/*! 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 What is the slope鈥搃ntercept fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 8

What is the slope鈥搃ntercept form of the equation of a line?

Short Answer

Expert verified
The slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step by step solution

01

Understanding Slope-Intercept Form

The slope-intercept form of a line is a way of expressing the equation of a line using its slope and y-intercept. The general formula is given by \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) represents the y-intercept, the point where the line crosses the y-axis.
02

Identifying Components

In the slope-intercept form \(y = mx + b\), \(m\) is the slope. It indicates how much \(y\) changes for a unit change in \(x\). \(b\) is the y-intercept which is the value of \(y\) when \(x = 0\). These components are used to fully describe the line.
03

Equation Derivation

To derive the equation in slope-intercept form, you need two main pieces of information: the slope \(m\) and the y-intercept \(b\). Once you have these values, you can directly substitute them into the formula \(y = mx + b\) to define the line.

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.

Linear Equations
Linear equations represent a straight line when plotted on a graph. They are foundational in algebra and involve variables that change continuously. In the general sense, a linear equation can be represented by the formula: \(y = mx + b\). Here, \(y\) and \(x\) are variables, and \(m\) and \(b\) are constants. Each unique set of \(m\) and \(b\) values will define a different straight line.
Linear equations are critical for:
  • Modeling real-life situations, such as calculating profit and loss over time.
  • Predicting trends and making projections.
  • Understanding relationships between two variables.
Remember, other forms of linear equations exist, like the standard form \(Ax + By = C\), but the slope-intercept form is often the most intuitive and practical for quick graphing.
Slope
The slope of a line is a numerical value that describes its steepness and direction. In the slope-intercept form \(y = mx + b\), the value of \(m\) represents the slope. The slope can be calculated by the formula \(m = \frac{\Delta y}{\Delta x}\), where \(\Delta y\) is the change in the vertical direction (rise) and \(\Delta x\) is the change in the horizontal direction (run).
The slope has different characteristics:
  • A positive slope \(m > 0\) means the line rises as it moves from left to right.
  • A negative slope \(m < 0\) means the line falls as it moves from left to right.
  • A slope of zero means the line is perfectly horizontal.
  • An undefined slope indicates a vertical line.
Understanding slope is crucial because it shows the relationship between variables. It indicates how a change in one variable impacts the other. A steeper slope signifies a larger effect and steeper ascent or descent.
Y-Intercept
The y-intercept is a fundamental part of the equation of a line in the slope-intercept form. It is represented by \(b\) in the formula \(y = mx + b\). The y-intercept is the point where the line crosses the y-axis. It essentially tells us the value of \(y\) when \(x = 0\).
Some key points about the y-intercept:
  • The y-intercept provides a starting point for graphing a line.
  • It can be used to quickly determine if a linear equation will pass through the origin (when \(b = 0\)).
  • The y-intercept helps visualize the point of intersection with the y-axis without needing a full graph.
The y-intercept is practical for understanding initial conditions in real-world scenarios, such as starting points or amounts, before time or variable change influences them.

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

In \(2000,\) there were approximately \(6,600\) students enrolled in dental assisting programs in the U.S. By \(2008,\) that number had steadily increased to about \(9,200\) students. Find the rate of change in the number of students studying to be dental assistants from 2000 to \(2008 .\) (Source: American Dental Education Association) (IMAGE CANNOT COPY)

$$ \text { Evaluate: }-4+5-(-3)-13 $$

Vacationing. The function \(C(d)=500+100(d-3)\) gives the cost in dollars to rent an RV motor home for \(d\) days. (The terms of the rental agreement require a 3 -day minimum.) Find the cost of renting the RV for a vacation that will last 7 days. CAN'T COPY THE IMAGE

Why is \(y=m x+b\) called the slope-intercept form of the equation of a line?

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((11,0)\) and \((11,-5)\) \((14,6)\) and \((25,6)\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

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.