/*! 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 22 Find three solutions to each of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 22

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$4 x+2 y=4$$

Short Answer

Expert verified
The solutions are (0, 2), (1, 0), and (2, -2). Plot these on a graph and connect them with a line.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

To make it easier to find points to plot, rewrite the equation in the form of \(y = mx + b\). Start by isolating \(y\) in the equation \(4x + 2y = 4\). Subtract \(4x\) from both sides to get \(2y = -4x + 4\). Next, divide each term by 2 to solve for \(y\), resulting in \(y = -2x + 2\).
02

Choose Values for "x"

Select three different values for \(x\) to find corresponding \(y\) values. Commonly chosen values are \(x = 0\), \(x = 1\), and \(x = 2\).
03

Calculate "y" for Each Chosen "x" Value

Use the equation \(y = -2x + 2\) to calculate \(y\) for each chosen \(x\). For \(x = 0\), \(y = -2(0) + 2 = 2\). For \(x = 1\), \(y = -2(1) + 2 = 0\). For \(x = 2\), \(y = -2(2) + 2 = -2\).
04

Plot the Points on the Graph

With the points calculated, plot them on a graph: (0, 2), (1, 0), and (2, -2). These points should align in a straight line because the equation represents a linear function.
05

Draw the Graph

Connect the points with a straight line to represent the graph of the linear equation \(4x + 2y = 4\). Extend the line in both directions and include arrows to indicate it continues indefinitely.

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 is a way of writing the equation of a line so it is very easy to understand and use. In this form, the equation of the line looks like this: \[ y = mx + b \]This might seem a bit abstract, so let's break it down:
  • \( y \) is the value on the y-axis that you are solving for, it changes depending on the x value.
  • \( m \) represents the slope of the line. The slope is the steepness or tilt of the line. It's essentially how much \( y \) increases or decreases as \( x \) increases.
  • \( x \) is the value on the x-axis, the independent variable that you choose.
  • \( b \) is the y-intercept. This is the point where the line crosses the y-axis. It's the value of \( y \) when \( x \) is zero.
To rewrite an equation in this form, you need to isolate \( y \), which allows you to read the slope \( m \) and y-intercept \( b \) straight from the equation. For example, the equation \( 4x + 2y = 4 \) becomes \( y = -2x + 2 \) after simplifying, with a slope of \(-2\) and y-intercept at 2.
Graphing Linear Equations
Graphing linear equations involves visualizing the solutions on a 2D grid. The slope-intercept form makes graphing straightforward because it directly shows you where the line crosses the y-axis and its slope.
The graph of a linear equation is always a straight line. To graph an equation like \( y = -2x + 2 \), start by identifying the y-intercept, which here is 2. Plot this point on the y-axis.
Next, use the slope
  • A slope of \(-2\) means for every step to the right (positive x direction), the line moves two steps down.
  • From the y-intercept (0,2), move to the right one unit and down two units to plot the next point (1,0).
Continue using the slope to find additional points. Once you've plotted a few points, draw a line through them. Extend the line in both directions, adding arrows at each end to show that it continues indefinitely.
Solving for y
Sometimes, equations are given in forms that aren't immediately useful for graphing directly. That's where solving for y becomes extremely handy.
To solve for y, your goal is to rearrange the equation so that \( y \) is by itself on one side of the equation. Consider the equation \( 4x + 2y = 4 \).
  • Start by isolating terms with \( y \). Subtract \( 4x \) from both sides to rewrite it as \( 2y = -4x + 4 \).
  • To get \( y \) by itself, divide every term by 2. This gives you \( y = -2x + 2 \).
By solving for \( y \), you uncover both the slope and y-intercept, turning the equation into a handy format for drawing graphs or analyzing the line's behavior.

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

Match each statement on the left with the property that justifies it on the right. a. Distributive property b. Associative property c. Commutative property d. Commutative and associative properties $$3+(4+1)=(3+4)+1$$

Add or subtract as indicated. $$\frac{2}{5}-\frac{5}{10}$$

Multiply. $$2 \cdot \frac{1}{2}$$

Perform the indicated operation. $$-\frac{5}{4}\left(\frac{8}{15}\right)$$

Simplify. $$\frac{1+\frac{1}{3}}{1-\frac{1}{3}}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Decision Maths

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.