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Triangular Form and Gaussian Elimination

Understand Triangular Form with leading terms and Gaussian Elimination to solve systems of linear equations. Learn the steps and examples to solve equations effectively.

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Triangular Form and Gaussian Elimination

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  1. Triangular Form and Gaussian Elimination Boldly on to Sec. 7.3a…

  2. Hey Bain, what the heck are they??? Well, I’m glad you asked… 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. Hey Bain, what the heck are they??? Well, I’m glad you asked… Gaussian Elimination – the process of transforming a system to triangular form 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. 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. Another Example – Solve using Gaussian Elimination: 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. Solve using Gaussian Elimination: Solution: (x, y, z, w) = (2, 1, 0, –1)

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