Determine the Equation of a Straight Line Perpendicular to 2x-7y6 and Passing Through -35

Introduction

In order to determine the equation of a straight line that is perpendicular to a given line and passes through a specific point, we need to follow a series of mathematical steps. This article will guide you through the process using a practical example: finding the equation of a line that is perpendicular to the line defined by the equation 2x-7y6 and passes through the point (-3, 5).

Understanding Perpendicular Lines

First, let's start by understanding what it means for two lines to be perpendicular. Two lines are perpendicular if the product of their slopes is -1. In other words, if one line has a slope (m_1), the slope of the line perpendicular to it ((m_2)) satisfies the condition (m_1 cdot m_2 -1).

Given Line and Point

The given line is defined by the equation:

2x - 7y 6

We can rewrite this equation in the standard form (ax by c 0), which gives us:

2x - 7y - 6 0

The given point through which the required line passes is (-3, 5).

Finding the Slope of the Given Line

To find the slope of the given line, we first need to get the slope in the form (y mx b). Rearrange the equation:

-7y -2x 6

y frac{2}{7}x - frac{6}{7}

Here, the slope (m_1) of the given line is (frac{2}{7}).

Determining the Slope of the Perpendicular Line

Since the required line is perpendicular to the given line, its slope (m_2) will be the negative reciprocal of (frac{2}{7}), which is (-frac{7}{2}).

Using the Point-Slope Form to Find the Equation of the Perpendicular Line

The general form of the line passing through a point ((x_1, y_1)) with a slope (m) is given by:

yy_1 m(xx_1)

Substituting the slope (-frac{7}{2}) and the point ((-3, 5)) into the equation:

y - 5 -frac{7}{2}(x 3)

Now, we simplify and convert the equation to the standard form:

y - 5 -frac{7}{2}x - frac{21}{2}

2y - 10 -7x - 21

7x 2y 11 0

Conclusion

The equation of the line that is perpendicular to 2x-7y6 and passes through the point (-3, 5) is:

7x 2y 11 0

Understanding these steps can be incredibly useful in various mathematical and practical applications, such as in geometry, physics, and engineering. By following this process, you can determine the equation of a line that is perpendicular to any given line that passes through a specific point.

Keywords: equation of a straight line, perpendicular lines, slope of a line