The method of solution where in it is based on addition/elimination, there is a systematized method for solving the three-or-more variable system. This method is called "Gaussian Elimination" (with the equations ending up in what is called "row-echelon form").
Elementary Row Operations:
1. Interchange two equations.
2. Multiply an equation by a nonzero constant.
3. Add a multiple of an equation to another equation.
Row-Echelon Form of a Matrix
*The term "echelon" refers to the stair-step pattern formed by the nonzero elements of the matrix.
Definition of Row-Echelon Form of a Matrix:
A Matrix that is in Row-Echelon Form has the following properties:
1. All rows consisting entirely of zeros occur at the bottom of the matrix.
2. For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1).
3. For two successive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.
Gaussian Elimination w/ Back Substitution:
1. Write the augmented matrix of the system of linear equations.
2. Use the elementary row operations to rewrite the augmented matrix in row-echelon form.
3. Write the system of linear equations corresponding to the matrix in row-echelon form, and use back substitution to find the solution.
Example #1.
Example #2.
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