Problem 39 - College Algebra

Question

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither. $$ \begin{aligned} 3 y+4 x &=12 \\ -6 y &=8 x+1 \end{aligned} $$

Step-by-Step Solution

Answer

The lines are parallel.

Step 1: Rewrite the Equations in Slope-Intercept Form

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope. First, solve the first equation \( 3y + 4x = 12 \) for \( y \). Subtract \( 4x \) from both sides to get \( 3y = -4x + 12 \), then divide by 3 to obtain \( y = -\frac{4}{3}x + 4 \). Thus, the slope \( m_1 \) is \(-\frac{4}{3}\).

Step 2: Convert the Second Equation into Slope-Intercept Form

For the second equation \(-6y = 8x + 1\), divide every term by \(-6\) to isolate \( y \). This gives \( y = -\frac{8}{6}x - \frac{1}{6} \) which simplifies to \( y = -\frac{4}{3}x - \frac{1}{6} \). Thus, the slope \( m_2 \) is also \(-\frac{4}{3}\).

Step 3: Compare the Slopes

For lines to be parallel, their slopes must be equal. For lines to be perpendicular, the product of their slopes should be \(-1\). In this case, the slopes \( m_1 \) and \( m_2 \) are both \(-\frac{4}{3}\). Since the slopes are equal, the lines are parallel.

Key Concepts

Slope-Intercept Form

The concept of slope-intercept form is crucial to understanding linear equations in a more visual way. This form is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. The slope \( m \) indicates the steepness and direction of the line, while \( b \) tells us where the line crosses the y-axis.

• Slope \( m \): It is the ratio of the change in \( y \) to the change in \( x \), often described as "rise over run".
• Y-intercept \( b \): This is the point where the line cuts through the y-axis, therefore \( x = 0 \).

When rewriting equations into slope-intercept form, the goal is to manipulate the equation such that it follows the format \( y = mx + b \). The process often involves isolating \( y \) on one side of the equation by using operations like addition, subtraction, multiplication, or division.

Perpendicular Lines

Understanding perpendicular lines can add another layer of complexity to working with linear equations. Perpendicular lines intersect at a right angle (90 degrees). In the context of their slopes, perpendicular lines have slopes that are negative reciprocals of each other.

• If one line has a slope \( m_1 \), a line perpendicular to it will have a slope \( m_2 \) such that \( m_1 \times m_2 = -1 \).
• For example, if one line's slope is \( 2 \), a perpendicular line's slope would be \( -\frac{1}{2} \).

Identifying perpendicular lines involves calculating the slopes and checking the negative reciprocal relationship. Remember, if the product of two slopes equals \(-1\), then the lines are definitely perpendicular.

Equation of a Line

The equation of a line can take various forms, but it's primarily a mathematical expression that describes all points along that line. Whether you're given a point and a slope or deciphering from a graph, understanding how these equations formulate the characteristics of lines is essential.

• Standard Form: This is another common form, represented as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.
• Point-Slope Form: Ideal for when you know a point on the line \((x_1, y_1)\) and the slope \( m \). It is written as \( y - y_1 = m(x - x_1) \).

Switching between these forms often involves algebraic manipulation, suited for various uses in different scenarios. For example, slope-intercept form \( y = mx + b \) is excellent for graphing, while point-slope form is practical when reconstructing an equation from a known set of points.