1. Systems of Linear Equations Let’s say you need to solve the following for x, y, & z: 2x + y – 2z = 10 3x + 2y + 2z = 1 5x + 4y + 3z = 4 Two methods –Gaussian elimination –Cramer’s rule

2. Gaussian Elimination For any system of independent linear equations, we can set up the following augmented matrix: and perform elementary row operations to reduce it to row-echelon form …

3. Our Example 1.Multiply row 1 by 0.5 … 2.Multiply row 1 by -3 and add to row 2 …

4. Our example (cont.) 3.Continue until we have row-echelon form …

5. Our example (concl.) 4.This corresponds to ____________________= ____ _____________= ____ ______= ____

6. Alternative Method – Cramer’s Rule 1.Convert A to A i where A i are the matrices obtained by replacing the i th column with B:

7. Cramer’s Rule (cont.) 2.Find the determinants of each of these matrices: D = det(A) = _______ N 1 = det(A 1 ) = _______ N 2 = det(A 2 ) = _______ N 3 = det(A 3 ) = _______

8. Cramer’s Rule (concl.) 3.The unique solution is now found by: x = N 1 /D = _______ y = N 2 /D = _______ z = N 3 /D = _______

9. Cramer’s Rule Works If and Only If … 1.Number of equations = number of unknowns 2.D ≠ 0

10. Homework Solve each of the following systems of linear equations, a) using Gaussian Elimination b) using Cramer’s Rule 1.2x + y – z = 3 x + y + z = 1 x – 2y – 3z = 4 2. x – 3y – 2z = 6 2x – 4y – 3z = 8 -3x + 6y + 8z = -5