/*! 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 1 What does it mean when we say th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 1

What does it mean when we say that two lines are parallel?

Short Answer

Expert verified
Two lines are parallel if they have the same slope and never intersect.

Step by step solution

01

Definition of Parallel Lines

Parallel lines are defined as lines in the same plane that do not intersect, regardless of how far they are extended.
02

Understanding the Slope

The slope of a line is a measure of its steepness. For two lines to be parallel, they must have the same slope.
03

Mathematical Condition for Parallel Lines

In mathematical terms, if two lines have slopes \(m_1\) and \(m_2\), they are parallel if and only if \(m_1 = m_2\), provided that the lines are in the same plane.

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.

Definition of Parallel Lines
Parallel lines are a fascinating concept to grasp. They are lines that exist in the same plane but never, ever cross each other. Imagine two railroad tracks extending endlessly beside each other; they make a perfect example of parallel lines. No matter how long you'd extend these lines in either direction, they would never intersect. This lack of intersection is what defines parallel lines, and it holds true for any pair of parallel lines in a plane.
Slope of a Line
To understand parallel lines, one should first grasp the concept of the slope of a line. The slope indicates how steep a line is, describing its angle relative to the horizontal axis. You can calculate a line's slope by taking the vertical change (rise) and dividing it by the horizontal change (run) between two points.
Some key points to remember about slopes:
  • A positive slope means the line rises as it moves to the right.
  • A negative slope means it falls as it moves to the right.
  • Zero slope signifies a perfectly horizontal line.
  • Undefined slope indicates a perfectly vertical line.
For lines to be parallel, their slopes must have a unique relationship, which is covered next.
Mathematical Condition for Parallelism
Mathematically speaking, parallel lines have a clear condition to fulfill. For two lines to be parallel, their slopes must be identical. This means if you have two lines with slopes represented by \(m_1\) and \(m_2\), then the lines are parallel only if \(m_1 = m_2\).
Why does this condition exist? Because having the same slope ensures two lines have the same angle with the horizontal axis, meaning they never converge or diverge. This is true as long as the lines lie within the same plane and the slopes are constant.
In summary, knowing that \(m_1 = m_2\) gives you the green light to call those lines parallel, simplifying your understanding and analysis of linear relationships.

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

For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions. $$ \sqrt{x-1}=x-7 $$

For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions. $$ \sqrt{t+1}+9=7 $$

\(\frac{x-5}{x+7} \leq 0, x \neq-7\)

Solve \(x^{2}-x>12\)

Input the left-hand side of the inequality as a \(\mathrm{Y} 1\) graph in your graphing utility. Enter \(\mathrm{Y} 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall ( \(2^{\text {nd }}\) CALC 5:intersection, 1st curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |x+2| \geq 5 $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.