Problem 40 - College Algebra

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 equations to slope-intercept form

To determine the relationship between the two lines, we first need to convert each equation to the slope-intercept form, which is given by \( y = mx + b \), where \( m \) is the slope.For the first equation, \(6x - 9y = 10\):\[ 9y = 6x - 10 \]\[ y = \frac{6}{9}x - \frac{10}{9} \]Simplifying, we find:\[ y = \frac{2}{3}x - \frac{10}{9} \]So, the slope of the first line is \( \frac{2}{3} \).For the second equation, \(3x + 2y = 1\):\[ 2y = -3x + 1 \]\[ y = -\frac{3}{2}x + \frac{1}{2} \]So, the slope of the second line is \( -\frac{3}{2} \).

Step 2: Compare the slopes

Now that we have both lines in the slope-intercept form, we compare their slopes to determine their relationship.The slope of the first line is \( \frac{2}{3} \), and the slope of the second line is \( -\frac{3}{2} \).

Step 3: Check for parallel lines

Two lines are parallel if their slopes are equal. We compare the slopes \( \frac{2}{3} \) and \( -\frac{3}{2} \). Since these two slopes are not equal, the lines are not parallel.

Step 4: Check for perpendicular lines

Two lines are perpendicular if the product of their slopes is \(-1\). We multiply the slopes of the two lines:\( \frac{2}{3} \times -\frac{3}{2} = -1 \).Since the product of the slopes is \(-1\), the lines are perpendicular.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that it's easy to see both the slope of the line and where it intercepts the y-axis. This form is expressed as \( y = mx + b \), where:

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

To convert a line's equation into the slope-intercept form, you need to solve the equation for \( y \). Let's look at an example. If you have an equation like \( 6x - 9y = 10 \):

• First, isolate \( y \) on one side. Start by subtracting \( 6x \) from both sides to get \( -9y = -6x + 10 \).
• Then, divide every term by \(-9\) to solve for \( y \). You'll end up with \( y = \frac{2}{3}x - \frac{10}{9} \).

This equation is now in slope-intercept form, with a slope of \( \frac{2}{3} \) and a y-intercept of \(-\frac{10}{9}\).

Parallel Lines

Parallel lines are lines in a plane that never meet; they are always the same distance apart. For two lines to be parallel, their slopes must be identical.

• This means that for equations in slope-intercept form \( y = m_1x + b_1 \) and \( y = m_2x + b_2 \), the lines are parallel if \( m_1 = m_2 \).

Let's use the equations from the solved example:

• Consider \( y = \frac{2}{3}x - \frac{10}{9} \) and \( y = -\frac{3}{2}x + \frac{1}{2} \).
• The slopes, \( \frac{2}{3} \) and \(-\frac{3}{2} \), are not equal.

Since their slopes are different, these two lines are not parallel. Whenever you're checking for parallel lines, just ensure the slopes are the same.

Perpendicular Lines

Perpendicular lines intersect at a right angle, which is 90 degrees. For two lines to be perpendicular, the product of their slopes must be -1.

• For lines with slopes \( m_1 \) and \( m_2 \), they are perpendicular if \( m_1 \times m_2 = -1 \).

Let's apply this to the given solutions:

• The slope of the first line is \( \frac{2}{3} \), and the second line has a slope of \( -\frac{3}{2} \).
• When multiplying these slopes, \( \frac{2}{3} \times -\frac{3}{2} = -1 \).

Since the product is -1, these lines are indeed perpendicular. In practice, once you determine the slopes' multiplication results in -1, you can confidently state that the lines are perpendicular.

Slope

The slope of a line characterizes its steepness and direction. It is a measure of the rate at which the line rises or falls. The slope \( m \) in the equation \( y = mx + b \) can be calculated using the formula:

• \( m = \frac{\text{rise}}{\text{run}} \)

Here’s how you can understand and calculate the slope:

• If a line moves upward from left to right, the slope is positive.
• If it moves downward from left to right, the slope is negative.
• The steeper the line, the larger the absolute value of the slope.

In the step-by-step solution, the first line has a slope of \( \frac{2}{3} \). This means for every 3 units it moves horizontally, it rises by 2 units vertically. The second line, with a slope of \( -\frac{3}{2} \), falls by 3 units for every 2 units it moves horizontally. Clearly understanding the slope helps determine the nature of relationships between the lines, such as parallel or perpendicular tendencies.