/*! 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 60 For each pair of points, a. find... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 60

For each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. (-1,4) and (3,-1)

Short Answer

Expert verified
The slope is \(-\frac{5}{4}\) and the line is decreasing.

Step by step solution

01

Understanding the Slope Formula

To find the slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \), we use the formula: \\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]\Here, the given points are \( (-1, 4) \) and \( (3, -1) \). Let's apply these coordinates to the formula.
02

Calculating the Numerator

Substitute the given \(y \) values into the formula: \\[ y_2 - y_1 = -1 - 4 \]\Calculate the expression: \(-1 - 4 = -5\).
03

Calculating the Denominator

Substitute the \(x \) values into the formula: \\[ x_2 - x_1 = 3 - (-1) \]\Simplify by adding the opposite: \(3 + 1 = 4\).
04

Finding the Slope

Now apply the results into the slope formula: \\[ m = \frac{-5}{4} \]\The slope of the line through the points \( (-1, 4) \) and \( (3, -1) \) is \( \frac{-5}{4} \).
05

Classifying the Line

Since the slope \(m = \frac{-5}{4}\) is a negative number, the line is decreasing. This means it moves down as it goes from left to right.

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
The slope formula is a fundamental tool in algebra that helps us understand the steepness and direction of a line on a coordinate plane.
To compute the slope given two points, \((x_1, y_1)\) and \((x_2, y_2)\), we use the following formula:
  • \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
The slope is represented by \(m\), and this formula calculates the change in y (vertical change) over the change in x (horizontal change).
This can also be described as "rise over run," which reflects how much the line rises or falls as it moves horizontally. To apply the slope formula correctly:
  • Identify the coordinates of the two points.
  • Substitute the y-values into the numerator and the x-values into the denominator.
  • Perform the arithmetic operations to simplify both the numerator and the denominator.
The slope formula is a quick way to gauge the line's direction and intensity.
Decreasing Line
A decreasing line is a type of linear graph with a negative slope. This means that as you move from left to right across the graph, the line goes downwards.
It signifies that there is an inverse relationship between the x-values and y-values. In simpler terms, when x increases, y decreases.Recognizing a Decreasing Line:
  • If the slope \(m\) is negative, the line is decreasing.
  • This can be visualized when plotting the line: the line will point downhill.
For example, in our previous exercise, the slope calculated was \( \frac{-5}{4} \).
This negative fraction clearly indicates a decreasing line. Understanding a line's increasing or decreasing nature is crucial for predicting behavior within a dataset or a function's graph.
Linear Equations
Linear equations are equations of the first order, which means they graph as a straight line on the coordinate plane. The general form of a linear equation is \( y = mx + b \), where:
  • \( m \) is the slope of the line.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
Linear equations can model a variety of real-world situations due to their simple and predictable nature.In practical applications:
  • If the slope \(m\) is positive, the line increases.
  • If \(m\) is negative, the line decreases.
  • The y-intercept \(b\) indicates the value of y when x equals zero.
Using linear equations, you can visually represent relationships between variables and use this to make predictions or identify trends. This makes understanding and applying linear equations a powerful tool in both academic and real-world contexts.

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

[T] A condominium in an upscale part of the city was purchased for \(\$ 432,000\) . In 35 years it is worth \(\$ 60,500\) . Find the rate of depreciation.

Find the degree, \(y\) -intercept, and zeros for the following polynomial functions. $$ f(x)=2 x^{2}+9 x-5 $$

[T] A company purchases some computer equipment for \(\mathrm{S} 20,500\) , At the end of a 3 -year period, the value of the equipment has decreased linearly to \(\$ 12,300\) . a. Find a function \(y=V(t)\) that determines the value V of the equipment at the end of \(t\) years. b. Find and interpret the meaning of the \(x\) - and \(y-\) intercepts for this situation. c. What is the value of the equipment at the end of 5 years? d. When will the value of the equipment be \(\$ 3000 ?\)

For the following exercises, use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. $$ \ln a^{3} \sqrt{b} $$

For the following exercises, write the equation in equivalent logarithmic form. $$ 4^{-3 / 2}=0.125 $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Probability and 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.