ALGEBRA TWO
CHAPTER THREE: SYSTEMS OF
LINEAR EQUATIONS AND
INEQUALITIES
Section 3.6 - Solving Systems of
Linear Equations in Three
Variables
LEARNING GOALS
Goal One - Solve systems of
linear equations in three
variables.
Goal Two - Use linear systems
in three variables to model reallife situations.
VOCABULARY
 A system of three linear equations
includes three equations in the same
variables.
 A solution of a linear system in three
variables is an ordered triple (x, y, z) that
satisfies all three equations. The linear
combination method you learned in Lesson
3.2 can be extended to solve a system of
linear equations in three variables.
ALGEBRAIC SOLUTIONS
THE LINEAR COMBINATION METHOD
(THREE VARIABLE SYSTEMS)
STEP 1: Use the linear combination method to
rewrite the linear system in three variables as a
linear system in two variables.
STEP 2: Solve the new linear system for both of its
variables.
STEP 3: Substitute the values found in Step 2 into
one of the original equations and solve for the third
variable.
ALGEBRAIC SOLUTIONS
THE LINEAR COMBINATION METHOD
(THREE VARIABLE SYSTEMS)
NOTE: If you obtain a false equation,
such as 0 = 1, in any step, then the
system has no solution. If you do not
obtain a false solution, but obtain an
identity, such as 0 = 0, then the system
has infinitely many solutions.
Using a Linear Combination
Method
Solve the system:
x+y+z=2
-x + 3y + 2z = 8
4x + y = 4
Equation 1
Equation 2
Equation 3
Since Equation 3 does not have a z-term, eliminate
the z from one of the other equations.
x + y + z = 2 times -2
-x + 3y + 2z = 8 times 1
Add the equations.
-2x - 2y - 2z = -4
-x + 3y + 2z = 8
-3x + y = 4
Using a Linear Combination
Method
Solve the system:
x+y+z=2
-x + 3y + 2z = 8
4x + y = 4
Equation 1
Equation 2
Equation 3
Now use this new equation with Equation 3 to solve
for x and y.
-3x + y = 4
4x + y = 4
times -1
times 1
Add the equations.
3x - y = -4
4x + y = 4
7x = 0
x =0
Using a Linear Combination
Method
Solve the system:
x+y+z=2
-x + 3y + 2z = 8
4x + y = 4
Equation 1
Equation 2
Equation 3
Substitute x = 0 and solve for y.
-3(0) + y = 4
y=4
Using a Linear Combination
Method
Solve the system:
x+y+z=2
-x + 3y + 2z = 8
4x + y = 4
Equation 1
Equation 2
Equation 3
Substitute x = 0, y = 4 and solve for z.
x+y+z=2
The solution is
the ordered
(0) + (4) + z = 2
triple (0, 4, -2).
z = -2
ALGEBRAIC SOLUTIONS
THE LINEAR COMBINATION METHOD
VARIABLE SYSTEMS)
(THREE
NOTE: This technique requires
careful, tedious attention to the detail
of the steps involved. The best way to
master the technique is to PRACTICE,
PRACTICE, PRACTICE!!!!!!!!!
Solve the system:
3x+2y+4z = 11
2x-y+3z = 4
5x-3y+5z= -1
equation 1.
equation 2.
equation 3.
Solve the system:
3x+2y+4z = 11
2x-y+3z = 4
5x-3y+5z= -1
equation 1.
equation 2.
equation 3.
ASSIGNMENT
READ & STUDY: pg. 177-180.
WRITE: pg. 181-184.
#13, #17, #19, #23, #25,
#29, #31, #33