Finding the Equation of a Perpendicular Line: A Comprehensive Guide

When faced with the task of identifying the equation of a line that is perpendicular to a given line, the first step is to determine the slope of the original line. In this article, we will walk through the process step-by-step to find the equation of a line that is both perpendicular and passes through a specified point. We'll also discuss the relationship between parallel lines and their equations.

Understanding the Given Equation

Let's start with the equation of the original line, 6x - 5y 10. This is a linear equation in the standard form, but to solve many problems related to slopes, it's often more insightful to convert it to the slope-intercept form, which is:

y 6/5x - 2

In the slope-intercept form, y mx b, the coefficient of x (here, 6/5) represents the slope of the line.

Calculating the Slope of the Original Line

The slope of the line 6x - 5y 10 is m 6/5. This can be derived from the equation in slope-intercept form, y mx b, where m is the slope.

Determining the Slope of the Perpendicular Line

A line that is perpendicular to a given line has a slope that is the negative reciprocal of the original line's slope. If the slope of the original line is m, the slope of the perpendicular line will be -1/m.

For our line, the slope is 6/5. Therefore, the slope of the perpendicular line is:

m_perpendicular -1 / (6/5) -5/6

Using Point-Slope Formula to Find Perpendicular Line

The point-slope formula is a useful tool for finding the equation of a line given its slope and a point that the line passes through. The formula is:

y - y1 m(x - x1)

Here, (x1, y1) is the point the line passes through, and m is the slope of the line.

We need to use the point (3, 5) and the slope we calculated, -5/6. Substituting these values into the point-slope formula:

y - 5 -5/6(x - 3)

Expanding and simplifying:

y - 5 -5/6x 5/2

y -5/6x 5/2 5

y -5/6x 15/2

y -5/6x 7.5

Therefore, the equation of the line that is perpendicular to 6x - 5y 10 and passes through the point (3, 5) is:

y -5/6x 7.5

Understanding Parallel Lines in Detail

Parallel lines have the same slope. If we already have the equation of a line, we can express it in the slope-intercept form to find the slope and then write the equation of a parallel line by using the same slope and substituting a different point.

From earlier, the parallel line passing through the point (3, 5) can be derived using the same slope of 6/5 as the original line. Substituting the point and slope into the point-slope formula:

y - 5 6/5(x - 3)

Expanding and simplifying:

y - 5 6/5x - 18/5

y 6/5x 25/5 - 18/5

y 6/5x 7/5

The equation in standard form is:

6x - 5y 7

Thus, the equation of the line parallel to 6x - 5y 10 and passing through the point (3, 5) is:

6x - 5y 7

Conclusion

Understanding the slope and the methods to find parallel and perpendicular lines is crucial in geometry and algebra. By converting equations into slope-intercept form, we can quickly determine the slope and use it to find the equations of other lines. This guide provides a step-by-step approach to solving such problems, ensuring that you can efficiently find the equations of lines in various scenarios.