3.6 Solving Systems of Linear
Equations in 3 Variables
p. 177
A system of lin. eqns. in 3 variables
Looks something like:
x+3y-z=-11
2x+y+z=1
5x-2y+3z=21
A solution is an ordered triple (x,y,z) that
makes all 3 equations true.
Steps for solving in 3 variables
1. Using the 1st 2 equations, cancel one of
the variables.
2. Using the last 2 equations, cancel the
same variable from step 1.
3. Use the results of steps 1 & 2 to solve for
the 2 remaining variables.
4. Plug the results from step 3 into one of
the original 3 equations and solve for the
3rd remaining variable.
5. Write the solution as an ordered triple
(x,y,z).
Solve the system.
1. x+3y-z=-11
2x+y+z=1
z’s are easy to cancel!
3x+4y=-10
2. 2x+y+z=1
5x-2y+3z=21
Must cancel z’s again!
-6x-3y-3z=-3
5x-2y+3z=21
-x-5y=18
2(2)+(-4)+z=1
4-4+z=1
z=1
x+3y-z=-11
2x+y+z=1
5x-2y+3z=21
3. 3x+4y=-10
-x-5y=18
Solve for x & y.
3x+4y=-10
-3x-15y+54
-11y=44
y=- 4
3x+4(-4)=-10
x=2
(2, - 4, 1)
Solve the system.
-x+2y+z=3
2x+2y+z=5
4x+4y+2z=6
2. 2x+2y+z=5
1. -x+2y+z=3
4x+4y+2z=6
2x+2y+z=5
z’s are easy to cancel! Cancel z’s again.
-4x-4y-2z=-10
-x+2y+z=3
4x+4y+2z=6
-2x-2y-z=-5
0=- 4
-3x=-2
Doesn’t make sense!
x=2/3
No solution
Solve the system.
1. -2x+4y+z=1
3x-3y-z=2
z’s are easy to cancel!
x+y=3
2. 3x-3y-z=2
5x-y-z=8
Cancel z’s again.
-3x+3y+z=-2
5x-y-z=8
2x+2y=6
-2x+4y+z=1
3x-3y-z=2
5x-y-z=8
3. x+y=3
2x+2y=6
Cancel the x’s.
-2x-2y=-6
2x+2y=6
0=0
This is true.
¸ many solutions
Assignment