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 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.
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
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.
curated content from YouTube
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
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.
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.
Important!
Don't Forget!!!
The solution to a system of equations is the point (x,y) that makes both equations true.
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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.
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). 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.
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.
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.
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
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.
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 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.
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.
curated content from YouTube
Another Special Case
This is another look at special cases of systems of equations.
curated content from YouTube
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:
TI-84 Plus Graphing Calculator for Dummies by C. C. Edwards
If you have a T1-84 Plus Graphing Calculator, you more...0 points
Texas Instruments TI-84 Plus Silver Edition Graphing Calculator
Texas Instruments TI-84 Plus Silver Edition Graphi more...0 points
Texas Instruments TI-30X IIS 2-Line Scientific Calculator
A calculator for science, math, algebra, trigonome more...0 points
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- Using a TI-83 or TI-84, you can solve nonlinear systems using the intersect technique.
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by MrT68
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 Counts... more »
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