Problem 38 - Algebra A Combined Function

Question

Determine whether each pair of lines is parallel, perpendicular, or neither. See Example 7. $$ \begin{array}{l} y=\frac{1}{5} x+20 \\ y=-\frac{1}{5} x \end{array} $$

Step-by-Step Solution

Answer

The lines are neither parallel nor perpendicular.

Step 1: Identify the Slopes of Both Lines

Identify the slope for each line from their equations in the slope-intercept form \(y = mx + c\), where \(m\) is the slope.- The slope of the first line \(y = \frac{1}{5}x + 20\) is \(m_1 = \frac{1}{5}\).- The slope of the second line \(y = -\frac{1}{5}x\) is \(m_2 = -\frac{1}{5}\).

Step 2: Test for Parallel Lines

Lines are parallel if their slopes are equal. Compare the slopes from Step 1.Since \(m_1 = \frac{1}{5}\) and \(m_2 = -\frac{1}{5}\), the slopes are not equal. Thus, the lines are not parallel.

Step 3: Test for Perpendicular Lines

Lines are perpendicular if the product of their slopes is \(-1\). Calculate the product of the slopes.Calculate the product: \(m_1 \times m_2 = \frac{1}{5} \times -\frac{1}{5} = -\frac{1}{25}\).Since \(-\frac{1}{25} eq -1\), the lines are not perpendicular.

Step 4: Conclusion

Since the lines are neither parallel (not having equal slopes) nor perpendicular (product of slopes not equal to \(-1\)), conclude they are neither parallel nor perpendicular.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so you can quickly identify both its slope and its y-intercept. The general formula for the slope-intercept form is \( y = mx + b \). In this formula:

• \( m \) represents the slope of the line, showing how steep the line is.
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

Knowing the slope and intercept makes it easy to sketch a line on a graph. For instance, in the equation \( y = \frac{1}{5}x + 20 \), the slope is \( \frac{1}{5} \) and the y-intercept is 20. This means the line goes up one unit for every five units it moves to the right, crossing the y-axis at 20. Likewise, the equation \( y = -\frac{1}{5}x \) demonstrates a slope of \(-\frac{1}{5}\), with a y-intercept of 0, indicating a downward slant.

Slopes of Lines

Slopes tell you the direction and steepness of a line. Calculating the slope involves finding the rise over run—the change in the vertical direction divided by the change in the horizontal direction. Here's what you can determine:

• A positive slope means the line goes upwards from left to right.
• A negative slope indicates the line goes downwards from left to right.
• A zero slope suggests a flat, horizontal line.
• An undefined slope means the line is vertical.

In our example, the slopes are \( m_1 = \frac{1}{5} \) for the first line and \( m_2 = -\frac{1}{5} \) for the second. Despite their similarities, the opposite signs imply different directions. Positive for one, negative for the other.

Identifying Line Relationships

Understanding the relationship between two lines involves examining the slopes. Here's how to determine if lines are parallel, perpendicular, or neither:

• Parallel Lines: Two lines are parallel if they have identical slopes. They will never intersect, like train tracks running in the same direction.
• Perpendicular Lines: These lines intersect at a right angle (90 degrees). This occurs if the product of their slopes is \(-1\).

In our case, the slopes were \( m_1 = \frac{1}{5} \) and \( m_2 = -\frac{1}{5} \). The product \( \frac{1}{5} \times -\frac{1}{5} = -\frac{1}{25} \), not \(-1\), shows they aren't perpendicular. Since their slopes differ, they aren't parallel either. Thus, these lines don't share a special relationship—they're neither parallel nor perpendicular.