/*! 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 5 Determine the slope of parallel ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 5

Determine the slope of parallel lines and perpendicular lines. $$ y=-58 x-12 $$

Short Answer

Expert verified
Slope of parallel lines: -58; slope of perpendicular lines: \(\frac{1}{58}\).

Step by step solution

01

Identify the Slope of the Given Line

The given equation is in the slope-intercept form, which is \( y = mx + b \), where \( m \) represents the slope. Here, the equation is \( y = -58x - 12 \). Thus, the slope \( m \) of this line is \( -58 \).
02

Determine the Slope of Parallel Lines

Parallel lines share the same slope. Therefore, any line parallel to the given line will have a slope of \( -58 \).
03

Determine the Slope of Perpendicular Lines

Perpendicular lines have slopes that are negative reciprocals of each other. To find the negative reciprocal of \( -58 \), flip the sign and take the reciprocal: The reciprocal of \( -58 \) is \( \frac{1}{58} \). Thus, the slope of a line perpendicular to the given line is \( \frac{1}{58} \).

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-Intercept Form
The slope-intercept form is a familiar and convenient way to express linear equations. It is written as \( y = mx + b \), where:
  • \( m \) represents the slope of the line. The slope determines the steepness and direction of the line. A positive slope moves upward, while a negative one moves downward as it goes from left to right.
  • \( b \) represents the y-intercept. This is the point where the line crosses the y-axis.
Given the equation \( y = -58x - 12 \), we can see it follows the slope-intercept form, making it easy to identify the slope as \( m = -58 \). This tells us that for every unit increase in \( x \), \( y \) decreases by 58 units, forming a steep descending line.
Negative Reciprocal
The concept of a negative reciprocal is crucial for understanding perpendicular lines.
  • A reciprocal of a number is simply flipping the numerator and the denominator. For example, the reciprocal of 2 is \( \frac{1}{2} \).
  • To find the negative reciprocal, you change its sign as well. So, the negative reciprocal of 2 is \( -\frac{1}{2} \).
In the context of our exercise, the slope of the given line is \( -58 \). To find the slope of the perpendicular line, calculate the negative reciprocal: The reciprocal is \( \frac{1}{58} \) and flipping the sign gives us \( \frac{1}{58} \) since the original slope is already negative. This means any line perpendicular to the original will rise gently to the right.
Perpendicular Lines
Perpendicular lines intersect at a right angle, which is 90 degrees.
  • In terms of slopes, perpendicular lines have slopes that are negative reciprocals of each other. This relationship ensures their intersection forms the precise angle needed.
  • If one line's slope is \( -58 \), as we calculated, a line perpendicular to it must have a slope of \( \frac{1}{58} \). This is derived by taking the negative reciprocal.
  • Therefore, even though these lines meet at one point, they have distinctly different directions due to how their slopes relate.
Understanding this concept is helpful not only in geometry but also in various real-life applications such as urban planning, architecture, and designing intersections where right angles are common.

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

Express in slope-intercept form and identify the slope and y-intercept. $$ 12 x-4 y=8 $$

Determine the slope, given two points. \((1 / 2,-2 / 3)\) and \((1 / 4,-1 / 3)\)

Find the intercepts and graph them. $$ 3 x-4 y=12 $$

The cost in dollars of producing pens with a company logo is given by the function \(C(x)=1.65 x+120,\) where \(x\) is the number of pens produced. Use the function to calculate the cost of producing 200 pens.

Graph the linear function and state the domain and range. $$ C(x)=50+100 x $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

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.