Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither. $$ \begin{array}{l} y=\frac{3 x+1}{4} \\ y=3 x+2 \end{array} $$

Step-by-Step Solution

Answer

The lines are neither parallel nor perpendicular.

Step 1: Identify the Slope of the First Equation

The first equation is given as \( y = \frac{3x + 1}{4} \). To identify the slope, rewrite it in the standard form \( y = mx + b \), where \( m \) is the slope. By distributing the division by 4, the equation becomes \( y = \frac{3}{4}x + \frac{1}{4} \). Hence, the slope \( m_1 \) is \( \frac{3}{4} \).

Step 2: Identify the Slope of the Second Equation

The second equation is already in the form \( y = mx + b \), which is \( y = 3x + 2 \). From this equation, the slope \( m_2 \) is 3.

Step 3: Compare Slopes to Determine Relationship

Check if the lines are parallel or perpendicular. For lines to be parallel, their slopes must be equal \( m_1 = m_2 \). For lines to be perpendicular, the product of their slopes must be \(-1\), \( m_1 \times m_2 = -1 \).

Step 4: Analyze Slope Conditions

The slopes \( m_1 = \frac{3}{4} \) and \( m_2 = 3 \) are not equal, thus the lines are not parallel. The product of the slopes is \( \frac{3}{4} \times 3 = \frac{9}{4} \), which is not equal to \(-1\). Therefore, the lines are neither parallel nor perpendicular.

Key Concepts

Graphing Linear Equations

To graph linear equations, it's crucial that we understand how they are structured, particularly in the slope-intercept form, given as \( y = mx + b \). This form allows us to identify two important components: the slope \( m \) and the y-intercept \( b \). These values help in sketching the line on a graph.
- **Slope \( m \):** Indicates the steepness and direction of the line. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
- **Y-Intercept \( b \):** Represents the point where the line crosses the y-axis.

• Start by plotting the y-intercept on the y-axis.
• Use the slope to find another point on the line. For example, a slope of \( \frac{3}{4} \) means you move up 3 units and right 4 units from the y-intercept.

Drawing a straight line through these points gives you the graph of the equation. Repeat the process for the second equation to see how the two lines relate on the graph.

Slope Identification

Identifying the slope in a linear equation is an essential step in understanding the behavior of a line. The slope \( m \) provides insights into how a change in one variable affects the other.
When the equation is in the form \( y = mx + b \), \( m \) is your slope. This equation divides the linear relationship into two parts:

• The coefficient of \( x \) (\( m \)) is the slope.
• The constant term (\( b \)) is the y-intercept.

If the equation is not already in this form, you may need to rearrange it by isolating \( y \). For instance, transforming \( y = \frac{3x + 1}{4} \) into \( y = \frac{3}{4}x + \frac{1}{4} \) clearly shows the slope as \( \frac{3}{4} \). Identifying and comparing slopes is crucial when determining relationships like parallelism or perpendicularity between lines.

Parallel and Perpendicular Lines

Understanding the relationship between lines is essential in geometry and algebra. When comparing two lines, their slopes can reveal if lines are parallel, perpendicular, or neither.
Parallel lines have the same slope. For example, two lines \( y = \frac{3}{4}x + c_1 \) and \( y = \frac{3}{4}x + c_2 \) are parallel because they share the slope \( m = \frac{3}{4} \).
Perpendicular lines, on the other hand, have slopes that are negative reciprocals. If one line’s slope is \( m_1 \), the perpendicular line's slope \( m_2 \) must satisfy \( m_1 \times m_2 = -1 \). For instance, a line with slope \( \frac{3}{4} \) will be perpendicular to a line with slope \( -\frac{4}{3} \).
If the slopes of the two lines don’t match these conditions, like in the given exercise where the slopes are \( \frac{3}{4} \) and \( 3 \), the lines are neither parallel nor perpendicular. This understanding helps in visualizing and predicting the orientation of lines in a graph.