/*! 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 39 Find the slope of the line conta... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 39

Find the slope of the line containing the given pair of points. If a slope is undefined, state that fact. $$(2,-3) \text { and }(-1,-4)$$

Short Answer

Expert verified
The slope is \frac{1}{3}\.

Step by step solution

01

- Understanding the Slope Formula

The slope of a line passing through two points \(x_1, y_1\) and \(x_2, y_2\) is given by the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}\].
02

- Identify the Coordinates

Here, the coordinates given are \(2, -3\) and \(-1, -4\). Assign \(x_1 = 2, y_1 = -3\) and \(x_2 = -1, y_2 = -4\).
03

- Substitute into the Slope Formula

Substituting the identified coordinates into the slope formula: \[m = \frac{-4 - (-3)}{-1 - 2} = \frac{-4 + 3}{-1 - 2} = \frac{-1}{-3}\].
04

- Simplify the Expression

Simplify the expression obtained: \[m = \frac{-1}{-3} = \frac{1}{3}\].
05

- State the Slope

The slope of the line passing through the points \(2, -3\) and \(-1, -4\) is \frac{1}{3}\.

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
Calculating the slope of a line is an essential skill in coordinate geometry. The slope of a line measures its steepness and direction. It's defined using two distinct points on the line. The slope formula is \[\begin{equation} m = \frac{y_2 - y_1}{x_2 - x_1}\tag{1}\end{equation}\]where
  • ( x_1, y_1 ) are the coordinates of the first point
  • ( x_2, y_2 ) are the coordinates of the second point
Using this formula, we can determine if a line rises, falls, or is horizontal or vertical. Remember:
  • A positive slope indicates a line rising from left to right
  • A negative slope indicates a line falling from left to right
  • A zero slope indicates a horizontal line
  • An undefined slope indicates a vertical line
The step-by-step solution applies this formula to the points ( 2, -3 ) and ( -1, -4 ), giving us a slope of \frac{1}{3}.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, describes the relationship between algebra and geometry through graphs and equations. Key concepts include:
  • Points: Defined by coordinates ( x, y ) on a plane
  • Lines: Defined by linear equations, slopes, and intercepts
  • Distance: Measures between points using the distance formula
  • Midpoint: Average of x and y coordinates of two points
In this exercise, you identify the coordinates of two points and use them within formulas to understand their geometric relationship. Recognizing coordinates allows us to compute various properties, like the slope, which tells us about the tilt or inclination of a line.For instance, in our problem, the points ( 2, -3 ) and ( -1, -4 ) are plotted on a coordinate plane. By connecting these points and applying the slope formula, we understand how steep the line is between them.
Linear Equations
Linear equations are the foundation of algebra and coordinate geometry. They represent straight lines and have a general format: \[\begin{equation}y = mx + c\tag{2}, \end{equation}\]where
  • y represents the dependent variable
  • x signifies the independent variable
  • m is the slope of the line
  • c is the y-intercept, where the line crosses the y-axis
The slope calculated in our example ( \frac{1}{3} ) and the points provided can also help us find that particular line's equation, but primarily, our focus here is on understanding slope.Linear equations help simplify complex relationships in algebra and geometry, making them easier to understand and visualize. By learning to calculate the slope, you gain a crucial step in deciphering and plotting these equations.

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 vertical-line test to determine whether each graph is that of a function. (The vertical dashed lines are not part of the graph.)

Find the equilibrium point for each pair of demand and supply functions. Demand: \(q=\frac{5}{x} ; \quad\) Supply: \(q=\frac{x}{5}\)

The amount of money, \(A(t),\) in a savings account that pays \(6 \%\) interest, compounded quarterly for \(t\) years, with an initial investment of \(P\) dollars, is given by $$A(t)=P\left(1+\frac{0.06}{4}\right)^{4 t}$$ If \(\$ 500\) is invested at \(6 \%\), compounded quarterly, how much will the investment be worth after 2 yr?

Solve for \(y\) in terms of \(x\). Decide whether the resulting equation represents a function. $$2 y^{2}+3 x=4 x+5$$

Find the equilibrium point for each pair of demand and supply functions. Demand: \(q=8800-30 x ;\) \(\quad\) Supply: \(q=7000+15 x\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.