/*! 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 Graph each linear equation using... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 28

Graph each linear equation using the slope and y-intercept. $$y=3 x+1$$

Short Answer

Expert verified
The line from the linear equation \(y = 3x + 1\) crosses the y-axis at the point \(0,1\) and has a slope of \(3\). For every step right along the x-axis, you move three steps up on the y-axis.

Step by step solution

01

Understand the y-intercept and the slope.

In the equation \(y = 3x + 1\), you have \(3\) as your slope and \(1\) as your y-intercept. The y-intercept is the point on the y-axis where the line crosses, and the slope is the steepness of the line.
02

Plot the y-intercept.

Start by plotting the y-intercept on your graph. The y-intercept for this equation is \(1\), which corresponds to the point \(0,1\). This is where the line will cross the y-axis.
03

Use the slope to draw the line.

Now, use the slope to determine the direction and steepness of the line. The slope of the line is \(3\), which means for each step right along the x-axis, you move three steps up on the y-axis. From the point \(0,1\), move to the point \(1,4\) and so on, as this adheres to the slope of \(3\). Draw the line through these points.

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 Slope-Intercept Form
In mathematics, the slope-intercept form is a straightforward and popular way to express the equation of a line. The form is given by the equation \( y = mx + b \). Here, \( m \) and \( b \) play crucial roles:
  • \( m \) represents the slope of the line, indicating how steep the line is.
  • \( b \) represents the y-intercept, the point at which the line crosses the y-axis.
This form is incredibly useful because you can graph a linear equation quickly with it. By identifying the slope and the y-intercept directly from the equation, you can easily plot the line on a graph.
In the example \( y = 3x + 1 \), identifying the slope and y-intercept allows you to graph the line with minimal effort. This is a helpful approach for students beginning to learn about graphing linear equations.
Identifying the Y-Intercept
The y-intercept is a key concept when graphing linear equations. It is the point where the line crosses the y-axis. In the equation \( y = 3x + 1 \), the y-intercept is \( 1 \). What this means is:
  • When \( x \) is zero, \( y \) equals \( 1 \).
  • This point is where you start plotting the graph.
To graph this equation, first plot the y-intercept as a point on the graph at \( (0, 1) \). This initial plotting provides a reference for positioning the rest of the line. The y-intercept simplifies the process of graph drawing, serving as the line's anchor.
Understanding the Slope
The slope of a line defines its steepness and direction. In the equation \( y = 3x + 1 \), the slope is \( 3 \). The slope indicates how much \( y \) changes with a change in \( x \). Here's what a slope of 3 means in practical terms:
  • For every 1 unit increase in \( x \), \( y \) increases by 3 units.
  • The slope provides the directional change of the line.
To create the full line from the initial point \( (0, 1) \) on the graph:
  • Move 1 unit to the right (along the x-axis).
  • Move 3 units up (along the y-axis).
Plot these new coordinates, such as \( (1, 4) \), and continue in this manner to extend the line. The slope is essential for constructing the linear equation graphically, helping you determine subsequent points along the line.

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

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The inequality \(2 x-3 y<6\) contains a "less than" symbol, so its graph lies below the boundary line.

Will help you prepare for the material covered in the first section of the next chapter. Is \((4,-1)\) a solution of both \(x+2 y=2\) and \(x-2 y=6 ?\)

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-\frac{5}{2} x+1$$

Solve for \(y\) and put the equation in slope-intercept form. $$y+3=-\frac{3}{2}(x-4)$$

Graph equation. \(12-3 x=0\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Pure Maths

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.