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Chapter 10: Problem 21

In Exercises 21 to \(24,\) state whether the lines are parallel, perpendicular, the same (coincident), or none of these. $$2 x+3 y=6 \text { and } 2 x-3 y=12$$

Short Answer

Expert verified
The lines are neither parallel nor perpendicular.

Step by step solution

01

Convert Equations to Slope-Intercept Form

First, we need to convert each equation into the slope-intercept form, \( y = mx + b \), where \( m \) is the slope. This helps us determine the relationship between the lines.\(\)The first equation is \( 2x + 3y = 6 \). Solve for \( y \):\(\)\( 3y = -2x + 6 \)\(\)\( y = -\frac{2}{3}x + 2 \)\(\)The second equation is \( 2x - 3y = 12 \). Solve for \( y \):\(\)\( -3y = -2x + 12 \)\(\)\( y = \frac{2}{3}x - 4 \).
02

Identify Slopes of the Lines

In the slope-intercept form \( y = mx + b \), \( m \) represents the slope.\(\)For the first equation, \( y = -\frac{2}{3}x + 2 \), the slope \( m_1 = -\frac{2}{3} \).\(\)For the second equation, \( y = \frac{2}{3}x - 4 \), the slope \( m_2 = \frac{2}{3} \).
03

Determine Relationship Using Slopes

We analyze the slopes to determine if the lines are parallel, perpendicular, or neither.\(\)Two lines are parallel if their slopes are equal: \( m_1 = m_2 \).\(\)Two lines are perpendicular if the product of their slopes is \(-1\), i.e., \( m_1 \cdot m_2 = -1 \).\(\)In this case, the slopes are \( -\frac{2}{3} \) and \( \frac{2}{3} \). They are not equal, so the lines are not parallel.\(\)The product of the slopes is \( -\frac{2}{3} \times \frac{2}{3} = -\frac{4}{9} \), which is not \(-1\), so the lines are not perpendicular.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form of a linear equation is a way to express the equation to easily identify the slope and the y-intercept. It is written as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept 鈥� the point where the line crosses the y-axis.
This form is particularly useful because it directly gives us the slope of the line, which is crucial for understanding the angle or steepness of the line.
Moreover, by knowing the slope and y-intercept, you can quickly graph the line on a coordinate plane.
  • To convert an equation to slope-intercept form, solve for \( y \) on one side of the equation.
  • This typically involves isolating \( y \) by following basic algebra rules, like dividing and subtracting terms on both sides.
Understanding the slope-intercept form is essential for analyzing and comparing lines easily.
Slope of a Line
The slope of a line is a measure of how steep the line is. It tells us how much the y-value changes for a unit change in x-value.
In the slope-intercept form \( y = mx + b \), the slope \( m \) is the coefficient of \( x \).
This number can be positive, negative, or even zero.
  • If the slope is positive, the line rises to the right.
  • If the slope is negative, the line falls to the right.
  • A slope of zero means the line is horizontal.
The slope is also referred to as the "rate of change" and is calculated as the ratio of vertical change to horizontal change between two points on the line.
It's expressed mathematically as \( m = \frac{{\Delta y}}{{\Delta x}} \), where \( \Delta y \) is the change in y and \( \Delta x \) is the change in x.
Relationship Between Lines
Understanding the relationship between lines involves comparing their slopes. The slope is the key factor in determining whether two lines are parallel, perpendicular, or neither.
Parallel lines have the same slope, meaning they run in the same direction but never meet.
This is expressed as \( m_1 = m_2 \).
Perpendicular lines, on the other hand, have slopes that are negative reciprocals of each other.
This relationship is represented by \( m_1 \times m_2 = -1 \).
  • If neither of these conditions is met, the lines are neither parallel nor perpendicular.
  • Identifying if lines are parallel or perpendicular is useful in geometry and when solving systems of equations.
These relationships are fundamental concepts in algebra and allow students to solve problems involving geometric shapes and graphs more effectively.

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