Finding the Equation of a Parallel Line: A Step-by-Step Guide

Understanding the equation of a line can be a foundational concept in coordinate geometry. One crucial aspect is determining the equation of a straight line that is parallel to a given line and passes through a specific point. In this article, we will explore how to find the equation of such a line, using a practical example.

Background and Problem Statement

We are given a line L1 with the equation y 2x - 4. Our task is to find the equation of a line L2 that is parallel to L1 and passes through the point (6, -3).

Understanding Parallel Lines

Two lines are parallel if their slopes are equal. The slope of line L1 is 2. Therefore, the equation of any line parallel to L1 will have the same slope, i.e., 2.

Formulating the Equation of L2

Let's denote the equation of the line L2 as y 2x c. Our objective is to find the value of c using the point (6, -3).

Substituting the Point

Since the line L2 passes through the point (6, -3), we can substitute x 6 and y -3 into the equation of L2:

-3 2(6) c

Let's solve this equation step-by-step:

Step 1: Multiply 2 by 6: Step 2: Add the result to -3 to solve for c.

Solving for c

-3 12 c

c -15

The Final Equation

Substituting the value of c into the equation of L2, we get:

y 2x - 15

Other Forms of the Equation

In addition to the slope-intercept form, we can also express the equation of the line in other forms:

Slope-Intercept Form: y 2x - 15 Standard Form: 2x - y - 15 0

Conclusion

By understanding the properties of parallel lines and utilizing the slope-intercept form, we can easily determine the equation of a line that is parallel to a given line and passes through a specified point. In this case, the equation of the line parallel to y 2x - 4 and passing through the point (6, -3) is:

y 2x - 15

Understanding these concepts is not only crucial for solving mathematical problems but also forms the backbone of many real-world applications in fields such as physics, engineering, and data science.