/*! 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 41 Find the slope of the line conta... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 41

Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact. $$ (3,-7) \text { and }(3,-9) $$

Short Answer

Expert verified
The slope is undefined because the line is vertical.

Step by step solution

01

Identifying Points

We are given two points: \((3, -7)\) and \((3, -9)\). These points are formatted as \((x_1, y_1)\) and \((x_2, y_2)\) respectively.
02

Finding the Change in X

Subtract the x-coordinates of the points: \(x_2 - x_1 = 3 - 3 = 0\). This gives the run between the points.
03

Finding the Change in Y

Subtract the y-coordinates of the points: \(y_2 - y_1 = -9 - (-7) = -9 + 7 = -2\). This gives the rise between the points.
04

Calculating the Slope

The formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the values, we get \(m = \frac{-2}{0}\).
05

Determining Undefined Slope

Since division by zero is undefined, the slope \(\frac{-2}{0}\) is undefined. This means the line is vertical, and its slope cannot be calculated.

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 Formula
When determining the slope of a line, we rely on the slope formula. This essential concept helps us understand how steep a line is and in which direction it slants.
The slope formula is expressed as:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
where:
  • \( m \) is the slope of the line,
  • \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of two distinct points on the line.
To calculate the slope, the difference between the y-coordinates is divided by the difference between the x-coordinates. This difference tells us two things:
  • The change in the vertical direction (rise),
  • The change in the horizontal direction (run).
By using this simple formula, we can quickly find out whether a line is rising, falling, horizontal, or possibly vertical.
Vertical Line
When we talk about a vertical line in the context of slopes, we mean a line where all points along it have the same x-coordinate. Such lines show a unique characteristic:
  • Every point on a vertical line shares the same x-value.
In our exercise, for the points \((3, -7)\) and \((3, -9)\), both points have the same x-coordinate, which is 3. This indicates that the line is vertical.
The change in x-coordinates, or "run," for a vertical line is zero.
This results in a key consequence when using the slope formula, as division by zero occurs, which is mathematically undefined.
Undefined Slope
An undefined slope occurs when attempting to determine the slope of a vertical line. This term simply means that we cannot define the slope numerically:
  • In mathematics, dividing by zero is not possible.
  • A vertical line has an undefined slope because the run (change in x) is zero.
  • Substituting into the slope formula, the denominator \(x_2 - x_1 = 0\), leading to division by zero.
This concept is crucial when distinguishing between various types of lines:
  • Horizontal lines have a zero slope,
  • Lines sloping upwards or downwards have positive or negative slopes respectively,
  • But a vertical line with a constant x-coordinate has an undefined slope.
Understanding the reasons behind an undefined slope helps clarify why vertical lines defy typical slope calculations.

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

Use the ZERO feature or the INTERSECT feature to approximate the zeros of each function to three decimal places. $$ f(x)=x^{4}+4 x^{3}-36 x^{2}-160 x+300 $$

In January 2005,3143 manatees were counted in an aerial survey of Florida. In January 2011,4834 manatees were counted. (Source: Florida Fish and Wildlife Conservation Commission.) a) Using the year as the \(x\) -coordinate and the number of manatees as the \(y\) -coordinate, find an equation of the line that contains the two data points. b) Use the equation in part (a) to estimate the number of manatees counted in January \(2010 .\) c) The actual number counted in January 2010 was 5067. Does the equation found in part (a) give an accurate representation of the number of manatees counted each year? Why or why not?

Determine the domain of each function. $$ f(x)=\frac{x^{2}-4}{x+2} $$

Explain why the vertical-line test works.

Graph. $$ f(x)=\left\\{\begin{array}{ll} -6, & \text { for } x=-3 \\ -x^{2}+5, & \text { for } x \neq-3 \end{array}\right. $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.