Problem 38 - College Algebra

Question

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither. $$ \begin{aligned} 3 y+x &=12 \\ -y &=8 x+1 \end{aligned} $$

Step-by-Step Solution

Answer

The lines are neither parallel nor perpendicular.

Step 1: Put the first equation in slope-intercept form

The slope-intercept form of a line is given by \( y = mx + b \), where \( m \) is the slope. Starting with the equation \( 3y + x = 12 \), we want to isolate \( y \).Subtract \( x \) from both sides: \( 3y = -x + 12 \).Then, divide every term by 3 to solve for \( y \):\[y = -\frac{1}{3}x + 4\].Here, the slope \( m_1 = -\frac{1}{3} \).

Step 2: Put the second equation in slope-intercept form

Starting with the equation \( -y = 8x + 1 \), the goal is to solve for \( y \).Multiply every term by -1: \( y = -8x - 1 \).Thus, the slope \( m_2 = -8 \).

Step 3: Compare the slopes to determine the relationship

Two lines are parallel if they have the same slope. They are perpendicular if the product of their slopes is \(-1\).First, compare the slopes from Step 1 and Step 2:\( m_1 = -\frac{1}{3} \) and \( m_2 = -8 \).Their product is:\[m_1 \cdot m_2 = \left(-\frac{1}{3}\right) \cdot (-8) = \frac{8}{3}.\]Since \( \frac{8}{3} eq -1 \), the lines are neither parallel nor perpendicular.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for working with linear equations in college algebra. The slope-intercept form of a line is expressed as \( y = mx + b \), where:

• \( y \) represents the dependent variable.
• \( x \) is the independent variable.
• \( m \) denotes the slope of the line, indicating its steepness and direction.
• \( b \) represents the y-intercept, which is the point where the line crosses the y-axis.

To convert an equation into slope-intercept form, you need to solve for \( y \). This involves mathematical operations such as addition, subtraction, multiplication, or division to isolate \( y \) on one side of the equation. For example, in transforming \( 3y + x = 12 \) into slope-intercept form, the goal is to make \( y \) stand alone:1. Subtract \( x \) from both sides to get \( 3y = -x + 12 \).2. Divide every term by 3, resulting in \( y = -\frac{1}{3}x + 4 \).The outcome is a clearer representation of the line's slope and intercept.

Parallel Lines

Parallel lines are significant in algebra and geometry because they maintain a constant distance between each other and will never intersect. How do we determine if two lines are parallel from their equations?

• Lines are parallel if their slopes are equal.
• Comparing the slopes of two equations in slope-intercept form helps identify if lines are parallel.

From the example given:- The first line, after simplification, has a slope \( m_1 = -\frac{1}{3} \).- The second line has a slope \( m_2 = -8 \).Since \( m_1 = -\frac{1}{3} \) is not equal to \( m_2 = -8 \), these lines are not parallel. Identifying parallel lines is straightforward when the equations are expressed in slope-intercept form.

Perpendicular Lines

Perpendicular lines intersect at a right angle, forming a perfect "T" shape. To determine if two lines are perpendicular:

• The product of their slopes should equal \(-1\).
To check if two lines are perpendicular, compare their slopes. If you calculate the slopes of two lines and multiply them together, the result tells you their relationship. In our example:- The slope of the first line \( m_1 = -\frac{1}{3} \).- The slope of the second line \( m_2 = -8 \).Multiply them:\[m_1 \times m_2 = \left(-\frac{1}{3}\right) \times (-8) = \frac{8}{3}\].Since \( \frac{8}{3} eq -1 \), the lines are not perpendicular. Recognizing perpendicular lines is key for solving many geometric problems in algebra.