Question

The equations of two lines are given. Determine whether the lines are parallel, perpendicular, or neither. $$ \begin{array}{l} y=4 x+5 \\ y=-4 x+2 \end{array} $$

Short answer

The lines are neither parallel nor perpendicular.

Step-by-step solution

- Identify the Slopes

The equations of the lines are given in slope-intercept form, which is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. From the first equation \(y = 4x + 5\), the slope \(m_1 = 4\). From the second equation \(y = -4x + 2\), the slope \(m_2 = -4\).

- Determine Parallelism

Lines are parallel if their slopes are equal. Compare the slopes of the two lines: \(m_1 = 4\) and \(m_2 = -4\). Since \(4 eq -4\), the lines are not parallel.

- Determine Perpendicularity

Lines are perpendicular if the product of their slopes is \(-1\). Calculate the product of the slopes: \(4 \times -4 = -16\). Since \(-16 eq -1\), the lines are not perpendicular.

- Conclusion

Since the lines are neither parallel nor perpendicular based on the slopes, they are classified as neither.

Key concepts

slope-intercept form

The slope-intercept form is a way of writing the equation of a line so that you can easily identify its slope and y-intercept. The slope-intercept form is written as follows:

\( y = mx + b \)

In this equation, \( m \) represents the slope of the line, and \( b \) represents the y-intercept, which is where the line crosses the y-axis. This form is very useful because it allows us to quickly see the steepness and direction of the line.

For example, in the equation \( y = 4x + 5 \), the slope \( m = 4 \), and the y-intercept \( b = 5 \). Likewise, in the equation \( y = -4x + 2 \), the slope \( m = -4 \), and the y-intercept \( b = 2 \). Different slopes will give lines different angles of inclination, and different y-intercepts will shift the line up or down on the coordinate plane.

slope comparison

When comparing slopes, we can determine the relationship between two lines. There are three main scenarios to consider:

• Parallel Lines: Lines are parallel if they have the same slope. They never intersect because they are always the same distance apart. For example, lines with slopes \(m_1 = 4 \) and \(m_2 = 4 \) are parallel.


• Perpendicular Lines: Lines are perpendicular if the product of their slopes is \(-1\). This means they create a 90-degree angle where they intersect. For instance, lines with slopes \( m_1 = 2 \) and \( m_2 = - \frac{1}{2} \) are perpendicular because \( 2 \times - \frac{1}{2} = -1 \).


• Neither Parallel Nor Perpendicular: If lines don't meet either of the above criteria, they are classified as neither parallel nor perpendicular. For example, lines with slopes \( m_1 = 4 \) and \( m_2 = -4 \) are neither parallel nor perpendicular because neither \( 4 = -4 \) nor \( 4 \times -4 = -1 \).

line classification

Classifying lines based on their slopes helps us understand their geometric relationships. Given the equations \( y = 4x + 5 \) and \( y = -4x + 2 \):

• Step 1: Identify the Slopes - From these equations, the slopes are \( m_1 = 4 \) and \( m_2 = -4 \).


• Step 2: Check for Parallelism - Compare if \( m_1 \) is equal to \( m_2 \). Since \( 4 ≠ -4 \), the lines are not parallel.


• Step 3: Check for Perpendicularity - Check if the product of \( m_1 \) and \( m_2 \) is \(-1\). Since \( 4 \times -4 = -16 ≠ -1 \), the lines are not perpendicular.


• Conclusion: The lines are neither parallel nor perpendicular.

This method of classification can be applied to any pair of lines to determine their relationship based on their slopes.