Solving a System of Equations
_Previously we have learned about solving for a variable
in an equation. If there are two variables in an equation, a solution
for one of those variables cannot be found. A system of equations is
two or more equations. If you have two of the same variables in both
equations and both are to the first power, a solution for the variables
can be found. When we solve for the variables, we find numbers that
work for both equations when they are substituted into the correct
variables.
The easiest way to solve the system is to graph the two linear equations and see where they intersect. The point of intersection is the solution, or where the x and y values will work for both equations.
There are two other ways to solve algebraically for systems. Here is a good explanation of those methods: http://tutorial.math.lamar.edu/Classes/Alg/SystemsTwoVrble.aspx. The two methods for solving systems of equations are substitution and elimination. Let's look at each one of those in depth individually.
Once you understand how to use both the Substitution and Elimination methods, you will need to determine when it is best to use each one. Visit this website for a good overview: http://www.biology.arizona.edu/BioMath/tutorials/Linear/
LinearEquations.html
Sometimes when you solve the systems, you get a very odd answer. If the answer is not true, such as 0 = 6, then the lines have no solution and in fact are parallel lines. If the answer you get is always true, such as 7 = 7, then both linear equations are the same line and have an infinite set of points that work in the system. Read about it on this website and then try the interactive card sorting activity: http://stufiles.sanjac.edu/THEA/THEAMathReviewforWebsite/
THEAMathReviewforWebsite8.html#systemsoflinearequations.
This is a pdf file that you can download or print and keep as a good reference on solving systems of equations: http://www.uamont.edu/Math_and_Sciences/mathematics/algebras/
College_Algebra/Efird_5.1.pdf. This link requires that you have Adobe Reader.
PRACTICE
Practice your skills with these problems: http://www.khanacademy.org/exercise/systems_of_equations
Combine your knowledge of solving systems and word problems by doing these problems and checking your answers: http://www.regentsprep.org/Regents/math/ALGEBRA/AE3/PracWord.htm
ASSESS
Now test yourself with this online quiz: http://math.uww.edu/~mcfarlat/141/141quiz7.htm.
The easiest way to solve the system is to graph the two linear equations and see where they intersect. The point of intersection is the solution, or where the x and y values will work for both equations.
There are two other ways to solve algebraically for systems. Here is a good explanation of those methods: http://tutorial.math.lamar.edu/Classes/Alg/SystemsTwoVrble.aspx. The two methods for solving systems of equations are substitution and elimination. Let's look at each one of those in depth individually.
- Solving a systems of equations using Substitution.
- Solving a system of equations using Elimination.
Once you understand how to use both the Substitution and Elimination methods, you will need to determine when it is best to use each one. Visit this website for a good overview: http://www.biology.arizona.edu/BioMath/tutorials/Linear/
LinearEquations.html
Sometimes when you solve the systems, you get a very odd answer. If the answer is not true, such as 0 = 6, then the lines have no solution and in fact are parallel lines. If the answer you get is always true, such as 7 = 7, then both linear equations are the same line and have an infinite set of points that work in the system. Read about it on this website and then try the interactive card sorting activity: http://stufiles.sanjac.edu/THEA/THEAMathReviewforWebsite/
THEAMathReviewforWebsite8.html#systemsoflinearequations.
This is a pdf file that you can download or print and keep as a good reference on solving systems of equations: http://www.uamont.edu/Math_and_Sciences/mathematics/algebras/
College_Algebra/Efird_5.1.pdf. This link requires that you have Adobe Reader.
PRACTICE
Practice your skills with these problems: http://www.khanacademy.org/exercise/systems_of_equations
Combine your knowledge of solving systems and word problems by doing these problems and checking your answers: http://www.regentsprep.org/Regents/math/ALGEBRA/AE3/PracWord.htm
ASSESS
Now test yourself with this online quiz: http://math.uww.edu/~mcfarlat/141/141quiz7.htm.