/*! 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 49 Graph the line with the given eq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 49

Graph the line with the given equation. \(y+2=2(x-1)\)

Short Answer

Expert verified
Graph the line of \( y = 2x - 4 \) using points (0,-4) and (1,-2).

Step by step solution

01

Simplify the Equation

Begin by simplifying the given equation to the slope-intercept form, which is \( y = mx + b \). Start by distributing the 2 on the right side: \[ y + 2 = 2(x - 1) \] This becomes: \[ y + 2 = 2x - 2 \] Next, subtract 2 from both sides to isolate \( y \): \[ y = 2x - 4 \] The equation is now in the form \( y = mx + b \) with \( m = 2 \) (the slope) and \( b = -4 \) (the y-intercept).
02

Identify the Y-Intercept

The y-intercept \( b \) is the point where the line crosses the y-axis. From the equation \( y = 2x - 4 \), we can see that \( b = -4 \). This tells us that the line crosses the y-axis at the point (0, -4). Plot this point on a graph.
03

Use the Slope to Find Another Point

The slope \( m = 2 \) means that for every 1 unit increase in \( x \), the value of \( y \) increases by 2 units. Starting from the y-intercept (0, -4), move 1 unit to the right to \( x = 1 \) and 2 units up to \( y = -2 \). This gives us another point on the line: (1, -2).
04

Draw the Line

With the points (0, -4) and (1, -2) plotted, draw a straight line through these points. Extend the line in both directions, using a ruler to ensure accuracy. This visualizes the graph of the equation \( y = 2x - 4 \).

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 slope-intercept form of a linear equation is particularly useful for graphing lines. It's written as \( y = mx + b \). Here, \( m \) represents the slope of the line, while \( b \) is the y-intercept. This form is straightforward because it clearly shows the slope and where the line crosses the y-axis. For beginners, converting any linear equation to this form simplifies the process of graphing. To identify the components quickly, look for the coefficient of \( x \) which gives you the slope, and the constant term which indicates the y-intercept. This makes plotting the line intuitive.
Slope
The slope of a line is a measure of its steepness and direction. From the equation \( y = mx + b \), \( m \) is the slope. A positive value indicates the line rises as it moves from left to right, while a negative value means it falls. A slope of zero creates a horizontal line, and an undefined slope corresponds to a vertical line.
Calculating the slope as 'rise over run' helps to visually plot the line. For instance, a slope of 2 means for every 1 unit you move to the right (run), you move 2 units up (rise). Understanding this concept assists in accurately plotting additional points once you know the y-intercept.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the equation \( y = mx + b \), \( b \) represents this intercept. For graphing, it acts as the initial point to start with on the graph. Simply locate this point on the y-axis. Knowing the y-intercept allows you to set the foundation for the line.
For example, if the y-intercept is -4, like in the exercise, you start plotting your line at point (0, -4). From there, using the slope, you can determine the direction of your line. Defining this intercept accurately is crucial for drawing the correct graph.
Graphing Techniques
Graphing a line involves plotting it accurately on the coordinate plane. After identifying the slope and y-intercept, you can begin sketching the line. Start at the y-intercept point and use the slope to find additional points. Plot the initial point (0, -4) based on the intercept. Then apply the slope, in this case, 2, to move from one point to the next.
Use a ruler to connect these points with a straight line. Extending the line in both directions showcases the linear equation fully. Ensuring precision when plotting results in more accurate and easy-to-read graphs. Small adjustments can lead to big differences in understanding linear relationships on a graph.

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 coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with the given equation. Then graph the hyperbola. $$ y^{2}-3 x^{2}+6 y+6 x-18=0 $$

REVIEW To begin a game, Nate must randomly draw a red, blue, green, or yellow game piece, and a tile from a group of 26 tiles labeled with all the letters of the alphabet. What is the probability that Nate will draw the green game piece and a tile with a letter from his name? $$ \begin{array}{ll}{\mathbf{F} \frac{1}{26}} & {\mathbf{H} \frac{3}{52}} \\\ {\mathbf{G} \frac{1}{13}} & {\mathbf{J} \frac{1}{2}}\end{array} $$

Find the coordinates of the vertices and foci and the equations of the asymptotes for the hyperbola with the given equation. Then graph the hyperbola. $$ \frac{(x+1)^{2}}{4}-\frac{(y+3)^{2}}{9}=1 $$

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{y^{2}+x^{2}=9} \\ {y=7-x}\end{array} $$

Graph the line with the given equation. \(y=\frac{1}{2} x\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.