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» » Addition Method

Solving Systems of Equations
Using Linear Combinations (Addition Method)



There are two ways to solve systems of equations without graphing. You can use the substitution method or linear combinations (which is also commonly known as the addition method).

This lesson is going to focus on using linear combinations, which is typically used when both equations are written in standard form.

The following steps are a guide for using Linear Combinations. Don't worry, it will make a lot more sense as we look at a few examples.




    Steps for Using Linear Combinations (Addition Method)


  • Arrange the equations with like terms in columns.

  • Analyze the coefficients of x or y. Multiply one or both equations by an appropriate number to obtain new coefficients that are opposites

  • Add the equations and solve for the remaining variable.

  • Substitute the value into either equation and solve.

  • Check the solution.



Ok... let's make these examples make sense by looking at some examples.



Our first example is the easiest example as it is already set up perfectly to use the linear combination method.


Example 1


system of equations examples




The next problem isn't set up so perfectly. It demonstrates the extra step that you need to take if your original problem doesn't have opposite terms. Look for that extra step.


Example 2


solving systems of equations



The next problem requires two extra steps! This time, you need to rewrite both equations in order to create opposite terms.




Example 3


linear combinations example



Now, let's see what happens if our system of equations happens to be the same line.



Example 4


using linear combinations to solve systems of equations



Last example! We are going to see what happens when you try to use linear combinations to solve a system that has parallel lines.




Example 5


using the addition method to solve systems of equations



Take a look at the questions that other students have submitted:

question and answers

Another example of the addition method


To ask a question, please visit my Math Forum.





Now, are you ready to practice a few problems on your own?Click here to go to the Combination Method Practice Problems.



system of equations example problems



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