Problem 40 - Algebra A Combined Function

Question

Determine whether each pair of lines is parallel, perpendicular, or neither. See Example 7. $$ \begin{array}{l} y=4 x-2 \\ 4 x+y=5 \end{array} $$

Step-by-Step Solution

Answer

The lines are neither parallel nor perpendicular.

Step 1: Write the equations in slope-intercept form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The first equation is already in slope-intercept form: \( y = 4x - 2 \).For the second equation \( 4x + y = 5 \), solve for \( y \):Subtract \( 4x \) from both sides to get \( y = -4x + 5 \).

Step 2: Identify the slopes of the lines

Now that both equations are in slope-intercept form, identify the slopes of the two lines from the equations:1. For the line \( y = 4x - 2 \), the slope \( m_1 = 4 \).2. For the line \( y = -4x + 5 \), the slope \( m_2 = -4 \).

Step 3: Determine the relationship between the slopes

To determine if lines are parallel, their slopes should be equal (\( m_1 = m_2 \)). To determine if lines are perpendicular, the slopes should be negative reciprocals of each other (\( m_1 \times m_2 = -1 \)). The slopes we have are 4 and -4:- Check if they are perpendicular: \( 4 \times (-4) = -16 \), which is not equal to -1.- Since the slopes are neither equal nor negative reciprocals, the lines are neither parallel nor perpendicular.

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express the equation of a line. It is written as \( y = mx + b \). In this formula:

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

This form is particularly useful for quickly sketching graphs of linear functions, as it provides a straightforward way to identify both the slope and the starting point on the graph. When a line is given in standard form, such as \( Ax + By = C \), it can be converted to slope-intercept form by isolating \( y \) on one side of the equation. This transformation allows for easy comparison of lines, which is crucial for determining relationships like parallelism or perpendicularity.

Equation of a Line

The equation of a line can be expressed in multiple forms, but the slope-intercept form \( y = mx + b \) is among the most intuitive for line analysis.
Converting equations into this form offers clarity and makes calculations easier.
For example, the equation \( y = 4x - 2 \) directly gives us:

• The slope \( m = 4 \)
• The y-intercept \( b = -2 \)

In comparison, an equation like \( 4x + y = 5 \) requires rearrangement. Start by subtracting \( 4x \) from both sides to get \( y = -4x + 5 \). Now, it’s easy to see that:

• The slope is \(-4\)
• The y-intercept is \(5\)

This manipulation allows not only easier graph plotting but also simplifies the process of comparing lines for properties such as being parallel or perpendicular.

Identifying Slopes

Identifying the slope of a line is a fundamental skill in understanding how two lines relate to each other. The slope tells us the direction and steepness of a line—positive slopes rise to the right, while negative slopes fall.
The slopes explain how a change in the x-value affects the y-value, with steeper slopes indicating more dramatic increases or decreases. Mathematically, parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals of each other. To identify the slopes from equations in slope-intercept form (\(y = mx + b\)), simply look at the coefficient of \(x\), which is \(m\).
For instance, from \(y = 4x - 2\), the slope is \(4\). And from \(y = -4x + 5\), the slope is \(-4\). With this information, you can determine that since \(4\) and \(-4\) are not equal, the lines are not parallel. Further, since their product is not \(-1\), they are not perpendicular either.

Mathematical Reasoning

Mathematical reasoning is all about methodically analyzing and solving problems using logical steps. When determining if lines are parallel or perpendicular, understanding and correctly applying mathematical properties is key.
First, analyze the slopes derived from converted line equations:

• If the slopes are equal, the lines are parallel.
• If the slopes are negative reciprocals (multiplying to \(-1\)), the lines are perpendicular.

In our exercise example:

• Slopes are \(4\) and \(-4\).
• Testing for perpendicularity, calculate \(4 \times (-4) = -16\), which does not equal \(-1\), ruling out perpendicular lines.
• The slopes are also not equal, ruling out parallel lines.

Concluding that they are neither parallel nor perpendicular shows your understanding of slope-related line properties, which is a crucial aspect of mathematical reasoning in geometry.