Problem 97 - Algebra for College Students

Question

Determine whether each pair of lines is parallel, perpendicular, or neither. $$2 x-4 y=9, \frac{1}{3} x=\frac{2}{3} y-8$$

Step-by-Step Solution

Answer

The lines are parallel.

Step 1 - Convert the equations to slope-intercept form

First, convert both equations into the slope-intercept form, which is given by: \[y = mx + b\]where \(m\) is the slope and \(b\) is the y-intercept.

Step 2 - Convert the first equation

Starting with the first equation \(2x - 4y = 9\), rearrange it to the form \(y = mx + b\):\[2x - 4y = 9 \-4y = -2x + 9 \y = \frac{1}{2}x - \frac{9}{4}\]Here, the slope \(m\) is \(\frac{1}{2}\).

Step 3 - Convert the second equation

Next, convert the second equation \(\frac{1}{3}x = \frac{2}{3}y - 8\) into the slope-intercept form:\[\frac{1}{3}x = \frac{2}{3}y - 8 \frac{1}{3}x + 8 = \frac{2}{3}y \8 = \frac{2}{3}y - \frac{1}{3}x \8 = \frac{2}{3}y - \frac{1}{3}x \2y = x - 24 \y = \frac{1}{2}x - 12\]Here, the slope \(m\) is also \(\frac{1}{2}\).

Step 4 - Compare the slopes

Compare the slopes of the two lines obtained:\(m_1 = \frac{1}{2}\) and \(m_2 = \frac{1}{2}\).Since the slopes are equal, the lines are parallel.

Key Concepts

slope-intercept form

To understand if lines are parallel or perpendicular, we first need to convert their equations into the slope-intercept form. The slope-intercept form is expressed as: \( y = mx + b \) where \( m \) represents the slope of the line, and \( b \) is the y-intercept.
By converting equations to this form, we can easily identify the slope and determine relationships between lines.
To convert any equation to slope-intercept form:

• Isolate the \( y \) variable on one side.
• Ensure \( y \) is by itself and in the form of \( y = mx + b \).

Once in this form, comparing slopes of different lines becomes simple. This allows us to determine if lines are parallel, perpendicular, or neither.

slopes of lines

The slope of a line in the slope-intercept form \( y = mx + b \) is the coefficient \( m \) of the \( x \) term. A slope indicates how steep a line is.
Here are key points about slopes when comparing lines:

• If two lines have the same slope, they are parallel. This means they never intersect and are equidistant.
  In the exercise, both lines had slopes of \( \frac{1}{2} \) indicating they are parallel.
• If the product of their slopes is \( -1 \), the lines are perpendicular. Perpendicular lines intersect at a right angle (90 degrees).
  For example, if one line has a slope of \( \frac{3}{4} \), a line perpendicular to it would have a slope of \( - \frac{4}{3} \).
• If the slopes are neither equal nor multiplied to \( -1 \), the lines are neither parallel nor perpendicular.
  They will intersect at some angle other than 90 degrees.

equation conversion

Converting an equation to slope-intercept form involves straightforward algebraic manipulation. Let's look at the provided exercise step by step.
First, take the initial equation:

• For \( 2x - 4y = 9 \)

1. Re-arrange terms to isolate \( y \) : \( -4y = -2x + 9 \)
2. Divide entire equation by -4: \( y = \frac{1}{2}x - \frac{9}{4} \)

Here, the slope is \( \frac{1}{2} \). Now, consider the second equation:

• For \( \frac{1}{3}x = \frac{2}{3}y - 8 \)

1. Re-arrange terms to isolate \( y \): \( \frac{1}{3}x + 8 = \frac{2}{3}y \)
2. Multiply through by 3 to clear fractions: \( x + 24 = 2y \)
3. Solve for \( y \) : \( y = \frac{1}{2}x - 12 \)

Again, the slope is \( \frac{1}{2} \). This exercise illustrated how to properly convert equations to identify slopes and understand line relationships.