/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 Determine whether the lines are ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 52

Determine whether the lines are parallel, perpendicular, or neither. See Examples 5 and 6. A line passing through \((-5,4)\) and \((5,4)\) A line passing through \((-6,-2)\) and \((3,-2)\)

Short Answer

Expert verified
The lines are parallel.

Step by step solution

01

Determine the Slope of the First Line

To find the slope of the line passing through \((-5, 4)\) and \((5, 4)\), we use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1) = (-5, 4)\) and \((x_2, y_2) = (5, 4)\). Calculating the slope: \( m = \frac{4 - 4}{5 - (-5)} = \frac{0}{10} = 0 \). Hence, the slope of the first line is 0.
02

Determine the Slope of the Second Line

For the line passing through \((-6, -2)\) and \((3, -2)\), we use the same slope formula. Here, \((x_1, y_1) = (-6, -2)\) and \((x_2, y_2) = (3, -2)\). The slope is calculated as \( m = \frac{-2 - (-2)}{3 - (-6)} = \frac{0}{9} = 0 \). Thus, the slope of the second line is 0.
03

Analyze the Slopes

Both lines have a slope of 0, which indicates that they are both horizontal lines. Lines with the same slope are parallel.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Slope of a Line
The concept of the slope of a line is essential in understanding its steepness and direction. The slope, often represented by the letter "m," is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \(x_1, y_1\) are the coordinates of the first point, and \(x_2, y_2\) are the coordinates of the second point on the line.
  • Positive Slope: The line rises as it moves from left to right.
  • Negative Slope: The line falls as it moves from left to right.
  • Zero Slope: The line is horizontal.
  • Undefined Slope: The line is vertical.
Knowing these categories helps in determining the behavior of the line quickly.
A zero slope, for instance, is indicative of a horizontal line, which is parallel to the x-axis.
Horizontal Line
A horizontal line is a line that runs left to right across a graph. Its defining feature is that it has a constant y-value for all x-values, leading to a slope of zero, since the change in y is zero.
In the problem, both lines are horizontal as they share the same y-coordinates for their respective endpoints:
  • The first line passes through points \((-5, 4)\) and \((5, 4)\).
  • The second line passes through points \((-6, -2)\) and \((3, -2)\).
These properties make horizontal lines particularly easy to identify and work with. They are always parallel to the x-axis and to each other when on the same plane.
Line Equations
Line equations provide a mathematical way to describe a line, and they typically come in the form of the slope-intercept equation: \[ y = mx + b \]where "m" is the slope and "b" represents the y-intercept, which is the point where the line crosses the y-axis.
For horizontal lines, the slope \(m\) is zero, thus the equation simplifies to:\[ y = b \]Here, 'b' is the y-coordinate of any point through which the horizontal line passes. Hence:
  • The first line has an equation of \(y = 4\).
  • The second line has an equation of \(y = -2\).
These simplified equations reflect the constant y-value across all x-values and affirm the parallel nature of the geometric features of the problem.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the slope of the line that passes through \((a, b)\) and \((-b,-a) .\) Assume \(a \neq 0\) and \(b \neq 0\)

Find \(h(5)\) and \(h(-2)\). \(h(x)=|x|+2\)

Explain why 8 is not in the domain of the function \(f(x)=\frac{5 x-7}{x-8}\).

Let \(f(x)=-2 x+5 .\) For what value of \(x\) does function \(f\) have the given value? \(f(x)=5\)

Consider the function defined by \(y=6 x+4 .\) Why do you think \(x\) is called the independent variable and \(y\) the dependent variable?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.