Problem 37 - 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. $$ \begin{array}{l} 3 x-2 y=5 \\ 6 y-9 x=6 \end{array} $$

Step-by-Step Solution

Answer

The lines are parallel as both have a slope of \( \frac{3}{2} \).

Step 1: Rewrite Equations in Slope-Intercept Form

To determine if the lines are parallel, perpendicular, or neither, we need the slope of each line. We achieve this by converting both equations to the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope.For the first equation, \( 3x - 2y = 5 \), solve for \( y \):\[ -2y = -3x + 5 \]\[ y = \frac{3}{2}x - \frac{5}{2} \]The slope \( m_1 \) is \( \frac{3}{2} \).For the second equation, \( 6y - 9x = 6 \), solve for \( y \):\[ 6y = 9x + 6 \]\[ y = \frac{3}{2}x + 1 \]The slope \( m_2 \) is \( \frac{3}{2} \).

Step 2: Compare Slopes

Now, compare the slopes from the converted equations. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals, the lines are perpendicular.We have calculated that both equations have the slope \( \frac{3}{2} \). Since \( m_1 = m_2 = \frac{3}{2} \), the lines are parallel.

Key Concepts

Slope-Intercept Form

Many linear equations can be easily understood when rewritten in the slope-intercept form, which is given by the expression \( y = mx + b \). Here, \( m \) represents the slope of the line, while \( b \) is the y-intercept, or the point where the line crosses the y-axis. This form is particularly useful because it directly shows how steep a line is and where it is positioned on the graph.
This simplicity allows for a clearer visual interpretation when plotting graphs.
For instance, in the example given, transforming an equation like \( 3x - 2y = 5 \) involves rearranging it to isolate \( y \), giving us \( y = \frac{3}{2}x - \frac{5}{2} \).
The slope here, \( \frac{3}{2} \), indicates a rise of 3 units for every 2 units moved horizontally to the right. This transformation is a fundamental step for graphing and analyzing linear equations.

Parallel Lines

Lines that run alongside each other, never intersecting, are known as parallel lines. A key characteristic of parallel lines in a Cartesian plane is that they have the same slope.
This means that in the equation \( y = mx + b \), two lines will be parallel if they share the same value of \( m \).
In the exercise provided, we transformed the original equations to obtain their slopes and found that both lines have a slope of \( \frac{3}{2} \). Since both lines have identical slopes, they run parallel to one another in the graph.
Parallel lines are significant in geometry and many real-world applications, ensuring constant distance between objects, like railway tracks or lanes on a highway.

Slope Comparison

The comparison of slopes is crucial in determining the relationship between two lines in a plane. By examining the slope values, we can quickly identify whether the lines are parallel, perpendicular, or neither.
If two lines have the same slope, they are parallel, as is the case in the given problem where both slopes are \( \frac{3}{2} \). This similarity shows that the lines run in the same direction without ever meeting. However, if the slopes were negative reciprocals, this would indicate that the lines are perpendicular, meaning they intersect at a right angle.
The slope comparison is a straightforward method to draw conclusions about line relationships, which is why it is a fundamental aspect of graph analysis.