${ \frac{3}{4} }$ and the y-intercept 45, we can write the equation of the line. The equation is ${ y = \frac{3}{4}x + 45 }$. Comparing this equation to the given options, we see that option C, ${ y = \frac{3}{4}x - 39 }$, is incorrect because it has the wrong y-intercept. However, a closer inspection reveals that none of the provided options exactly match our solution. This suggests there might be an error in the options provided or in the initial problem setup. The correct approach remains the same: identify the slope, use the given point to find the y-intercept, and then write the equation. It's possible that a typo exists in the options, and the correct equation should have been ${ y = \frac{3}{4}x + 45 }$. In such cases, it's crucial to double-check the calculations and the problem statement to ensure accuracy.

Analyzing the Given Options

Let's take a closer look at the options provided to understand why they are incorrect and to reinforce our understanding of how to identify parallel lines. Option A, ${ y = -\frac{4}{3}x + 20 }$, has a slope of ${ -\frac{4}{3} }$. This slope is the negative reciprocal of the slope of the given line (${ \frac{3}{4} }$), which means that this line is perpendicular, not parallel, to the given line. Perpendicular lines intersect at a 90-degree angle, so this option is incorrect. Option B, ${ y = -\frac{4}{3}x + 36 }$, also has a slope of ${ -\frac{4}{3} }$, making it perpendicular to the given line as well. The y-intercept of 36 is irrelevant when determining parallelism; it's the slope that matters. Therefore, option B is also incorrect. Option C, ${ y = \frac{3}{4}x - 39 }$, has the correct slope (${ \frac{3}{4} }$), which means it is parallel to the given line. However, it has a different y-intercept (-39) than what we calculated (45). To confirm that this option is incorrect, we can plug in the point (-12, 36) into the equation and see if it holds true. Substituting ${ x = -12 }$ into ${ y = \frac{3}{4}x - 39 }$ gives us ${ y = \frac{3}{4}(-12) - 39 = -9 - 39 = -48 }$, which is not equal to 36. This confirms that option C does not pass through the point (-12, 36) and is therefore incorrect. In summary, none of the provided options correctly represent a line that is parallel to ${ y = \frac{3}{4}x + 7 }$ and passes through the point (-12, 36). This exercise highlights the importance of carefully calculating and verifying the equation of the line, ensuring that both the slope and the y-intercept are correct. It also underscores the need to double-check the given options and problem statement for any potential errors.

Conclusion: Identifying Parallel Equations

In conclusion, the process of finding the equation of a line parallel to a given line involves a clear understanding of slopes and y-intercepts. The key takeaway is that parallel lines share the same slope, and the y-intercept determines the line's position on the coordinate plane. When given a line and a point, we can use the slope-intercept form of a linear equation (${ y = mx + b }$) to find the specific equation that satisfies the conditions. By identifying the slope from the given equation and substituting the coordinates of the point into the equation, we can solve for the y-intercept and thus determine the unique equation of the parallel line. In this particular problem, we correctly identified the slope as ${ \frac{3}{4} }$ and used the point (-12, 36) to calculate the y-intercept as 45. This led us to the equation ${ y = \frac{3}{4}x + 45 }$. However, none of the provided options matched this equation, suggesting a possible error in the options themselves. This situation emphasizes the importance of not only understanding the mathematical concepts but also critically evaluating the given information and solutions. Problem-solving in mathematics often involves a combination of applying learned principles and verifying results to ensure accuracy. In cases where the provided options do not match the calculated solution, it is crucial to double-check the calculations, the problem statement, and the options for any potential mistakes. Furthermore, this exercise reinforces the understanding of the relationship between the slope and y-intercept in linear equations and how they define the characteristics of a line. The ability to identify and manipulate these components is essential for solving a wide range of problems in algebra and geometry, making it a fundamental skill in mathematical proficiency.