/*! 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 66 What is the slope-intercept form... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 66

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

Short Answer

Expert verified
The slope-intercept form is \( y = mx + b \).

Step by step solution

01

Identify the Formula

The slope-intercept form of a linear equation is a standard way of representing a straight line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept.
02

Understand the Slope \( m \)

The slope \( m \) refers to the steepness and direction of the line. It is calculated as the change in \( y \) divided by the change in \( x \) between two distinct points on the line, often written as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
03

Recognize the Y-Intercept \( b \)

The y-intercept \( b \) is the value of \( y \) when the line crosses the y-axis. This occurs when \( x = 0 \).
04

Formulating the Equation

Once you have the slope \( m \) and the y-intercept \( b \), substitute these values into the slope-intercept formula to construct the linear equation: \( y = mx + b \).

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
The concept of slope is fundamental to understanding how lines behave on a graph. In the slope-intercept form, represented as \( y = mx + b \), \( m \) denotes the slope. Think of it as the measure of how steep a line is.

The slope tells us how much \( y \) increases or decreases as \( x \) increases by one unit. This is often described with the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on the line.

  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls as it moves from left to right.
  • If the slope is zero, the line is horizontal.
  • An undefined slope indicates a vertical line.
Y-Intercept
The y-intercept is another key part of the slope-intercept equation. It's a simple, yet crucial concept to grasp: it represents the point where the line crosses the y-axis, which is the point where \( x = 0 \).

In the slope-intercept equation \( y = mx + b \), \( b \) is the y-intercept. Visually, it provides a starting point from which the slope can direct the line.

  • If \( b > 0 \), the intercept is above the origin.
  • If \( b < 0 \), the intercept is below the origin.
  • If \( b = 0 \), the line passes through the origin.
Linear Equation
A linear equation represents a straight line and is one of the simplest forms of algebraic expressions.

The slope-intercept form \( y = mx + b \) is a standardized way of writing a linear equation. Each linear equation has exactly one slope and one y-intercept, making it predictable and easy to graph.

  • Linear equations model real-world situations involving constant rates of change.
  • They can be used to describe relationships between variables or to predict future values based on existing data.
  • Understanding the linear equation helps in solving problems related to distance, time, speed, and more.

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. $$ \left\\{\begin{array}{l} y=\frac{-2 x+1}{3} \\ 3 x-2 y=8 \end{array}\right. $$

The owner of a candy store wants to mix some peanuts worth \(\$ 3\) per pound, some cashews worth \(\$ 9\) per pound, and some Brazil nuts worth \(\$ 9\) per pound to get 50 pounds of a mixture that will sell for \(\$ 6\) per pound. She uses 15 fewer pounds of cashews than peanuts. How many pounds of each did she use?

Mixing Coffee. How many pounds of regular coffee (selling for \(\$ 4\) per pound) and how many pounds of Kona coffee (selling for \(\$ 11.50\) per pound) must be combined to get 20 pounds of a mixture worth \(\$ 6\) per pound?

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 2 x+3 y=0 \\ 4 x-6 y=-4 \end{array}\right. $$

Are the lines described by \(y=2 x-7\) and \(x-2 y=7\) perpendicular?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.