Find The Parallel Line Equation

Dec 5, 2025 by Alex Johnson 32 views

Have you ever wondered how to determine if two lines are parallel? In the world of mathematics, parallel lines are lines that never intersect, no matter how far they are extended. They maintain the same distance from each other throughout their entire length. A key characteristic of parallel lines is that they share the exact same slope. This fundamental concept is what we'll be exploring today as we tackle a specific problem: finding the equation of a line that is parallel to another line defined by two points. We'll be given two points, $(8,3)$ and $(-8,1)$ , and our mission is to identify which of the provided options represents a line parallel to the one passing through these coordinates. This journey into parallel lines will not only solidify your understanding of slope but also enhance your ability to work with linear equations.

Understanding the Slope

The slope of a line is a measure of its steepness and direction. It's often represented by the letter 'm' and is calculated as the "rise over run" between any two distinct points on the line. Mathematically, if you have two points $(x_1, y_1)$ and $(x_2, y_2)$ , the slope $m$ is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ . This formula tells us how much the y-coordinate changes for every unit change in the x-coordinate. For our problem, the two given points are $(8,3)$ and $(-8,1)$ . Let's label them: $(x_1, y_1) = (8,3)$ and $(x_2, y_2) = (-8,1)$ . Plugging these values into the slope formula, we get: $m = \frac{1 - 3}{-8 - 8}$ . Simplifying the numerator, $1 - 3 = -2$ , and simplifying the denominator, $-8 - 8 = -16$ . So, the slope $m$ becomes $m = \frac{-2}{-16}$ . When we divide $-2$ by $-16$ , we get a positive result because a negative divided by a negative is a positive. Therefore, $m = \frac{2}{16}$ , which simplifies further to $m = \frac{1}{8}$ . This means that for every 8 units we move to the right along the x-axis, the line rises by 1 unit along the y-axis. The slope of the line passing through $(8,3)$ and $(-8,1)$ is $\frac{1}{8}$ . This value is crucial because, as we'll see, any line parallel to this one must have the exact same slope.

The Condition for Parallel Lines

Now that we've calculated the slope of the line passing through the given points, let's revisit the core property of parallel lines: they have the same slope. This is the fundamental principle that will guide us in selecting the correct equation from the given options. If a line is parallel to the line with a slope of $\frac{1}{8}$ , then that parallel line must also have a slope of $\frac{1}{8}$ . It's like two identical twins walking side-by-side; they move at the same pace and in the same direction. In mathematical terms, this means that the 'm' value in the equation of the parallel line must be equal to $\frac{1}{8}$ . The equations provided are all in the slope-intercept form, which is $y = mx + b$ , where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis). We need to examine each option and see which one has a slope ( $m$ ) that matches our calculated slope of $\frac{1}{8}$ . This comparison is straightforward and allows us to quickly eliminate incorrect options and pinpoint the correct one. This step is critical in solving the problem, as it directly applies the definition of parallel lines to the given equation formats.

Analyzing the Options

We've established that the target slope for our parallel line is $\frac{1}{8}$ . Now, let's scrutinize each of the provided options to find the one that shares this slope. Remember, we are looking for the equation in the form $y = mx + b$ where $m = \frac{1}{8}$ .

Option A: $y = -\frac{1}{8} x + 16$ In this equation, the slope $m$ is $-\frac{1}{8}$ . This is not equal to $\frac{1}{8}$ . Therefore, this line is not parallel to our original line.
Option B: $y = -8 x + 16$ Here, the slope $m$ is $-8$ . Again, this is not equal to $\frac{1}{8}$ . This line is not parallel.
Option C: $y = 8 x + 16$ In this equation, the slope $m$ is $8$ . This is also not equal to $\frac{1}{8}$ . This line is not parallel.
Option D: $y = \frac{1}{8} x + 16$ For this equation, the slope $m$ is $\frac{1}{8}$ . This exactly matches the slope we calculated for the line passing through $(8,3)$ and $(-8,1)$ . Therefore, this line is parallel to the original line.

By carefully comparing the slope of each option with the calculated slope, we can confidently identify option D as the correct answer. The y-intercept ( $b=16$ ) in this case doesn't affect the parallelism; only the slope matters for determining if lines are parallel. It simply means this parallel line crosses the y-axis at the same point as potentially other lines with a slope of $\frac{1}{8}$ , but it is the slope that defines its parallel relationship to our initial line.

Conclusion

In conclusion, the process of identifying a parallel line involves two main steps: first, calculating the slope of the line defined by the given points, and second, finding the equation among the options that possesses the identical slope. We successfully calculated the slope of the line passing through $(8,3)$ and $(-8,1)$ to be $\frac{1}{8}$ . By examining each provided equation, we determined that Option D: $y = \frac{1}{8} x + 16$ is the only equation with a slope of $\frac{1}{8}$ . This confirms that the line represented by this equation is indeed parallel to the line passing through the given points. Understanding the relationship between slopes and parallel lines is a fundamental concept in algebra, empowering you to solve a variety of problems involving linear equations. Keep practicing, and you'll become a pro at spotting parallel lines in no time!

For further exploration into the fascinating world of linear equations and their properties, you can visit Khan Academy's comprehensive resources on algebra. They offer in-depth explanations, practice exercises, and tutorials that can help solidify your understanding of concepts like slope, parallel lines, and perpendicular lines. Another excellent resource is Math is Fun, which provides clear and engaging explanations of mathematical topics suitable for learners of all levels. These platforms are invaluable tools for anyone looking to deepen their mathematical knowledge.