Review: Solve a system of 3 equations using Substitution

1. Review: Solve a system of 3 equations using Substitution

2. 10.2 Systems of Linear Equations: Matrices Objectives • Write the Augmented Matrix • Write the System from the Augmented matrix • Perform Row Operations • Solve a System of Linear Equations using Matrices • Row operations • Row-echelon form (Gauss Elimination) • Reduced Row-echelon (Gauss-Jordan)

3. 1. Augmented Matrix Given the system: The coefficient matrix is: The augmented matrix is:

4. 1. Augmented Matrix write the augmented matrix for the warm-up

5. 2. Write System from augmented matrix

6. 3. Row Operations Notation: new row after row operations are applied original row Multiply row i by a constant k Interchange row 1 and row 2

7. 3. a) An example of row operations Example: Perform the following operations. (Interchange row 1 and row 2) (Add row 1 to row 2) (Add -2 times row 1 to row 2)

8. 3 b) Row-Echelon Form Augmented matrix reduced to a form with 1’s on diagonal and 0’s beneath diagonal is called row-echelon form A 2x2 system would have the form: A 3x3 system would have the form: The method for solving a system using row-echelon form is also known as Gaussian Elimination.

9. Note: The augmented matrix from warm-up is in row-echelon form Example 1: Solve the 2x2 system p. 755 #37 Handout 10.2: Solving Linear Systems using Matrices