/*! 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 8 Find the slope of the line passi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 8

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. (-4,-3) \text { and }(-2,-5)

Short Answer

Expert verified
The slope is \( m = -1 \).

Step by step solution

01

- Identify the Points

The given points are (-4, -3) and (-2, -5). Let's label them as Point 1 \(x_1, y_1\) and Point 2 \(x_2, y_2\). Therefore, \(x_1 = -4, y_1 = -3, x_2 = -2\), and \(y_2 = -5\).
02

- Write the Slope Formula

Recall that the formula for the slope \(m\) of a line passing through two points \( (x_1, y_1)\) and \( (x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
03

- Substitute the Values into the Formula

Substitute \(x_1 = -4, y_1 = -3, x_2 = -2\), and \(y_2 = -5\) into the slope formula: \[ m = \frac{-5 - (-3)}{-2 - (-4)} \]
04

- Simplify the Numerator and Denominator

Simplify the expressions in the numerator and denominator: \[ m = \frac{-5 + 3}{-2 + 4} \]
05

- Perform the Subtraction

Calculate the differences: \[ m = \frac{-2}{2} \]
06

- Divide to Find the Slope

Complete the division to find the slope: \[ m = -1 \]

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.

coordinate geometry
Coordinate geometry, also known as analytic geometry, allows us to describe geometric figures using a coordinate system. In a coordinate plane, every point is defined by an ordered pair \(x, y\), which represents its location. For instance, the point (-4, -3) means that -4 is the x-coordinate (horizontal direction) and -3 is the y-coordinate (vertical direction). Understanding points and their locations is essential for solving problems related to lines, distances, and slopes.

Coordinate geometry forms the basis for calculating various properties of geometric shapes, such as slopes, which indicate the steepness and direction of a line.
slope calculation
The slope of a line is a measure of its steepness. Mathematically, the slope is the ratio between the vertical change (rise) and the horizontal change (run) between two points on a line. The slope formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

In our example, the points are \(x_1, y_1 = \ (-4, -3)\) and \(x_2, y_2 = \ (-2, -5)\). By substituting these coordinates into the slope formula, we calculate:

\[ m = \frac{-5 - (-3)}{-2 - (-4)} = \frac{-5 + 3}{-2 + 4} = \frac{-2}{2} = -1 \]

Thus, the slope (m) is -1, indicating that for every unit we move horizontally to the right, we move one unit downward.
algebraic expressions
An algebraic expression represents mathematical relationships using variables, numbers, and operation symbols. In the context of slope calculation, we use algebraic expressions to substitute the given coordinates into the slope formula.

For example:
  • Given \ x_1, y_1 = (-4, -3) \ and \ x_2, y_2 = (-2, -5)

The slope formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] becomes an algebraic expression when we replace the variables with actual coordinates. This substitution and subsequent simplification help us solve for the slope, as seen when we evaluated:
  • Simplified numerator and denominator: \ \frac{-5 + 3}{-2 + 4} = \ \frac{-2}{2}
  • Final division to find the slope: \ -1
two-point formula
The two-point formula is a fundamental method in coordinate geometry for calculating the slope of a line passing through two distinct points. This formula is pivotal when you need to determine the slope given only two coordinates. It is expressed as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using our example:
  • Points: (-4, -3) and (-2, -5)

Substitute the values into the formula:
\[ m = \frac{-5 - (-3)}{-2 - (-4)} \]
Simplify the expression:
\[ m = \frac{-5 + 3}{-2 + 4} = \frac{-2}{2} = -1 \] The negative slope indicates the line is decreasing about the x-axis, meaning it goes downwards as we move from left to right.

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

What is the slope of a line that is parallel to the line whose equation is \(4 y-5 x=12 ?\)

Solve for \(x: \frac{x}{3}-\frac{x-1}{4}=\frac{x+1}{2}\)

Suppose that the monthly cost of maintaining a car is linearly related to the number of miles driven during that month, and that for a particular model, on average, it costs 160 dollar per month to maintain a car driven 540 miles and 230 dollar per month to maintain a car driven 900 miles. (a) Write an equation relating the cost \(c\) of monthly maintenance and the number \(n\) of miles driven that month. (b) What would be the average cost of maintaining a car driven 750 miles per month? (c) On the basis of this relationship, if your average monthly maintenance is 250 dollar how many miles do you drive per month? (d) Sketch a graph of this equation using the horizontal axis for \(n\) and the vertical axis for \(c\) (e) What is the \(c\) -intercept of this graph? How would you interpret the fact that even if no miles are driven there is still a monthly maintenance cost?

This exercise discusses the relationship between the slopes of perpendicular lines. (a) Sketch the graphs of \(y=2 x+4\) and \(y=-\frac{1}{2} x+4\) on the same coordinate system. (b) On the basis of your graph, does it appear that these lines are perpendicular? (c) What is the relationship between the slopes of these two lines? (d) It is a fact that the slopes of perpendicular lines are negative reciprocals of each other (provided that neither of the lines is vertical). What is the slope of a line perpendicular to the line whose equation is \(y=\frac{2}{5} x+7 ?\)

Round off to the nearest hundredth when necessary. In physiology a jogger's heart rate \(N\), in beats per minute, is related linearly to the jogger's speed \(s .\) A certain jogger's heart rate is 80 beats per minute at a speed of \(15 \mathrm{ft} / \mathrm{sec}\) and 82 beats per minute at a speed of \(18 \mathrm{ft} / \mathrm{sec} .\) (a) Write an equation relating the jogger's speed and heart rate. (b) Predict this jogger's heart rate if she jogs at a speed of \(20 \mathrm{ft} / \mathrm{sec}\). (c) According to the equation obtained in part (a), what is the jogger's heart rate at rest? [Hint: At rest the jogger's speed is 0.]
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.