Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Elementary Algebra » Solutions by Graphing

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Endorsed by Endorsed (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
  • College Open Textbooks display tagshide tags

    This collection is included inLens: Community College Open Textbook Collaborative
    By: CC Open Textbook Collaborative

    Comments:

    "Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"

    Click the "College Open Textbooks" link to see all content they endorse.

    Click the tag icon tag icon to display tags associated with this content.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This collection is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange Grove

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Comments:

    "Elementary Algebra covers traditional topics studied in a modern elementary algebra course. Written by Denny Burzynski and Wade Ellis, it is intended for both first-time students and those […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Solutions by Graphing

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Beginning with the graphical solution of systems, this chapter includes an interpretation of independent, inconsistent, and dependent systems and examples to illustrate the applications for these systems. The substitution method and the addition method of solving a system by elimination are explained, noting when to use each method. The five-step method is again used to illustrate the solutions of value and rate problems (coin and mixture problems), using drawings that correspond to the actual situation. Objectives of this module: be able to recognize a system of equations and a solution to it, be able to graphically interpret independent, inconsistent, and dependent systems, be able to solve a system of linear equations graphically.

Overview

  • Systems of Equations
  • Solution to A System of Equations
  • Graphs of Systems of Equations
  • Independent, Inconsistent, and Dependent Systems
  • The Method of Solving A System Graphically

Systems of Equations

Systems of Equations

A collection of two linear equations in two variables is called a system of linear equations in two variables, or more briefly, a system of equations. The pair of equations

{ 5x2y=5 x+y=8 { 5x2y=5 x+y=8
is a system of equations. The brace { { is used to denote that the two equations occur together (simultaneously).

Solution to A System of Equations

Solution to a System

We know that one of the infinitely many solutions to one linear equation in two variables is an ordered pair. An ordered pair that is a solution to both of the equations in a system is called a solution to the system of equations. For example, the ordered pair ( 3,5 ) ( 3,5 ) is a solution to the system

{ 5x2y=5 x+y=8 { 5x2y=5 x+y=8
since ( 3,5 ) ( 3,5 ) is a solution to both equations.

5x2y = 5 x+y = 8 5(3)2(5) = 5 Is this correct? 3+5 = 8 Is this correct? 1510 = 5 Is this correct? 8 = 8 Yes, this is correct. 5 = 5 Yes, this is correct. 5x2y = 5 x+y = 8 5(3)2(5) = 5 Is this correct? 3+5 = 8 Is this correct? 1510 = 5 Is this correct? 8 = 8 Yes, this is correct. 5 = 5 Yes, this is correct.

Graphs of Systems of Equations

One method of solving a system of equations is by graphing. We know that the graph of a linear equation in two variables is a straight line. The graph of a system will consist of two straight lines. When two straight lines are graphed, one of three possibilities may result.

Example 1

The lines intersect at the point ( a,b ) ( a,b ) . The point ( a,b ) ( a,b ) is the solution to the corresponding system.
A graph of two lines; 'line one' and 'line two,' intersecting at a point labeled with coordinates (a, b) and with a second label with x-coordinate negative one and one-half, and y-coordinate negative one and one-half. Line one is passing through a point with coordinates zero, one over two, and line two is passing through a point with coordinates negative four and one half, zero.

Example 2

The lines are parallel. They do not intersect. The system has no solution.
A graph of two parallel lines; 'Line one' and 'Line two'. Line one is passing through two points with the coordinates zero, one, and five, negative two. Line two is passing through two points with the coordinates zero, three, and five, zero.

Example 3

The lines are coincident (one on the other). They intersect at infinitely many points. The system has infinitely many solutions.
A graph of two conincident lines; 'Line one' and 'Line two'. The lines are passsing through the same two points with the coordinates negative three, negative one, and four, three. Since the lines are coincident lines they have the same graph.

Independent, Inconsistent, and Dependent Systems

Independent Systems

Systems in which the lines intersect at precisely one point are called independent systems. In applications, independent systems can arise when the collected data are accurate and complete. For example,

The sum of two numbers is 10 and the product of the two numbers is 21. Find the numbers.

In this application, the data are accurate and complete. The solution is 7 and 3.

Inconsistent Systems

Systems in which the lines are parallel are called inconsistent systems. In applications, inconsistent systems can arise when the collected data are contradictory. For example,

The sum of two even numbers is 30 and the difference of the same two numbers is 0. Find the numbers.

The data are contradictory. There is no solution to this application.

Dependent Systems

Systems in which the lines are coincident are called dependent systems. In applications, dependent systems can arise when the collected data are incomplete. For example.

The difference of two numbers is 9 and twice one number is 18 more than twice the other.

The data are incomplete. There are infinitely many solutions.

The Method of Solving A System Graphically

The Method of Solving a System Graphically

To solve a system of equations graphically: Graph both equations.

  1. If the lines intersect, the solution is the ordered pair that corresponds to the point of intersection. The system is independent.
  2. If the lines are parallel, there is no solution. The system is inconsistent.
  3. If the lines are coincident, there are infinitely many solutions. The system is dependent.

Sample Set A

Solve each of the following systems by graphing.

Example 4

{ 2x+y=5 x+y=2 ( 1 ) ( 2 ) { 2x+y=5 x+y=2 ( 1 ) ( 2 )
Write each equation in slope-intercept form.

( 1 ) 2x+y=5 ( 2 ) x+y=2 y=2x+5 y=x+2 ( 1 ) 2x+y=5 ( 2 ) x+y=2 y=2x+5 y=x+2
Graph each of these equations.
A graph of two lines; ‘one’ and ‘two.’ The lines are intersecting at a point with coordinates negative one, three. Line one is passing through a point with coordinates zero, five. Line two is passing through two points with coordinates zero, two, and one, one.
The lines appear to intersect at the point ( 1,3 ) ( 1,3 ) . The solution to this system is ( 1,3 ) ( 1,3 ) , or

x=1, y=3 x=1, y=3

Check:  Substitute x=1,y=3 x=1,y=3 into each equation.

(1) 2x+y = 5 (2) x+y = 2 2(1)+3 = 5 Is this correct? 1+3 = 2 Is this correct? 2+3 = 5 Is this correct? 2 = 2 Yes, this is correct. 5 = 5 Yes, this is correct. (1) 2x+y = 5 (2) x+y = 2 2(1)+3 = 5 Is this correct? 1+3 = 2 Is this correct? 2+3 = 5 Is this correct? 2 = 2 Yes, this is correct. 5 = 5 Yes, this is correct.

Example 5

{ x+y=1 x+y=2 ( 1 ) ( 2 ) { x+y=1 x+y=2 ( 1 ) ( 2 )
Write each equation in slope-intercept form.

( 1 ) x+y = 1 ( 2 ) x+y = 2 y = x1 y = x+2 ( 1 ) x+y = 1 ( 2 ) x+y = 2 y = x1 y = x+2
Graph each of these equations.
A graph of two parallel line; 'one' and 'two'. Line one is passing through two points with the coordinates zero, two, and one, three. Line two is passing through two points with the coordinates zero, negative one, and one, zero.
These lines are parallel. This system has no solution. We denote this fact by writing inconsistent.

We are sure that these lines are parallel because we notice that they have the same slope, m=1 m=1 for both lines. The lines are not coincident because the y y -intercepts are different.

Example 6

{ 2x+3y=2 6x+9y=6 ( 1 ) ( 2 ) { 2x+3y=2 6x+9y=6 ( 1 ) ( 2 )
Write each equation in slope-intercept form.

( 1 ) 2x+3y = 2 ( 2 ) 6x+9y = 6 3y = 2x2 9y = 6x6 y = 2 3 x 2 3 y = 2 3 x 2 3 ( 1 ) 2x+3y = 2 ( 2 ) 6x+9y = 6 3y = 2x2 9y = 6x6 y = 2 3 x 2 3 y = 2 3 x 2 3
A graph of two conincident lines; 'one' and 'two'. The lines are passsing through the same two points with the coordinates zero, negative two over three, and three, one and one third. Since the lines are coincident, they have the same graph.
Both equations are the same. This system has infinitely many solutions. We write dependent.

Practice Set A

Solve each of the following systems by graphing. Write the ordered pair solution or state that the system is inconsistent, or dependent.

Exercise 1

{ 2x+y=1 x+y=5 { 2x+y=1 x+y=5
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

x=2,y=3 x=2,y=3
A graph of two lines intersecting at a point with the coordinates two, negative three. One of the lines is passing through a point with  the coordinates one over two, zero, and the other line is passing through a point with the coordinates zero, negative five.

Exercise 2

{ 2x+3y=6 6x9y=18 { 2x+3y=6 6x9y=18
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

dependent
A graph of two coincident lines passing through the same two points with the coordinates zero, two, and three, four. Since the lines are coincident, they have the same graph. The graph is labeled as 'coincident lines.'

Exercise 3

{ 3x+5y=15 9x+15y=15 { 3x+5y=15 9x+15y=15
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

inconsistent
A A graph of two parallel lines. One of the lines is passing through two points with coordinates zero, one and one and two third, zero. The other line is passing through two points with coordinates zero, three, and five, zero. The graph is labeled as 'parallel lines.'

Exercise 4

{ y=3 x+2y=4 { y=3 x+2y=4
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

x=2,y=3 x=2,y=3
A graph of two lines intersecting at a point with the coordinates two, negative three. One of the lines is passing through a point with  the coordinates one zero, negative two. The other line is parallel to x axis, and is passing  through a point with the coordinates negative three, negative three.

Exercises

For the following problems, solve the systems by graphing. Write the ordered pair solution, or state that the system is inconsistent or dependent.

Exercise 5

{ x+y=5 x+y=1 { x+y=5 x+y=1
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

( 3,2 ) ( 3,2 )
A graph of two lines intersecting at a point with coordinates negative three, negative two. One of the lines is passing through a point with coordinates zero, negative five and, the other line is passing through two points with coordinates negative one, zero; and zero, one.

Exercise 6

{ x+y=4 x+y=0 { x+y=4 x+y=0
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Exercise 7

{ 3x+y=5 x+y=3 { 3x+y=5 x+y=3
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

( 1,2 ) ( 1,2 )
A graph of two lines intersecting at a point with coordinates negative one, two. One of the lines is passing through a point with coordinates zero, five, and the other line is passing through two points with coordinates zero, three; and one, four.

Exercise 8

{ xy=6 x+2y=0 { xy=6 x+2y=0
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Exercise 9

{ 3x+y=0 4x3y=12 { 3x+y=0 4x3y=12
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

( 12 13 , 36 13 ) ( 12 13 , 36 13 )
A graph of two lines intersecting at a point with coordinates twelve over thirteen, negative thirty-six over thirteen. One of the lines is passing through a point with coordinates zero, zero and the other line is passing through two points with coordinates zero, negative four; and three, zero.

Exercise 10

{ 4x+y=7 3x+y=2 { 4x+y=7 3x+y=2
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Exercise 11

{ 2x+3y=6 3x+4y=6 { 2x+3y=6 3x+4y=6
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

These coordinates are hard to estimate. This problem illustrates that the graphical method is not always the most accurate. ( 6,6 ) ( 6,6 )
A graph of two lines intersecting at a point with coordinates negative six, six. One of the lines is passing through a point with coordinates zero, three over two and the other line is passing through two points with coordinates zero, two; and three, zero.

Exercise 12

{ x+y=3 4x+4y=12 { x+y=3 4x+4y=12
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Exercise 13

{ 2x3y=1 4x6y=4 { 2x3y=1 4x6y=4
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

inconsistent
A graph of two parallel lines. One of the lines is passing through two points with coordinates zero, negative two over three and three, zero. The other line is passing through a point with coordinates zero, negative one over three.

Exercise 14

{ x+2y=3 3x6y=9 { x+2y=3 3x6y=9
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Exercise 15

{ x2y=6 3x6y=18 { x2y=6 3x6y=18
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

dependent
A graph of two coincident lines passing through the same two points with coordinates zero, negative three; and two, negative two. Since the lines are coincident, they have the same graph.

Exercise 16

{ 2x+3y=6 10x15y=30 { 2x+3y=6 10x15y=30
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Exercises For Review

Exercise 17

((Reference)) Express 0.000426 0.000426 in scientific notation.

Solution

4.26× 10 4 4.26× 10 4

Exercise 18

((Reference)) Find the product: ( 7x3 ) 2 . ( 7x3 ) 2 .

Exercise 19

((Reference)) Supply the missing word. The

          
of a line is a measure of the steepness of the line.

Solution

slope

Exercise 20

((Reference)) Supply the missing word. An equation of the form a x 2 +bx+c=0,a0 a x 2 +bx+c=0,a0 , is called a

          
equation.

Exercise 21

((Reference)) Construct the graph of the quadratic equation y= x 2 3. y= x 2 3.
An xy coordinate plane with gridlines, labeled negative five and five with increments of one unit for both axes.

Solution

A graph of a parabola passing through four points with coordinates negative two, one; negative one, negative two; one, negative two; and two, one.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks