1. Triangular Form and Gaussian Elimination Boldly on to Sec. 7.3a… HW: p. 602 1-19 odd

2. Triangular Form – of a system of equations has the leading term of each equation with coefficient 1, the final equation has only one variable, and each higher equation has one additional variable Example:

3. Gaussian Elimination – the process of transforming a system to triangular form. Gaussian elimination will lead to another type of matrices manipulation. Steps that can be used in Gaussian Elimination (all of which produce equivalent systems of linear equations): 1. Interchange any two equations of the system. 2. Multiply (or divide) one of the equations by any nonzero real number. 3. Add a multiple of one equation to any other equation in the system. 4. Replace the equation, and continue the process.

4. Back to our original example  Solve by substitution! Solution:

5. Another Example – Solve using Gaussian Elimination: Multiply the first equation by –3 and add the result to the second equation, replacing the second equation

6. Another Example – Solve using Gaussian Elimination: Multiply the first equation by –2 and add the result to the third equation, replacing the third equation.

7. Multiply the second equation by –2 and add the result to the third equation, replacing the third equation.  This is our first example!!!

8. Solve using Gaussian Elimination: Steps: This last equation is never true…  No Solution!!!

9. Solve using Gaussian Elimination: Solution: (x, y, z) = (5/13, 10/13, 74/13)

10. Solution: (x, y, z, w) = (2, 1, 0, –1)