Introduction to Parallel Lines and Their Equations

In the realm of geometry and algebra, understanding the properties of parallel lines is essential. Two lines are said to be parallel if they never intersect, meaning they maintain a constant distance from each other throughout their infinite expanse. This article will guide you through the process of finding the equation of a line that is parallel to a given line and passes through a specific point. Follow along to understand the underlying concepts and apply the step-by-step methodology effectively.

Understanding the Concept: Parallel Lines and Slope

The fundamental concept of parallel lines lies in their shared slope. The slope of a line determines its steepness and direction. For two lines to be parallel, their slopes must be identical. This means if you know the slope of one line, you know the slope of the parallel line without needing any further calculations.

Step-by-Step Process to Find the Equation of a Parallel Line

Let's dive into the detailed process of finding the equation of a line that passes through a given point and is parallel to another line. This example will illustrate the application of the point-slope form and the use of slopes in defining parallel lines.

Given Points and Slope Calculation

First, let's identify the given points and calculate the slope of the line passing through these points. We start with two points: Point A(-3, 5) and Point B(3, 2).

Calculating the Slope

To find the slope, we use the slope formula:

m frac{y_2 - y_1}{x_2 - x_1}

Substituting the coordinates of the points:

x_1, y_1 -3, 5 and x_2, y_2 3, 2

We get:

m frac{2 - 5}{3 - (-3)} frac{-3}{6} -frac{1}{2}

Using the Slope to Form the Equation

Now that we have the slope, we can use the point-slope form of the equation of a line:

y - y_1 m(x - x_1)

We substitute the calculated slope and the given point (2, 8) into the equation:

y - 8 -frac{1}{2}(x - 2)

Expanding the equation:

y - 8 -frac{1}{2}x 1

Adding 8 to both sides:

y -frac{1}{2}x 9

Conclusion

Thus, the equation of the line that passes through the point (2, 8) and is parallel to the line passing through the points (-3, 5) and (3, 2) is:

y -frac{1}{2}x 9

Additional Examples

Let's look at an additional example. We will find the slope of the line that passes through points (-2, 2) and (4, 5). Using the same slope formula:

m frac{5 - 2}{4 - (-2)} frac{3}{6} frac{1}{2}

The slope is (frac{1}{2}). Applying the point-slope form with the point (2, 8) as before, we get:

y - 8 frac{1}{2}(x - 2)

Expanding and simplifying:

y - 8 frac{1}{2}x - 1

2y - 16 x - 2

x - 2y -14

Conclusion

This step-by-step guide demonstrates the importance of understanding the properties of parallel lines and how to use slopes and the point-slope form to find the equation of a parallel line passing through a given point. By following these steps, you can easily identify and work with parallel lines in various applications and real-world scenarios.