/*! 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 71 Find the equation of each line. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 71

Find the equation of each line. Write the equation in standard form unless indicated otherwise. Through \((5,-6) ;\) perpendicular to \(y=9\)

Short Answer

Expert verified
The equation of the line is \(x = 5\).

Step by step solution

01

Understand the Given Line

The given line is \( y = 9 \), which is a horizontal line. This means it has an equation of the form \( y = c \) where \( c \) is a constant. The slope of a horizontal line is zero.
02

Find the Slope of the Perpendicular Line

Lines that are perpendicular to horizontal lines are vertical lines. Vertical lines have undefined slopes and are described by equations of the form \( x = a \), where \( a \) is a constant.
03

Use the Given Point

The given point is \((5, -6)\). Since the line is vertical (perpendicular to a horizontal line), the equation of the line will be \( x = 5 \). A vertical line passing through the point \((5, -6)\) will have this equation because its x-coordinate is constant.

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.

Equation of a line
Lines can be represented by equations that describe their slope and position in a coordinate plane. The most common format for a line's equation is the slope-intercept form:
  • The slope-intercept form is given by \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept, the point where the line crosses the y-axis.
  • The slope defines the steepness or incline of the line, indicating how much the line rises or falls for every unit it progresses horizontally.
  • Another form is the point-slope form \( y - y_1 = m(x - x_1) \), which is useful when you know a point on the line \((x_1, y_1)\) and the slope \( m \).
In the context of the original exercise, we need a line perpendicular to \( y = 9 \). Horizontal lines like \( y = 9 \) have no slope change, leading to undefined perpendicular slopes, indicating a vertical line. Thus, our perpendicular line's equation will focus solely on a constant x-value. Understanding how to identify when a line is horizontal or vertical allows us to write appropriate linear equations.
Vertical lines
Vertical lines run up and down the graph parallel to the y-axis. Unlike other lines, vertical lines span all values of y for a single x-value without deviation:
  • The equation of a vertical line is written as \( x = a \), where \( a \) is the constant x-value for every point on the line.
  • Vertical lines are significant because they differ from any other line in that they have an undefined slope; this is due to the fact that a vertical rise does not result in any horizontal change.
  • These lines are perpendicular to horizontal lines, making them quite crucial in geometry problems that involve perpendicularity.
In solving the step-by-step problem, recognizing the perpendicularity of the vertical line to the horizontal line \( y = 9 \) is essential. Specifically, the line that is perpendicular to \( y = 9 \) and passes through the point \((5, -6)\) is simply \( x = 5 \), as it remains constant no matter what the y-coordinate is.
Standard form equation
Writing equations in standard form provides a clean, organized way to present linear equations. In standard form, the equation of a line appears as:
  • \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A > 0 \).
  • Unlike the slope-intercept form, standard form does not provide direct information about the slope or y-intercept. However, it is useful for easily identifying vertical or horizontal lines and simplifying addition or subtraction of two lines.
  • While our focus was primarily on vertical lines in this exercise, converting between forms is a handy skill. In the case of the equation \( x = 5 \), converting it to standard form results in \( 1x + 0y = 5 \).
Although writing vertical lines in standard form can sometimes feel redundant, especially when used as \( x = a \), acknowledging and understanding this form enhances comprehension of the different ways to express a line's equation. By recognizing the utility and characteristics of standard form, geometric problem-solving becomes more versatile and structured.

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

Discuss whether a horizontal line ever has an \(x\) -intercept.

The dosage in milligrams D of Ivermectin, a heartworm pre ventive, for a dog who weighs \(x\) pounds is given by $$D(x)=\frac{136}{25} x$$ Find the proper dosage for a dog that weighs 50 pounds.

From the Chapter 3 opener, we have two functions to describe the percent of college students taking at least one online course. For both functions, \(x\) is the number of years since 2000 and \(y\) (or \(f(x)\) or \(g(x))\) is the percent of students taking at least one online course. $$f(x)=2.7 x+4.1 \text { or } g(x)=0.07 x^{2}+1.9 x+5.9$$ Assume the trend of \(f(x)\) continues. Find \(f(16)\) and describe in words what this means.

Broyhill Furniture found that it takes 2 hours to manufacture each table for one of its special dining room sets. Each chair takes 3 hours to manufacture. A total of 1500 hours is available to produce tables and chairs of this style. The linear equation that models this situation is \(2 x+3 y=1500\) where \(x\) represents the number of tables produced and \(y\) the number of chairs produced. a. Complete the ordered pair solution \((0,)\) of this equation. Describe the manufacturing situation this solution corresponds to. b. Complete the ordered pair solution \((, 0)\) for this equation. Describe the manufacturing situation this solution corresponds to. c. If 50 tables are produced, find the greatest number of chairs the company can make.

The yearly cost of tuition (in-state) and required fees for attending a public two-year college full time can be estimated by the linear function \(f(x)=64 x+2083,\) where \(x\) is the number of years after 2000 and \(f(x)\) is the total cost. (Source: The College Board) a. Use this function to approximate the yearly cost of attending a two-year college in the year \(2016 .\) [Hint: Find \(f(16) .]\) b. Use the given function to predict in what year the yearly cost of tuition and required fees will exceed 3200 dollars. I Hint: Let \(f(x)=3200,\) solve for \(x,\) then round your solution up to the next whole year. \(]\) c. Use this function to approximate the yearly cost of attending a two-year college in the present year. If you attend a two-year college, is this amount greater than or less than the amount that is currently charged by the college you attend?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

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.