Solving Systems of Equations
If you are a teacher looking for resoureces check out my math teaching website. There are some very good worksheets.
If you are a student and looking to get all of your math help in one place this math learning is for you.
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 25 minutes for FREE and is something you should check out.
Video Learning
Table of Contents
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.
- 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
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
5175 views
7 Comments:
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
1766 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
Procedure
1) Solve one of the equations for one of the variables (occasionally this is done for you). Usually you want to solve for the variable that is the easiest. For instance if the equation is 3x + y = 12 then you should solve for y because it does not have a coefficient. Solving for x would mean you would have to divide everything by three.
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
3016 views
8 Comments:
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
8927 views
10 Comments:
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
1717 views
6 Comments:
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
2618 views
6 Comments:
What is your favorite way to solve systems of equations.
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Solving Systems Equations by Elimination
x + y = 12
-x +3y = 4
If we add these together vertically:
x + y = 12
+-x + 3y = 4
=0x + 4y = 16So 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
1586 views
0 Comments:
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
6234 views
8 Comments:
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
2331 views
5 Comments:
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: 3:28
1460 views
2 Comments:
Solving Systems of Equations by Elimination Example 5
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
1265 views
1 Comments:
Another Special Case
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
1755 views
2 Comments:
Other Youtube videos on Solving Systems of Equations
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.
Here are some other recommendations
If you loved Texas Instruments TI-84 Plus Graphing Calculator, you might also enjoy:
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yuki wrote
thanks for all the effort!!
I completely understood it!
Posted July 16, 2008
Stephene wrote...
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 wrote...
You've illustrated yoru subject really well - squid angel blessing
Posted January 08, 2008
gods_grace_notes wrote...
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
by MrT68
My name is Trent Tormoehlen and I am a math teacher at Sycamore School in Indianapolis Indiana. I will also be helping coach the schools Math Co...
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