• A solution of a system of linear equations in three variables is an ordered triple of real numbers that satisfies all equations of the system.
• In other words, the solution is the point of intersection of three planes.

Solving – Linear Combination Method:
1. Use the linear combination method to rewrite the linear system in three variables as a linear system in two variables.
▫ *Combine two of the equations to eliminate one variable, then combine another two equations to eliminate the same variable.
2. Solve the new linear system for both of its variables.
3. Substitute the values found in step 2 into one of the original equations and solve for the remaining variable.

Solve the system of linear equations.
1. x

2 y

3 z
 
3
2 x

5 y

4 z

13
5 x

4 y
 z

5

Solve the system of linear equations.
2. 3 x

2 y

4 z

11
2 x
 y

3 z

4
5 x

3 y

5 z
 
1

• A system of linear equations in three variables can also have infinite solutions. This happens when one of the following:
1. three different planes intersect in a single line
2. two coinciding planes intersect in a third plane
3. three coinciding planes.

• A system of linear equations in three variables can also have no solution. This happens when one of the following:
1. three parallel planes
2. two coinciding planes parallel to a third plane
3. three planes intersecting in three parallel lines
4. two parallel lines intersecting a third plane in two parallel lines

Solve the system of linear equations.
3. x
 y
 z

2
3 x

3 y

3 z

14 x

2 y
 z

4

Solve the system of linear equations.
4. x
 y
 z

2 x
 y
 z

2
2 x

2 y
 z

4