1. 3.6 Solving Systems with Three Variables

2. 3-variable Systems
Systems with 3 variables will have 3 equations. These type of systems are in three dimensions! So it is not going to be easy to find their solution by graphing.

3. We can solve Systems with 3 variables, using Elimination OR Substitution.

4. Solve by Elimination

5. SO MUCH WORK!!!
Luckily, we have an easier way to do this! When solving system of the equations we can use Matrices!!

6. Writing Systems as a Matrix Equation
For Matrix Equations in the form
AX = B
A is called the COEFFICIENT MATRIX
X is called the VARIABLE MATRIX
B is called the CONSTANT MATRIX

7. A Coefficient is a number INFRONT of a variable.
A Variable is a value represented by a letter or symbol
A Constant is a number WITHOUT a variable.

8. Write the following System as a Matrix:

9. Remember that when solving matrix equations:
If AX=B then X = A-1B

10. Solve the Matrix Equation
(4,-10,1)

11. Write as a Matrix Equations and Solve!
(1.25, 2.5, -1.75)

12. Write the follow systems as Matrix Equations. Then Solve! 1. 2.
(-1,0,3) (4,3,4)

13. Unique Solutions
Remember, Systems can have 1 solution, NO solutions, or MANY solutions. IF matrix A’s Determinant is 0 then the matrix does NOT have an inverse and the systems does NOT have a unique solution. IF matrix A’s Determinant is NOT 0 then the matrix has an inverse and the system has a unique solutions!

14. Unique Solution
Determine if there is a unique solution.

15. Example
The sum of three numbers is 12. The 1st is 5 times the 2nd. The sum of the 1st and 3rd is 9. Find the numbers.
Represent as a system
Represent as a matrix -> majority of the time, easiest to solve as a system
(15, 3, -6)

16. HW 3.6/ Classwork
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