1. 3.5 Solving Systems of Equations in Three Variables
Objectives:
1. Solve system of linear equations in three variables.

2. 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.

3. Example
Use 1st and 3rd equations to eliminate z 5x+3y+2z=2 5x+3y+2z=2 x+4y+2z=16 -x-4y-2z=-16 4x-y=-14 Use 2nd and 3rd to eliminate z again. (mult 1st by 2) 2x+y-z=5 4x+2y-2z=10 x+4y+2z=16 x+4y+2z=16 5x+6y=26 Use these 2 equations to eliminate x or y.
Solve
5x+3y+2z=2
2x+y-z=5
x+4y+2z=16
Use two pairs of equations to create a system of 2 equations with 2 unknowns.

4. Continued
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= z=2 8+2z=2 2z=-6 z=-3 Solution: (-2, 6, -3)
4x-y=-14 5x+6y=26 Multiply 1st equation by 6 24x-6y=-84 29x=-58 x=-2 Plug in to find y 4(-2)-y= y=-14 -y=-6 y=6

5. Another Example
-x-3y=19 -3x-9y=57 Multiply 1st by -3 to eliminate y 3x+9y=-57 0=0 Infinitely many solutions
x-2y+z=8 2x+y+z=-11 3x-6y+3z=24 x-2y+z=8 x-2y+z=8 2x+y+z=-11 -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

6. Try one
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)
x+2y=12 3y-4z=25 x+6y+z=20 x+2y=12 -x-2y=-12 x+6y+z=20 x+6y+z=20 4y+z=8 Use this with the second equation

7. Homework page 142 13-21 odd (5 problems!)