Solving Linear Systems of Equations Triangular Form Consider

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. . . Table of Contents

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. Table of Contents Slide 2

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. . . Table of Contents Slide 3

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. . . Table of Contents Slide 4

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. Table of Contents Slide 5

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. Table of Contents Slide 6

Solving Linear Systems of Equations - Triangular Form Table of Contents Slide 7

Solving Linear Systems of Equations - Triangular Form Table of Contents Slide 8

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 Table of Contents Slide 9

Solving Linear Systems of Equations - Triangular Form Table of Contents