Problem 98 - Algebra for College Students

Question

Determine whether each pair of lines is parallel, perpendicular, or neither. $$\frac{1}{4} x-\frac{1}{6} y=\frac{1}{3}, \frac{1}{3} y=\frac{1}{2} x-2$$

Step-by-Step Solution

Answer

The lines are parallel.

Step 1 - Convert equations to slope-intercept form

First, each line equation should be converted to the slope-intercept form, which is given by: \(y = mx + b\) where \(m\) is the slope.

Step 2 - Convert first equation to slope-intercept form

Given the first equation: \(\frac{1}{4} x - \frac{1}{6} y = \frac{1}{3}\). Rearrange terms to isolate \(y\): \[ \frac{1}{4} x - \frac{1}{6} y = \frac{1}{3} \] Multiply both sides by 12 (the least common multiple of 4 and 6): \[ 12 * \left( \frac{1}{4} x - \frac{1}{6} y \right) = 12 * \left( \frac{1}{3} \right) \] This simplifies to: \[ 3x - 2y = 4 \] Next, solve for \(y\): \[ -2y = -3x + 4 \] Divide by -2: \[ y = \frac{3}{2} x - 2 \] Here, the slope \(m_1 = \frac{3}{2}\).

Step 3 - Convert second equation to slope-intercept form

Given the second equation: \(\frac{1}{3} y = \frac{1}{2} x - 2\). Multiply both sides by 3 to isolate \(y\): \[ y = \frac{3}{2} x - 6 \] Here, the slope \(m_2 = \frac{3}{2}\).

Step 4 - Compare slopes to determine relationship

The slopes of both lines are \(m_1 = \frac{3}{2}\) and \(m_2 = \frac{3}{2}\). Since the slopes are equal, the lines are parallel.

Key Concepts

slope-intercept form

When working with linear equations, we often use the slope-intercept form because it's straightforward and tells us a lot about the line. This form is given by the formula:
\( y = mx + b \)

• The letter \( y \) represents the value on the y-axis.
• The letter \( x \) represents the value on the x-axis.
• The letter \( m \) is used for the slope.
• The letter \( b \) stands for the y-intercept.

The slope-intercept form makes it easy to graph the line or understand its behavior.
For example, if we have an equation \( y = 2x + 3 \), the slope \( m \) is 2, and the line crosses the y-axis at 3. This means as the x value increases by 1, the y value increases by 2.

Using this method for the equations given in the exercise simplifies our calculations and makes identifying relationships between lines easier.

slope

The slope is a measure of how steep a line is. It tells you how much y changes for a change in x. In simpler terms, it describes how the line tilts.
[Slope Formula]
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula calculates the slope based on two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line.

To determine if lines are parallel, perpendicular, or neither, comparing their slopes is crucial.

• Parallel Lines: Same slope.
• Perpendicular Lines: Slopes are negative reciprocals (e.g., \( m_1 = \frac{3}{2}\) and \( m_2 = -\frac{2}{3} \)).
• Neither: Any other case.

For instance, in the provided solution, both lines have a slope of \( \frac{3}{2} \), making them parallel.

line equations

Line equations represent the relationship between the x and y coordinates for points on a line. Common forms include the slope-intercept form \( y = mx + b \), point-slope form \( y - y_1 = m(x - x_1) \), and standard form \( Ax + By = C \).

By converting to slope-intercept form, we make it easier to find the slope and y-intercept.
For example, let's look at the equation \( 3x - 2y = 4 \). We isolate \( y\) to get \( y = \frac{3}{2}x - 2 \), showing the slope (\( \frac{3}{2}\)) and y-intercept (-2).
This form is particularly useful when graphing lines or when determining the relationship between two lines, as seen in the exercise where it's used to determine whether the lines are parallel or perpendicular.