Determining the Equation of a Perpendicular Line

In this article, we will explore how to find the equation of a line that is perpendicular to a given line and passes through a specific point. This process involves understanding the properties of perpendicular lines, the slope-intercept form, and the point-slope form of a line's equation.

Step-by-Step Guide

Let's consider the given line 2y - x - 8 0, and a point (5, -7) through which the new line must pass.

Step 1: Find the Slope of the Given Line

The first step is to rearrange the given equation into the slope-intercept form, y mx b, where m is the slope.

2y - x - 8 0

First, isolate y:

Move x and 8 to the other side:

2y x 8

Divide by 2 to solve for y:

y (1/2)x 4

The slope m is (1/2).

Step 2: Determine the Slope of the Perpendicular Line

A line that is perpendicular to another line has a slope that is the negative reciprocal of the original slope. Thus, the slope of the perpendicular line is:

m_{perpendicular} -1 / (1/2) -2

Step 3: Use the Point-Slope Form

Use the point-slope form to write the equation of the new line. The point-slope form is:

y - y_1 m(x - x_1)

Here, (x_1, y_1) (5, -7) and m -2 will be used.

Substitute the values into the equation:

y - (-7) -2(x - 5)

Of simplify:

y 7 -2x 10

y -2x 10 - 7

y -2x 3

Conclusion

The equation of the line that is perpendicular to 2y - x - 8 0 and passes through the point (5, -7) is y -2x 3.

Additional Insights

Understanding the concept of perpendicular lines and their slopes is crucial for solving a variety of algebraic problems and for deepening your knowledge of linear equations. By applying the formula for the slope of a perpendicular line and using the point-slope form, you can easily find the equation of any line that is perpendicular to a given line.