Introduction

When dealing with lines in a plane, there are several key concepts and formulas that are useful in analyzing and manipulating their equations. This article discusses finding the equation of a line that is parallel to a given line and passes through a specific point. We will explore various methods and provide detailed examples to illustrate each step.

Understanding the Problem

Given the line 2x - y - 2 0 and a point (-2, 3), we aim to find the equation of a line that is parallel to the given line and passes through the specified point. Parallel lines share the same slope, which is a fundamental principle in solving such problems.

Step-by-Step Solution: Method 1

First, we need to determine the slope of the given line. We start with the equation:

2x - y - 2 0

Let's rearrange this equation into the slope-intercept form y mx b where m is the slope and b is the y-intercept.

Rearrange the equation: y 2x - 2 Identify the slope:

The slope m is 2.

Use the point-slope form: y - y_1 m(x - x_1) Substitute the known values: y - 3 2(x - (-2)) 2(x 2) Simplify the equation: y - 3 2x 4 y 2x 7

Thus, the equation of the line through the point (-2, 3) and parallel to the line 2x - y - 2 0 is:

y 2x 7

Step-by-Step Solution: Additional Methods

Method 2: Algebraic Substitution

Another valid approach is to use the point-slope form directly:

Substitute the point and the slope: y - 3 2(x - (-2)) Simplify the equation: y - 3 2x 4 Final form: y 2x 7

Method 3: Using the Parallel Line Formula

A third method involves using the formula for a line parallel to ax by c 0:

ax by - (ax_0 by_0) 0

Given the line x - 5y - 7 0, where the slope is 1/5, and the point (-2, 3):

Substitute the values: (x - 5y) - (x_0 - 5y_0) 0 (x - 5y) - (-2 - 5*3) 0 Simplify: x - 5y 17 0

Method 4: Direct Calculation

We can also calculate the equation step-by-step:

Given the slope and point: y - 3 2(x - (-2)) Substitute and simplify: y - 3 2(x 2) 2x 4 y 2x 7

Conclusion

In conclusion, finding the equation of a line that is parallel to a given line and passes through a specific point involves determining the slope of the given line and then using the point-slope form to find the new equation. The steps can be varied, but the core concept remains the same: parallel lines have the same slope.

By understanding and applying these methods, you can confidently solve similar problems involving parallel lines and points.