How to Solve Linear Equations in Three Variables

You can use the Gaussian method to solve a system of three linear equations simultaneously, if the system has a solution. The basic idea is to add integer multiples of two equations to produce a new equation with fewer variables. The new equation will substitute for one of the two equations from which you calculated it. You'd then repeat this process until all variables are determined. This approach is also called the "elimination method."

Instructions

1. Place the variable terms on the left side of the equal signs, and the constants on the right. Order the variables to match up vertically.

   For example,
   2x+3y+2z=0
   3x+2y+3z=0
   3x+3y+2z=1

2. Now add integer multiples of two equations to each other to eliminate at least one variable. Replace one of the two equations used to make this calculation.

   Continuing with the above example, note that the first and last equations have two terms that are the same. Therefore, subtracting the first equation from the last will produce an equation with only one variable left. Using E1, E2 and E3 to denote the current rows, replace E1 with E3-E1 to give you
   _x+__+__=1
   3x+2y+3z=0
   3x+3y+2z=1
   (The underscores are used here merely to keep the variables aligned between rows.) So x=1.

3. Replace any variables with their values as they become known.

   Continuing with the above example, inserting x=1 reduces the system above to two equations in two unknowns:
   3+2y+3z=0
   3+3y+2z=1
   or
   2y+3z=-3
   3y+2z=-2

4. Repeat Step 2 again to remove another variable, again replacing one of the two equations with the new one.

   Continuing with the above example, replacing the second equation with 3E1-2E2 gives
   2y+3z=-3
   __+5z=-5
   Therefore z=-1.

5. Repeat Step 3 as variable values become known.

   Continuing with the above example, replacing z=-1 in E1 while simplifying E2 gives
   2y+-3=-3
   __+_z=-1
   Therefore y=0. So the solution is x=1,y=0, z=-1.

Tips & Warnings

• Sometimes, the equations in the system don't lead to a unique solution but instead are redundant or contradictory. In the former case, there will be more than one solution. In the latter, there are no solutions. An example of redundancy is

• 2x+3y=4

• 4x+6y=8

• because one equation is a constant multiple of the other. In this case, an infinite number of solutions is possible. An example of contradictory equations is

• x+y=1

• x+y=0

• because the equations say x+y equals two different numbers, which it can't. See the eHow articles "Infinite Solution Elimination Method" and "About the Gauss-Jordan Method" by the same author for further reading on these two situations.

References

• Numerical Analysis; Richard L. Burden and J. Douglas Faires; 1989
• Harvey Mudd: Row Reduction