Meeting 19 System of Linear Equations

Linear Equations A solution of a linear equation in n variables is a sequence of n real numbers s 1, s 2,..., s n arranged to satisfy the equation when you substitute the values into the equation.

SYSTEMS OF LINEAR EQUATIONS A system of m linear equations in variables is a set of m equations, each of which is linear in the same n variables:

A solution of a system of linear equations is a sequence of numbers s 1, s 2,..., s n that is a solution of each of the linear equations in the system. For example, the system

Examples Solve and graph each system of linear equations. Solution:

Example Solve the system Solution: By using substitution,

From Equation 3, you know the value of To solve for substitute into Equation 2 to obtain

Rewriting a system of linear equations in row-echelon form usually involves a chain of equivalent systems, each of which is obtained by using one of the three basic operations. This process is called Gaussian elimination.

Gaussian Elimination and Gauss- Jordan Elimination One common use of matrices is to represent systems of linear equations. The matrix derived from the coefficients and constant terms of a system of linear equations is called the augmented matrix of the system. The matrix containing only the coefficients of the system is called the coefficient matrix of the system.

Example

The last matrix in the previous slide is said to be in row-echelon form. The term echelon refers to the stair-step pattern formed by the nonzero elements of the matrix. To be in this form, a matrix must have the following properties.

Examples Determine whether each matrix is in row-echelon form. If it is, determine whether the matrix is in reduced row-echelon form.

Example Solve the system Solution: The augmented matrix for this system is

Obtain a leading 1 in the upper left corner and zeros elsewhere in the first column.

Now that the first column is in the desired form, change the second column as follows.

The matrix is now in row-echelon form, and the corresponding system is as follows.

Gauss-Jordan Elimination Apply elementary row operations to a matrix to obtain a (row-equivalent) row-echelon form and continues the reduction process until a reduced row-echelon form is obtained.

Example Use Gauss-Jordan elimination to solve the system. Solution: We used Gaussian elimination to obtain the row-echelon form

Now, apply elementary row operations until you obtain zeros above each of the leading 1’s, as follows.

The matrix is now in reduced row-echelon form. Converting back to a system of linear equations, we have

Exercises