1. Section 7-3 Solving 3 x 3 systems of equations

2. Solving 3 x 3 Systems  substitution (triangular form)  Gaussian elimination  using an augmented matrix (algebraic)  using an inverse matrix (calc.)

3. Substitution – Triangular Form  if you can simplify a system of equations into the triangular form seen below, you can solve the system by substitution x – 2y + z = 7 y – 2z = -7 z = 3

4. Gaussian Elimination  this method is used to change a 3 equation-3 unknown system of equations into triangular form x – 2y + z = 7 3x – 5y + z = 14 2x – 2y – z = 3  x – 2y + z = 7 y – 2z = -7 z = 3

5. Gaussian Elimination The following operations can be used: 1. interchange any two equations 2. multiply (or divide) one of the equations by a real number 3. add a multiple of one equation to any other equation

6. Augmented Matrix  using an augmented matrix does the same work as elimination without having to re-write the equations and variables over and over again  the following is the augmented matrix for the same system used earlier

7. Augmented Matrix  the goal is to use the same techniques as for elimination to change the matrix into “triangular form” so it can be finished off with substitution  “triangular form” for an augmented matrix is called row echelon form

8. Row Echelon Form  rows consisting of all 0’s are on the bottom  the first non-zero entry of a row is 1  1’s are along the diagonal  the row echelon form of a system is not unique

9. Row Operations The row operations for a matrix are the same as for Gaussian elimination 1. Interchange any two rows 2. Multiply all elements of a row by a number 3. Add a multiple of one row to any other row

10. Reduced Row Echelon Form  instead of stopping to use substitution after you get to row echelon form, you can keep going using the augmented matrix to get the solution  this new form is called reduced row echelon form The solution of the system is (a, b, c)

11. Order of Attack

12. Special Cases  when using either technique and the variables disappear from a row, it leads to a special case:  if you are left with... false statement = no solution true statement = infinitely many solutions