1. Systems of Linear Equations
Three Variables

2. 3 variables, 3 answers
Because there are three variables, the solution will have three answers
They are shown in an ordered triple
(x,y,z)

3. Example 1 X+2y-3z=-3 2x-5y+4z=13 5x+4y-z=5
There is one answer. What is it?
(2,-1,1)
Simple right?

4. Linear Combination and Substitiution
Use linear combination to rewrite system with three variables into system with two.
Solve the new linear system for both variables.
Substitute your 2 answers into one of the original problems.
Solve for last variable.

5. Example two 3x+2y+4z=11 2x-y+3z=4 Solve 5x-3y=5z=-1
3x+2y+4z=11 add 2 times 2nd to 1st
4x-2y+6z=8
7x+10z=19 new equation #1

6. continued 5x-3y=5z=-1 add -3 times 2nd to 3rd -6x+3y-9z=-12
-x-4z=-13 New equation #2

7. Take new equations and combine them
7x+10z=19 add7 times new eq. 2
-7x-28z=-91
-18z=-72 4
Solve for z and insert it into equation to find x=-3.

8. LAST STEP, smh Now insert X and Z into original equation Solve for y
2(-3)-y+3(4)=4 y=2
Final answer is (-3,2,4)

9. And now you know Remember, Combination then substitution.
It does not matter which variable or equation you solve for first, second, or last.
And now you know