3.5 Solving Systems of Equations
in Three Variables
Objectives:
1. Solve system of linear equations in
three variables.
Systems in Three Variables
• The graph of an equation in 3 variables where
all variables are to the first power is a plane.
• The solution to a system with 3 equations and
3 variables is an ordered triple.
• Ordered Triple – (x, y, z)
• Three possible solutions – one solution, no
solution and infinitely many solutions.
• Solve by substitution or elimination.
Use 1st and 3rd equations to
Example
eliminate z
• Solve
5x+3y+2z=2
5x+3y+2z=2
• 5x+3y+2z=2
x+4y+2z=16 -x-4y-2z=-16
• 2x+y-z=5
4x-y=-14
• x+4y+2z=16
Use 2nd and 3rd to eliminate
Use two pairs of equations z again. (mult 1st by 2)
to create a system of 2 2x+y-z=5
4x+2y-2z=10
equations with 2
x+4y+2z=16
x+4y+2z=16
unknowns.
5x+6y=26
Use these 2 equations to
eliminate x or y.
Continued
4x-y=-14
5x+6y=26
Multiply 1st equation by 6
24x-6y=-84
5x+6y=26
29x=-58
x=-2
Plug in to find y
4(-2)-y=-14
-8-y=-14
-y=-6
y=6
Use the x and y you found to
find z. Plug into any of
the original equations.
5x+3y+2z=2
5(-2)+3(6)+2z=2
-10+18+2z=2
8+2z=2
2z=-6
z=-3
Solution: (-2, 6, -3)
Another Example
x-2y+z=8
2x+y+z=-11
3x-6y+3z=24
x-2y+z=8
2x+y+z=-11
x-2y+z=8
-2x-y-z=11
-x-3y=19
2x+y+z=-11
-6x-3y-3z=33
3x-6y+3z=24 3x-6y+3z=24
-3x-9y=57
-x-3y=19
-3x-9y=57
Multiply 1st by -3 to
eliminate y
3x+9y=-57
-3x-9y=57
0=0
Infinitely many solutions
Try one
x+2y=12
3y-4z=25
x+6y+z=20
x+2y=12
x+6y+z=20
-x-2y=-12
x+6y+z=20
4y+z=8
Use this with the second
equation
3y-4z=25
4y+z=8
Multiply 2nd by 4
3y-4z=25
16y+4z=32
19y=57
y=3
3(3)-4z=25
9-4z=25
-4z=16
z=-4
x+6(3)+(-4)=20
x+14=20
x=6
Solution: (6, 3, -4)
Homework
page 142
13-21 odd
(5 problems!)