
To explain: the Gaussian elimination

Answer to Problem 4RCC
The Gaussian elimination work was explained.
Explanation of Solution
Gaussian elimination is the process used to solve a linear system of equations in n variables which are of the form
The linear system of equations is represented by an augmented matrix
The augmented matrix consists of all the coefficients of the variables in the system, written on the left, and the values which the equations are equalled to, written on the right.
Use the elementary row operations - addition/subtraction of rows, multiplication of a row by nonzero constants, and Switching rows - the augmented matrix is rewritten in row-echelon form.
A matrix in row-echelon form satisfies the following properties:
• The first nonzero entry is called leading entry which is 1
• The leading entry should be the right of the previous leading entry
• At the bottom of the matrix, there will be a row of zeros
If each entry immediately above and below the leading entry in each row is 0, then the matrix is said to be in reduced row-echelon form.
This process can be carried out in a systematic fashion:
• First, make the leading entry in the first row equal to 1 using elementary row operations.
• Next, use the first row to obtain a 0 in the first entry of the next row and then make the leading entry of this next row equal to 1.
• Repeat this process, making sure that the leading 1 of each row is to the right of the leading 1 in the preceding row.
• Once the matrix has the triangular form characterized by row-echelon form, the process is complete.
Now that the matrix is in a triangular form, use back-substitution to solve for each variable. To do this, rewrite the new matrix as a system of equations where the entries of the matrix are coefficients for the variables in each equation.
Solve for one variable at a time and use that information to solve for the remaining variables.
Chapter 10 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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