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Solving Linear Systems of Equations - Triangular Form

Solving Linear Systems of Equations - Triangular Form. Consider the following system of equations . . . The system is easily solved by starting with equation #3 and solving for z ... . Then use equation #2 and z = -1 to solve for y .

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Solving Linear Systems of Equations - Triangular Form

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  1. Solving Linear Systems of Equations - Triangular Form • Consider the following system of equations ...   • The system is easily solved by starting with equation #3 and solving for z ... • Then use equation #2 and z = -1 to solve for y ...

  2. Solving Linear Systems of Equations - Triangular Form • Finally, use equation #1, y = 2 and z = -1 to solve for x ...  • Thus, the solution to the system is (1, 2, -1). • The process just used is called back substitution. Slide 2

  3. Solving Linear Systems of Equations - Triangular Form • Now consider the following augmented matrix representing a system of linear equations ... • Note that it is the same system used earlier ... Slide 3

  4. Solving Linear Systems of Equations - Triangular Form • If the goal was to solve the system represented by the matrix, we would proceed as before. • Write equation #3 as ... and solve ... • Write equation #2 as ... and solve using z = -1 ... • Write equation #1 as ... and solve using y = 2, z = -2 ... Slide 4

  5. Solving Linear Systems of Equations - Triangular Form • The augmented matrix at the right is considered to be in triangular form. The letters a - f represent real numbers. • Along the diagonal, all entries are 1’s. • The bottom left corner forms a triangle of 0’s. • While the 0’s are essential, the author feels that the diagonal of 1’s is not necessary. Often to get the 1’s, fractions are introduced. Slide 5

  6. Solving Linear Systems of Equations - Triangular Form • Example: • Use the augmented matrix at the right to solve the system. • Note that the matrix is in triangular form • (not considering the diagonal). • Solve the system using back substitution as before. Slide 6

  7. Solving Linear Systems of Equations - Triangular Form Slide 7

  8. Solving Linear Systems of Equations - Triangular Form Slide 8

  9. Solving Linear Systems of Equations - Triangular Form • The solution to the system is ... • Note that with an augmented matrix in triangular form, the solution is arrived at very quickly and easily. • So how do we get the triangular form of a matrix? That process is discussed in the presentation titled: • Solving Linear Systems of Equations - Gaussian Elimination Slide 9

  10. Solving Linear Systems of Equations - Triangular Form END OF PRESENTATION Click to rerun the slideshow.

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