Algebra II Quarter 2: Handout #4
Name ________________________ Date _____________ Period _____
3.6 Solving Systems of Linear Equations in Three Variables

When we add a third variable, we are going into a third dimension. We are
adding a “z” axis! So our ordered pair is now in the form of (x, y, z).

There are two ways to solve linear systems algebraically:
1. Substitution Method:
a. Solve one of the equations for one of its variables.
b. Substitute the expression from step “a” into the other
equation and solve for the other variable. You will not
get a number yet.
c. Substitute the expression from step “b” into the third
equation and solve. You will get a number answer here.
d. Substitute the number from step “c” back into your
modified second equation and solve. You will get a
number answer here.
e. Lastly, take the two numbers already found and
substitute back into the first equation to solve for the
remaining variable.
f. Write your ordered pair, (x,y,z).
2. Linear Recombination Method – adding equations to get rid of
one variable and solve for the other; then plug the one solved
for back into either of the original equations to solve for the
remaining variable
a. Use the linear recombination method (found in 3.2) to
rewrite the linear system in three variables as a linear
system in two variables.
b. Solve the new linear system for both of its variables.
c. Substitute the values found in step “b” into one of the
original equations and solve for the remaining variable.
d. Write you answer in (x, y, z) form. NOTE: If you obtain
a false equation (such as 0 = 1), in any of the steps then
it’s no solution. If you obtain an identity (such as 0 = 0),
then it’s all solutions.
Algebra II Quarter 2: Handout #4
Linear System:
x + 2y – 3z = -3
2x – 5y + 4z = 13
5x + 4y – z = 5
Solve using Substitution Method:
Solve using Linear Combination Method:
Final Answer:
(x, y, z) = (
,
,
)
Algebra II Quarter 2: Handout #4
Linear System:
3x + 2y + 4z = 11
2x – y + 3z = 4
5x – 3y + 5z = -1
Solve using Substitution Method:
Solve using Linear Combination Method:
Final Answer:
(x, y, z) = (
,
,
)
Algebra II Quarter 2: Handout #4
Linear System:
x + y – 2z = 5
x + 2y + z = 8
2x + 3y – z = 1
Solve using Substitution Method:
Solve using Linear Combination Method:
Final Answer:
(x, y, z) = (
,
,
)
Algebra II Quarter 2: Handout #4
Linear System:
x+y+z=2
x+y–z=2
2x + 2y + z = 4
Solve using Substitution Method:
Solve using Linear Combination Method:
Final Answer:
(x, y, z) = (
,
,
)
Algebra II Quarter 2: Handout #4