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Introduction

Parallel and perpendicular lines have unique characteristics in geometry and algebra, particularly when represented by linear equations. Understanding these properties helps in analyzing relationships between lines on the Cartesian plane.

In this lesson, you will learn to identify and work with the slopes of parallel and perpendicular lines, as well as to write equations for them.

Understanding Parallel Lines

Two lines are parallel if they have the same slope but different y-intercepts. This means they never intersect. For example, lines with equations \( y = 2x + 3 \) and \( y = 2x - 4 \) are parallel because they both have a slope of 2.

Understanding Perpendicular Lines

Two lines are perpendicular if the product of their slopes is -1. In other words, the slopes are negative reciprocals of each other. For example, a line with slope \( m = 3 \) is perpendicular to a line with slope \( m = -\frac{1}{3} \).

To find a line perpendicular to another, take the negative reciprocal of the given line's slope.

Practice Questions

1. Given the line \( y = 4x + 1 \), write the equation of a line that is parallel to it and passes through the point (2, 5).
   Solution

   Since the line is parallel, it has the same slope of 4. Use point-slope form \( y - y_1 = m(x - x_1) \):

   \[ y - 5 = 4(x - 2) \]

   Expanding, we get:

   \[ y = 4x - 3 \]

   The equation of the parallel line is \( y = 4x - 3 \).

2. Find the slope of a line that would be perpendicular to the line represented by \( y = -\frac{1}{2}x + 4 \).
Solution

The slope of the given line is \( m = -\frac{1}{2} \). The negative reciprocal of \( -\frac{1}{2} \) is 2, so the slope of a line perpendicular to it would be 2.

3. Write the equation of a line that is perpendicular to \( y = 3x - 7 \) and passes through the point (1, 4).
   Solution

   The slope of the given line is 3. The negative reciprocal of 3 is \( -\frac{1}{3} \).

   Using point-slope form \( y - y_1 = m(x - x_1) \):

   \[ y - 4 = -\frac{1}{3}(x - 1) \]

   Expanding, we get:

   \[ y = -\frac{1}{3}x + \frac{13}{3} \]

   The equation of the perpendicular line is \( y = -\frac{1}{3}x + \frac{13}{3} \).

4. Determine if the lines given by \( y = 5x + 6 \) and \( y = -\frac{1}{5}x + 2 \) are parallel, perpendicular, or neither.
   Solution

   The slope of the first line is 5, and the slope of the second line is \( -\frac{1}{5} \).

   The product of the slopes is \( 5 \times -\frac{1}{5} = -1 \), which means the lines are perpendicular.

Additional Practice

Try these on your own:

1. Write the equation of a line that is parallel to \( y = -3x + 4 \) and passes through the point (-2, 1).
2. Find the equation of a line perpendicular to \( y = \frac{1}{4}x - 5 \) and passing through (0, 2).
3. Given the line \( y = 7x + 3 \), determine if a line with slope \( -\frac{1}{7} \) would be parallel, perpendicular, or neither.