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Solving Systems of Equations

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My name is Trent Tormoehlen and I am a math teacher at Franklin Community High School in Franklin Indiana.  It is my goal to create the best free onl...  (more...)

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Solving Systems of Equations

 

This Lens explores the three ways to solve a system of equations in two variables.  It includes various videos explaining examples and an explanation on how to use each method.  Feel free to ask any specific questions in the comments section. Scroll down or use the Table of Contents to find the topics that interest you.

If you are looking for worksheets on Systems of Equations check out Systems of Equations WORKSHEETS.

Though I think this lens is valuable and can really help you understand systems of equations sometimes it isn't enough. One of the most effective ways to learn math is one on one tutoring. Tutor.com offers 60 minutes for only $5.00 and is something you should check out.


Tutor.com Direct -- Leader in On Demand Tutoring

Video Learning 

This page is designed around a series of videos I have created about systems of equations. The following is an example of one. You can either scroll down or use the table of contents to find the rest.

A17.1 Solving Systems of Equations by Graphing

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A17.3 Solving a System of Equations by Substitution

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700 views
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A17.8 Solving Systems of Equations by Elimination

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Mr. T on the Web 

Polynomials
A lens showing how to add, subtract, multiply and factor polynomials.
Solving Quadratic Equations
A lens showing how to solve quadratic equations.
Intercept Method
A tutorial on using the x and y intercept to graph lines.
My Math Blog
My various lessons will be linked and organized here.
Solving Inequalities
A lens on solving inequalities.
Linear Inequalities
A lens on solving linear inequalities.
Exponent Properties
A lens explaining the basic exponent properties of Algebra 1.

Solving Systems by Graphing 

The solution to a system of equations is the ordered pair (x, y) that makes both equations true. For example (2, 3) is the solution of the system:

2x + y = 7
4x - 2y = 2

Because:

2(2) + 3 = 7

and

4(2) - 2(3) = 2

It is also important to remember that the graph of a linear equation consists of all the points that make that equation true.

Thus to solve a system of equations by graphing you find the point where the two lines intersect. Because this point is on both lines it thus makes both equations true. So the intersection point represpents the solution to the system of equations.

Special Cases

1) The lines are parallel to each other - By definition two lines that are parallel never intersect each other. Thus they have no points in common and as a result we say the answer is "no solution". The best way to confirm this result is to put both equation in slope intercept form (y = mx + b). If the slopes are the same and the y - intercepts are different then the lines are parallel.

2) The lines are the same - Occasionally the two equations will look different but actually be the exact same line. When you graph them you will notice that they are the exact same line. As with before the simplest way to determine this is to put both equations in slope intercept form.

Some Other Things to Consider

1) Graphing by hand is not very precise. Thus occasionally it is difficult to determine the exact solution. If an exact solution is necessary then it would be best to use one of the other methods described below.

2) If you are having trouble graphing then check out some of the links below. I will be designing a lens on graphing, but it will be a few months before it is done. I primarily use the slope-intercept method and then x, y - intercept method.

Don't Forget!!!

The solution to a system of equations is the point (x,y) that makes both equations true.

Solving Systems by Graphing Example 1 

A17.1 Solving Systems of Equations by Graphing

From http://www.squidoo.com/systemsofequations this is an example of solving a system of equation by graphing. For more examples check out http://www.teachingandlearningmath.blogspot.com/

Runtime: 2:56
986 views
0 Comments:

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Solving Systems by Graphing Example 2 

A17.2 Solving a System of Equations by Graphing

From http://www.squidoo.com/systemsofequations this is an example of a system of equations being solved using graphing. To see more videos check out http://teachingandlearningmath.blogspot.com/

Runtime: 4:23
453 views
0 Comments:

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What Every High School Student Needs 

A graphing calculator is becoming an essential part of high school math classrooms. They not only help in solving problems but they can help you visualize things as well. Ebay is the best place to purchase quality graphing calculators for cheap. TI-84 are the most commonly used.

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Solving Systems of Equations by Substitution 

The solution to a system of equations is the ordered pair that makes both equations true. One way to discover this point is to use a method called substitution. There are 4 very specific steps that are followed when using this method.

Procedure

1) Solve one of the equations for one of the variables (occasionally this is done for you).

2) Look at the other equation, it should contain the variable you just solved for. The next step is to substitute for that variable what it equaled when you solved for it. This is accomplished by "replacing" the variable with the expression it equals. There should only be one variable remaining in the new equation.

3) Solve the new equation for the variable that is remaining. This will usually involve performing the distributive property first.

4) Subsitute that value into one of the two original equations and solve for the other variable.

The following examples show these four steps in action. The only thing that varies is step one. Steps 2 through 4 are pretty much the same everytime. At the end of each example the solution is checked. This is a very, very good habit to get into.

Solving Systems of Equations by Substitution Example 1 

A17.3 Solving a System of Equations by Substitution

From http://www.squidoo.com/systemsofequations this is an example problem of solving systems of equations by graphing. To see more videos check out http://teachingandlearningmath.blogspot.com/

Runtime: 2:11
700 views
0 Comments:

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Solving Systems of Equations by Substitution Example 2 

A17.4 Solving a System of Equations by Substitution

From http://www.squidoo.com/systemsofequations this is an example problem on how to solve system of equations using substitution. See more examples at http://teachingandlearningmath.blogspot.com/

Runtime: 2:46
1605 views
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Solving Systems of Equations by Substitution Example 3 

A17.5 Solving Systems of Equations by Substitution

From http://www.squidoo.com/systemsofequations the following is an example of a system of equations that is solved using the substitution method. For more examples and tutorials go to http://teachingandlearningmath.blogspot.com/

Runtime: 2:56
425 views
3 Comments:

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Solving Systems of Equations by Substitution Example 4 

A17.6 Solving Systems of Equations by Substitution

From http://www.squidoo.com/systemsofequations this is an example of a system of equations being solved using substitution. For more examples check out http://teachingandlearningmath.blogspot.com/

Runtime: 2:57
658 views
2 Comments:

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What is your favorite way to solve systems of equations. 

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Amazon Items 

A graphing calculator is an essential tool for a high school math student. You can get by without it in Algebra 1, but when you get to Algebra 2, PreCalculus and Calculus it becomes very important. The following calculators are the most common used and Amazon has them at a very good price.

Texas Instruments TI-84 Plus Graphing Calculator

Amazon Price: $112.35 (as of 05/11/2008)

Texas Instruments TI83 Graphing Calculator

Amazon Price: (as of 05/11/2008)

Algebra for Dummies

Amazon Price: $13.59 (as of 05/11/2008)

Algebra Workbook For Dummies

Amazon Price: $11.55 (as of 05/11/2008)

Practical Algebra: A Self-Teaching Guide, 2nd Edition

Amazon Price: $13.37 (as of 05/11/2008)

Solving Systems Equations by Elimination 

We have learned two methods for solving systems of equations. Graphing and subsitution. The goal of subsitution was to put the equations together in such a way that there was only one variable left. This allowed us to solve for the variable. The elimination method is similar. Our goal is to eliminate one of the variables by adding the two equations together. Consider the following two equations:

x + y = 12
-x +3y = 4

If we add these together vertically:

x + y = 12

+-x + 3y = 4

=0x + 4y = 16

So by adding the two equations together we eliminated the x variable. The key is that in both equations the x variable had the same coefficient (1) just an opposite sign. Thus when they are added together the coefficient of x is zero.

So with this in mind we break elimination down to the following steps.

1) The first step is to manipulate the equations such that one of the two variables will be eliminated.

2) Add the two equation together to get one equation with one variable.

3) Solve the new equation for the remaining variable.

4) Plug that value in for the variable in one of the two remaining equations and then solve for the other variable.

If you notice, steps 2-4 are very similar to steps 2-4 when using substitution. The key step is to multiply the original equations in a way that allows the variable to be eliminated when the two equations are added together.

Solving Systems of Equations by Elimination Example 1 

A17.7 Solving Systems of Equations by Elimination

From http://www.squidoo.com/systemsofequations this is an example of system of equations being solved using elimination. For more examples check out http://teachingandlearningmath.blogspot.com/

Runtime: 2:13
548 views
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Solving Systems of Equations by Elimination Example 2 

A17.8 Solving Systems of Equations by Elimination

From http://www.squidoo.com/systemsofequations this is an example of a system of equations being solved by the elimination method. For more example check out http://teachingandlearningmath.blogspot.com/

Runtime: 3:00
1427 views
4 Comments:

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Solving Systems of Equations by Elimination Example 3 

A17.9 Solving Systems of Equations using Elimination

From http://www.squidoo.com/systemsofequations this is an example of a system of equations being solved by the elimination method. For more videos and tutorials check out http://www.teachingandlearningmath.blogspot.com/

Runtime: 3:30
402 views
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Solving Systems of Equations using Elimination Example 4 

A17.10 Solving Systems of Equations by Elimination

From http://www.squidoo.com/systemsofequations this is an example of solving systems of equations by substitution. For more tutorials and videos check out http://www.teachingandlearningmath.blogspot.com/

Runtime: 3:27
353 views
1 Comments:

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Solving Systems of Equations by Elimination Example 5 

These two problems are examples of elimination where both variables are eliminated. The result is one of two things.

1) a = a Where a is a real number. Essentially you are looking for the equation to be true - When this happens the answer is "All Solutions" or something similar. It means that every ordered pair that is a solution to one of the equations is a solution to the other. If you solved both equations for "y" you would discover they are the same equation.

2) 0 = (A # other than zero). This time you are looking for an equation that is not true. When this happens the answer is "no solution." Which means an ordered pair that is the solution to one of the equations would not be the solution to the other equation. This also means that the lines are parallel.

A17.11 Solving Systems of Equations by Elimination

From http://www.squidoo.com/systemsofequations this is an example of a systems of equations problem solved by elimination. For more examples check out http://www.teachingandlearningmath.blogspot.com/

Runtime: 2:24
349 views
1 Comments:

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Another Special Case 

This is another look at special cases of systems of equations.

A17.14 Solving Systems of Linear Equations

From http://www.squidoo.com/systemsofequations this is an example of a system of linear equations. For more math help check out http://www.teachingandlearningmath.blogspot.com/

Runtime: 4:15
341 views
1 Comments:

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Systems of Equations Worksheet(s) 

Check back for more worksheets involving systems of equations.
Video Examples
These problems are the problems from the videos in this lens. There are a couple from my linear inequalities lens as well.
Solving Systems of Equations by Elimination
A worksheet with solutions requiring students to solve systems of equations using elimination.
Solving Systems of Equations by Substitution
A worksheet with solutions requiring students to solve systems of equations using substitution

Other Youtube videos on Solving Systems of Equations 


Solving System of Equations

Runtime: 4:57
10360 views
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Systems of Equations-Elimination-Part1

Runtime: 3:29
3797 views
10 Comments:


Systems of Equations-Elimination-Part2

Runtime: 2:56
2899 views
10 Comments:


Solving Systems of Equation via Substitution

Runtime: 4:39
5284 views
10 Comments:


Solving Simultaneous Equations by Substitution Example 1

Runtime: 2:01
2018 views
4 Comments:


Solving Simultaneous Equations by Graph

Runtime: 3:10
1025 views
4 Comments:


Solving Simultaneous Equations by Elimination Example 2

Runtime: 2:32
1638 views
5 Comments:


Solving Simultaneous Equations by Elimination Example 3

Runtime: 2:43
895 views
3 Comments:


Junior Certificate Maths - Simultaneous Equations

Runtime: 6:01
1994 views
8 Comments:

Other Websites Discussing Systems of Equations 

Purple Math
A good look at solving systems of equations by graphing.
Word Problems
A good website discussing word problems involving systems of equations.

New Guestbook 

StephenC

Good stuff!
TEACH911.com
StephenC

Posted March 16, 2008

Stephene

Thks for useful information.. i like your lense very much.. i hv bookmarked this lense and joined your fan club.. (^o^)

Posted March 16, 2008

flaminglacer

You've illustrated yoru subject really well - squid angel blessing

Posted January 08, 2008

gods_grace_notes

Greetings, Trent!
I just wanted to fly on by and rate your lens...Math is not one of my better subjects, so this is a terrific resource for me! Thanks for all your hard work! Welcome to Squidoo, and to our Unit Studies Group. It's nice to have you aboard!
Connie...aka Squid Angel
:

Posted January 07, 2008

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My name is Trent Tormoehlen and I am a math teacher at Franklin Community High School in Franklin Indiana.  It is my goal to create the best free online math help possible. 

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