/*! 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 23 Graph each linear equation. Plot... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 23

Graph each linear equation. Plot four points for each line. $$y=-2 x+3$$

Short Answer

Expert verified
Plot the points (0,3), (1,1), (2,-1), and (3,-3) and draw a line through them.

Step by step solution

01

Understand the equation

The given equation is a linear equation in the slope-intercept form, which is written as \( y = mx + b \). In this equation, \( m \) represents the slope and \( b \) represents the y-intercept.
02

Determine the slope and y-intercept

In the equation \( y = -2x + 3 \), the slope \( m \) is -2 and the y-intercept \( b \) is 3. This means the line will cross the y-axis at the point (0, 3).
03

Plot the y-intercept

Start by plotting the y-intercept (0, 3) on the graph.
04

Use the slope to find additional points

Since the slope is -2, it means that for every 1 unit increase in x, y decreases by 2 units. Use the slope to plot three more points. Start from (0, 3):- From (0,3), move right 1 unit to (1,3). Then move down 2 units to (1, 1). Plot this point.- From (1, 1), move right 1 unit to (2, 1) and then down 2 units to (2, -1). Plot this point.- From (2, -1), move right 1 unit to (3, -1) and then down 2 units to (3, -3). Plot this point.
05

Draw the line

Draw a straight line through all four points to graph the equation \( y = -2x + 3 \). Ensure the line crosses through each plotted point.

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-Intercept Form
The equation given in the exercise is in slope-intercept form. This form of a linear equation is written as \( y = mx + b \).
Here:
  • \(m\) stands for the slope of the line, which indicates its steepness or tilt.
  • \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.
For example, in the equation \( y = -2x + 3 \):
  • The slope \(m = -2\).
  • The y-intercept \(b = 3\).
Starting by understanding this form is crucial, as it allows you to quickly identify the slope and y-intercept, making the graphing process easier.

Plot Points
To graph a linear equation, we need points on the graph to connect them and form a line. In this exercise, we start by plotting the y-intercept. For \( y = -2x + 3 \), the y-intercept is (0, 3).
From this y-intercept, use the slope to find additional points.
Since the slope is -2, this means for every 1 unit increase in \(x\), \(y\) decreases by 2 units.
You start at (0, 3):
  • Move right 1 unit to (1, 3), then down 2 units to (1, 1).
  • Move right 1 unit to (2, 1), then down 2 units to (2, -1).
  • Move right 1 unit to (3, -1), then down 2 units to (3, -3).
Plot these points (1,1), (2,-1), and (3,-3) along with the y-intercept. Connecting these points with a straight line will give you the graph of the equation.

Slope
The slope of a line in the equation \( y = mx + b \) is represented by \(m\). The slope indicates how the value of \(y\) changes as \(x\) changes.
More explicitly:
  • 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.
  • A slope of 0 means the line is horizontal.
  • An undefined slope (where the equation is in the form \( x = c \)) means the line is vertical.
In the given equation \( y = -2x + 3 \), the slope is -2. This negative slope implies that for every increase of 1 unit in \(x\), \(y\) decreases by 2 units. Understanding the slope helps in plotting the line accurately and interpreting the relationship between the variables. By mastering the basic concept of slope, you can graph any linear equation with ease!

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 the equation of line \(l\). Write the answer in standard form with integral coefficient with a positive coefficient for \(x .\) See Example 8. Line \(l\) goes through \((4,-2)\) and is parallel to \(4 x+2 y=5\).

Graph each pair of lines in the same coordinate system using the slope and y-intercept. $$ \begin{array}{l} y=3 x+1 \\ y=-\frac{1}{3} x+1 \end{array} $$

Determine whether each pair of lines is parallel, perpendicular, or neither. $$9-x=3, \frac{1}{2} x=8$$

Graph \(y=1 x, y=2 x, y=3 x,\) and \(y=4 x\) together in the standard viewing window. These equations are all of the form \(y=m x .\) What effect does increasing \(m\) have on the graph of the equation? What are the slopes of these four lines?

Find \(k\) so that the line through \((2, k)\) and \((-3,-5)\) has slope \(\frac{1}{2}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Pure Maths

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.