/*! 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 28 Determine the slope of the line ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 28

Determine the slope of the line from the given equation of the line. \(y=3 x-2\)

Short Answer

Expert verified
The slope of the line represented by the equation \(y = 3x - 2\) is 3.

Step by step solution

01

Identify the standard form

Ensure that the provided equation is in the standard slope-intercept form, which is \(y = mx + c\). In this case, the equation \(y = 3x - 2\) is already in the standard form.
02

Identify the slope

From the standard form, we can identify the slope (\(m\)) and the y-intercept (\(c\)). The slope is the coefficient of \(x\), which in this case is 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.

Equation of a Line
An equation of a line is a mathematical representation that describes all the points along a particular straight line on a coordinate plane. There are various forms to represent this equation, such as:
  • Slope-Intercept Form (which we'll delve into more deeply later)
  • Point-Slope Form
  • Standard Form
These forms allow us to easily interpret crucial characteristics of the line, such as its steepness (slope) and where it intersects the axes. The primary purpose of the equation is to define the relationship between the x-coordinate and the y-coordinate for points that lie on the line.
In this content, we'll focus on the slope-intercept form, which is especially useful for quickly identifying the slope of a line.
Slope-Intercept Form
The slope-intercept form of a line's equation is written as: \[ y = mx + c \]In this format:
  • The letter \(m\) represents the slope of the line, which shows how much the line ascends or descends as we move along the x-axis.
  • The letter \(c\) represents the y-intercept, pointing out where the line crosses the y-axis.
This form is especially useful because it provides direct insights into the line's direction and position, simplifying the process of sketching the line on a graph. It also makes it easier to compare different lines and to calculate intersecting points between them.
Understanding this form is essential in mathematics problem-solving, making it possible to transition easily from an abstract equation to a visual graph.
Identifying Slope
Spotting the slope in a line's equation is a straightforward yet crucial task. The slope can be identified as the coefficient of the \(x\) term when an equation is expressed in the slope-intercept form: \(y = mx + c\).
The slope, indicated by \(m\), is the measure of how steep the line is and indicates whether it rises or falls:
  • A positive slope means the line ascends as it moves from left to right.
  • A negative slope indicates the line descends.
  • A zero slope signifies a perfectly horizontal line.
  • An undefined slope corresponds to a vertical line.
For the equation provided \(y = 3x - 2\), the slope \(m\) is \(3\), which tells us that the line rises 3 units for every 1 unit it moves to the right. This knowledge is foundational in assessing the behavior and interactions of lines, making it a key component in mathematics problem-solving.
Mathematics Problem-Solving
Effective mathematics problem-solving involves understanding and applying mathematical concepts to find solutions. When determining the equation of a line or its slope, a methodical approach aids vastly in ensuring accuracy and clarity. Here鈥檚 a simple problem-solving framework:
  • **Comprehend the Problem:** Understand what is given and what is required. For instance, identifying that you need to find the slope and that you have been given a line equation.
  • **Plan the Solution:** Decide on the approach, such as putting the equation in slope-intercept form to easily pinpoint the slope.
  • **Apply and Analyze:** Perform calculations to retrieve the slope and intercept. In our original exercise, we did this by checking the coefficient of \(x\) in the equation \(y = 3x - 2\).
  • **Review and Reflect:** Check if the solution is logical and whether all steps align with mathematical rules. Reflecting on the process can help in developing faster and more intuitive problem-solving strategies in the future.
Mastering these steps not only helps in solving problems efficiently but also fosters a deeper understanding of mathematical concepts, turning seemingly complex problems into manageable tasks.

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

Find an equation of the \(x\) -axis.

Write an equation of the line satisfying the following conditions. Write the equation in the form \(\mathrm{Ax}+\mathrm{By}=C\). It passes through (1,-3) and (-5,5) .

A company's revenue and cost in dollars are given by \(R=25 x\) and \(C=10 x+9,000\), where \(x\) represents the number of items. Find the number of items that must be produced to break-even.

Graph the line that passes through the given point and has the given slope. (2,3) and \(m=0\)

Both Jenny and Masur are sales people for Athletic Shoes. Jenny gets paid $$\$ 8$$ per hour plus $$4 \%$$ commission on the sales. Masur gets paid $$\$ 10$$ per hour plus $$8 \%$$ commission on the sales in excess of $$\$ 1,000$$. If they work 8 -hour days, for what sales amount would they both earn the same daily amounts?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

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.