Systems of Linear Equations

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 • Multiply row 1 by 0.5 … • Multiply row 1 by -3 and add to row 2 …

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

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

6. Alternative Method – Cramer’s Rule • Convert A to Ai where Ai are the matrices obtained by replacing the ith column with B:

7. Cramer’s Rule (cont.) • Find the determinants of each of these matrices: D = det(A) = _______ N1 = det(A1) = _______ N2 = det(A2) = _______ N3 = det(A3) = _______

8. Cramer’s Rule (concl.) • The unique solution is now found by: x = N1/D = _______ y = N2/D = _______ z = N3/D = _______

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

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