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Three variables

Three variables. Systems of Equations and Inequalities. Matrices and Systems of Equations. We can translate a given system of equations into an augmented matrix . With two rows and two columns, this matrix is a 2x3 matrix. Gaussian Elimination.

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Three variables

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  1. Three variables Systems of Equations and Inequalities

  2. Matrices and Systems of Equations We can translate a given system of equations into an augmented matrix. With two rows and two columns, this matrix is a 2x3 matrix.

  3. Gaussian Elimination To solve a system of equations using Gaussian elimination with matrices, we use the same rules as before. • Interchange any two rows. • Multiply each entry in a row by the same nonzero constant. • Add a nonzero multiple of one row to another row.

  4. Example 1 Solve: Matrix:

  5. Example • Solve the system of equations using Gauss-Jordan Method

  6. Example • Solve the system of equations using Gauss-Jordan Method

  7. Example • Solve the system of equations using Gauss-Jordan Method

  8. Example • Solve the system of equations using Gauss-Jordan Method

  9. Example • Solve the system of equations using Gauss-Jordan Method

  10. Example • Solve the system of equations using Gauss-Jordan Method

  11. Example • Solve the system of equations using Gauss-Jordan Method

  12. Example • Solve the system of equations using Gauss-Jordan Method

  13. Example • Solve the system of equations using Gauss-Jordan Method

  14. Example • Solve the system of equations using Gauss-Jordan Method

  15. Example • Solve the system of equations using Gauss-Jordan Method

  16. Example • Solve the system of equations using Gauss-Jordan Method

  17. Example • Solve the system of equations using Gauss-Jordan Method

  18. Example • Solve the system of equations using Gauss-Jordan Method

  19. Example • Solve the system of equations using Gauss-Jordan Method (0, 2, 1)

  20. Systems of Equations: Matrices Definition: An m X nmatrix is a rectangular array of numbers with mrows and ncolumns. The numbers are the entries of the matrix. The subscript on the entry indicates that it is in the ith row and the jth column

  21. Augmented Matrix Linear System Augmented Matrix

  22. Elementary Row Operations • Add a multiple of one row to another. • Multiply a row by a nonzero constant. • Interchange two rows. Symbol Description Change the ith row by adding k times row j to row i, putting the result back in row i. Multiply the ith row by k. Interchange row i and row j.

  23. Example Solve: Matrix:

  24. Row-Echelon Form and Reduced Row-Echelon Form A matrix is in row-echelon form if it satisfies the following conditions. • The first nonzero entry in each row (left to right) is 1. This is called a leading 1. • The leading entry in each row is to the right of the leading entry in the row immediately above it. • Every number above and below each leading entry is a zero. This is called reduced row-echelon form.

  25. Inconsistent and Dependent Systems A leading variable is a linear system is one that corresponds to a leading entry in the row-echelon form of the matrix of the system. Suppose the system has been transformed into row-echelon form. Then exactly one of the following is true. • No solution. There is a row that represents 0 = C, where C is not zero. The system has no solution and is inconsistent. • One solution. If each variable is a leading variable, then the system has exactly one solution. • Infinitely many solutions. If there is at least one row of all zeros, the system has infinitely many solutions. The system is called dependent.

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