/*! 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 75 Find the slope of each line. T... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 75

Find the slope of each line. The line that passes through \((0,-8)\) and \((-5,0)\)

Short Answer

Expert verified
The slope of the line is \(-\frac{8}{5}\).

Step by step solution

01

Identify Coordinates

First, identify the given coordinates of the two points. In this exercise, the points are \((x_1, y_1) = (0, -8)\) and \((x_2, y_2) = (-5, 0)\).
02

Use the Slope Formula

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

Substitute the Coordinates into the Formula

Substitute the coordinates \((0,-8)\) and \((-5,0)\) into the slope formula: \[m = \frac{0 - (-8)}{-5 - 0} = \frac{8}{-5}\]
04

Simplify the Fraction

Simplify the fraction obtained:\[m = \frac{8}{-5} = -\frac{8}{5}\].The negative sign indicates the slope is negative.

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.

Understanding Coordinates
Coordinates are a fundamental concept in geometry and mathematics, used to pinpoint the position of a point on a plane. They are often written as ordered pairs \((x, y)\), where "x" represents a position along the horizontal axis and "y" represents a position along the vertical axis.
  • The first number in the ordered pair鈥攌nown as the x-coordinate鈥攖ells us how far to move horizontally from the origin.
  • The second number, or y-coordinate, tells us how far to move vertically.
In our example with the points \((0, -8)\) and \((-5, 0)\), the coordinates tell us exactly where these two points are on the coordinate plane. The first point is directly on the y-axis, and the second is on the x-axis.
Understanding how to read and locate these points helps immensely when calculating distances, slopes, and other geometric values.
Applying the Slope Formula
The slope of a line is a measure of its steepness and direction. To find it, you use the slope formula, which compares the "rise" (change in y) to the "run" (change in x) of the line.
The formula is expressed as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line.
  • The numerator \((y_2 - y_1)\) represents how much the line goes up or down.
  • The denominator \((x_2 - x_1)\) represents how much the line goes left or right.
Inserting the coordinates \((0, -8)\) and \((-5, 0)\) into the formula gives us: \[ m = \frac{0 - (-8)}{-5 - 0} \] This simplifies to \( m = \frac{8}{-5} \), showing that the line is falling as you move from left to right.
Simplifying Fractions in Slope
Simplifying fractions is crucial for expressing numbers in their simplest form. When you compute the slope, you often find yourself with a fraction that needs simplifying.
Let's revisit our slope calculation example: \[ m = \frac{8}{-5} \] This fraction is already simplest in terms of numerator and denominator, but the negative sign can be confusing.
  • A negative sign in a fraction can be placed with either the numerator, the denominator, or in front of the entire fraction, so \(\frac{8}{-5} = -\frac{8}{5}\).
  • This means that for every 8 units the line rises, it moves 5 units to the left, indicating the line's downward slope.
Keeping the slope in its simplest form helps in interpreting the mathematical relationship it represents, making it easier to graph and understand.

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 Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 2 x+y+z=5 \\ x-2 y+3 z=10 \\ x+y-4 z=-3 \end{array}\right. $$

Digital Photography. A digital camera stores the black and white photograph shown on the right as a \(512 \times 512\) matrix. Each element of the matrix corresponds to a small dot of grey scale shading, called a pixel, in the picture. How many elements does a \(512 \times 512\) matrix have?

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 4 x-3 y=-1 \\ 8 x+3 y=4 \end{array}\right. $$

Is the graph of a circle the graph of a function?

What is the slope-intercept form of the equation of a line?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

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.