Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither. $$ \begin{array}{l} 6 x-9 y=10 \\ 3 x+2 y=1 \end{array} $$

Step-by-Step Solution

Answer

The lines are perpendicular.

Step 1: Convert the Equations to Slope-Intercept Form

Start by rearranging each equation to the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope. For the first equation, \( 6x - 9y = 10 \), solve for \( y \):\[ 9y = 6x - 10 \]\[ y = \frac{6}{9}x - \frac{10}{9} \]\[ y = \frac{2}{3}x - \frac{10}{9} \].For the second equation, \( 3x + 2y = 1 \), solve for \( y \):\[ 2y = -3x + 1 \]\[ y = -\frac{3}{2}x + \frac{1}{2} \].

Step 2: Identify the Slopes of the Lines

From the slope-intercept form of both equations, identify the slopes. For the first line, it's \( \frac{2}{3} \). For the second line, it's \( -\frac{3}{2} \).

Step 3: Determine the Relationship Between the Lines

To determine if the lines are parallel, check if their slopes are equal. Here, they are not, as \( \frac{2}{3} eq -\frac{3}{2} \). To determine if they are perpendicular, the slopes need to be negative reciprocals. Check: \( \frac{2}{3} \times -\frac{3}{2} = -1 \), so the lines are perpendicular.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to express a straight line on a graph easily. It is given by the equation \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept. This form makes it simple to graph a line, as you can start with the intercept and use the slope to determine the direction and steepness of the line.
To transform any linear equation into slope-intercept form, arrange it so that \( y \) is isolated on one side. This process involves rearranging and simplifying terms. For instance, consider the equation \( 6x - 9y = 10 \). To convert it, solve for \( y \): bring terms involving \( x \) to the other side and adjust coefficients so that \( y \) is by itself.
Converting to this format helps in identifying relationships between lines and graphing them with ease. It reveals quickly whether two lines are parallel or perpendicular simply by examining their slopes.

Linear Equations

Linear equations are equations of the first degree, meaning they have the general form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. They graph as straight lines in a Cartesian plane. They express a constant rate of change and are fundamental to understanding relationships between variables.
When working with linear equations, understanding different forms is key. Besides slope-intercept form, standard form is another way, where both variables and constants appear together on one side: \( Ax + By = C \). Transforming between these forms is a useful skill. It allows flexibility in solving problems and reveals different properties of the line.
Linear equations form the basis for more complex mathematical concepts and solving them efficiently helps in various mathematical and real-world applications like economics, physics, and engineering.

Slopes of Lines

The slope of a line measures its steepness and direction on a graph. It is represented as \( m \) in the slope-intercept form \( y = mx + b \). The slope is calculated as the change in \( y \) divided by the change in \( x \) (often described as "rise over run").
Identifying the slope helps determine if lines are parallel, perpendicular, or neither. Parallel lines share the same slope, which means they never intersect. Perpendicular lines have slopes that are negative reciprocals (for example, \( \frac{2}{3} \) and \( -\frac{3}{2} \)).
In the given problem, by converting the equations into slope-intercept form, you identified the slopes \( \frac{2}{3} \) and \( -\frac{3}{2} \). Checking if one slope is the negative reciprocal of the other clarifies that these lines intersect at a 90-degree angle, proving they are perpendicular.