Systems with Three Variables

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Solving Systems of Equations in Three Variables

Background

Graphical methods for solving linear equations in three variables are unsatisfactory, because a three-dimensional coordinate system is required.  Since the substitution methods becomes awkward for most systems of more than two variables, we will use the elimination method.  The process is basically the same for systems of three equations as for systems of two equations.

A solution of a system of equations in three variables is an ordered triple (x, y, z) that makes all three equations true.

The Process


Solve:
Begin by adding 1 and 3 together.  This will eliminate an x-term.
Next, multiply 1 by –2, then add  1 and 2 together  to eliminate the x-term.

Use 4 and 5 together to solve for z.


By using 4 and 6 together we can solve for y.

 
Substitute 6 and 7 into 1 to solve for x.

Write the solution as an ordered triple.
 (1, 2, 3)

Techniques to Use

One method for solving three equations with three unknowns is stated below.

1.    Choose a variable to eliminate. In the example shown above, we 
       chose to eliminate the x variable first.

2.    Use any two of the equations from the original system and use 
       the elimination method learned in Section 3–2.  Call the result 
       equation #4.

3.    Go back to the original system and use two different equations
       from the system and eliminate the same variable by using the
       elimination method.  Call the result equation #5.

5.    Use equations #4 and #5 together to eliminate a second variable
       by the elimination method.  This will give you one of your 
       unknowns.

6.    Take your solution for step 5 and substitute back into equation
       #4 or #5 to find a second unknown.

7.    Return to the original system and choose an equation that
       contains all three variables.  Substitute your two solution points
       into any one of the equations to find the third unknown.

8.    Check your ordered triple (solution) in each of the equations of 
       the original system to verify your solution is valid and you have
       not made any mistakes.

Triangularization Algorithm

Our goal is to obtain an equivalent system of equations in the following form.

1.    First, if possible, interchange equations to make each 
       x-coefficient a multiple of the first.

2.    Second, if (1) is not possible, multiply where appropriate to
       make each x-coefficient a multiple of the first.

3.    Multiply and add to eliminate x from the second and third
       equations.

4.    Interchange equations or multiply so the y-coefficient of the third
       equation is a multiple of the y-coefficient of the second
       equation.

5.    Multiply and add to eliminate y from the third equation.

6.    Solve the third equation for z, substitute in the second equation
       to find y, and then substitute y and z in the first equation to find
       x.


Solve using the Triangularization Algorithm:
Begin by multiplying 1 and 2 by 3 to make each x-coefficient a multiple of equation 3.  
Next, multiply 1 by –1, then add it to 2. Also multiply 1 by –1 and add it to 3 to eliminate the x-term.

Multiply 3 by –9 and add it to 2 to eliminate the y-coefficient.



Using 3 solve for z.
Substitute 4 into 2 to solve for y.

Substitute 4 and 5 into 1 to solve for x. 
Write the solution as an ordered triple.
 (1, – 2, 3)