Systems of Equations

by Aden Kranz

Graphing Method

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Substitution Method

1392171771 w

Elimination Method


  1. Divide the coefficients of a variable (whichever looks easiest) from one equation by coefficient of the same variable from the other equation.
  2. Take the result and multiply that entire other equation by it. That should make the two coefficients of the variable equivalent.
  3. Subtract one equation from the other. Make sure that the variable that you chose earlier is on the same side in both of the equations before you do this! This should leave you with an equation with only one variable.
  4. Solve the resulting equation for that remaining variable.
  5. Plug the solution for that variable back into one of the original equations, and then solve for the remaining variable.

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Which Method to Use

  • Use the Graphing Method when you have a graphing calculator at hand (and your allowed to use it!)
  • Use the Elimination Method when you can easily multiply one equation by a certain number to make the coefficients of one of the variables equivalent or at least inverse so you can eliminate that variable
  • Use the Substitution Method when neither of the above apply

Writing Systems of Equations


  1. identify what the each of variables represent in the word problem.
  2. Figure out which pieces of information applies to each equation or whether a piece applies to both of them.
  3. Determine what components each piece of information would be represented as their equation.
  4. Put all these components together to make each equation, using the necessary operations, in a way that correctly represents the work problem. Make sure to apply the order of operations.
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Word Problems


  1. identify the piece of information the problem wants you to find.
  2. write a system of equations using the steps from the previous entry. It's okay if neither of the variables is the piece of information that the problem wants you to find.
  3. solve the system of equations using whichever method you see fit.
  4. If one or both of the variables is the piece of information that the problem wants you to find, then you have your answer. If not, then use the variables' solutions to find that piece of information. you're probably going to need the variable solutions to finding the information anyways. (If you didn't, then, frankly, that must have been a pretty dumb question.)

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The 3 Types of Solutions

One Solution- when there's only one solution. Example:



Solution: (6,1)

Infinitely Many Solutions- when there are infinity solutions. Example:



Solution: infinitely many

No Solution: when there isn't any solution. Example:



Solution: None