6 -6 Factoring by Grouping

SAT Problem of the Day

4.5 Using Matrix Row OperationsObjectives: Represent a system of equations as an augmented matrixPerform elementary row operations on matricesMatrix Row OperationsThe row-reduction method of solving a system allows you to determine whether the system is independent, dependent, or inconsistent.

The row-reduction method of solving a system is performed on an augmented matrix.An augmented matrix consists of the coefficients and constant terms in the system of equations.System

Augmented MatrixMatrix Row OperationsThe goal of the row-reduction method is to transform, if possible, the coefficient columns into columns that form an identity matrix.

This is called the reduced row-echelon form of an augmented matrix if the matrix represents an independent system. The resulting constants will represent the unique solution to the system.Elementary Row OperationsThe following operations produce equivalent matrices, and may be used in any order and as many times as necessary to obtain reduced row-echelon form. -Interchange two rows -Multiply all entries in one row by a nonzero number -Add a multiple of one row to another row Row Operations and their Notations-Interchange rows 1 and 2 -Multiply each entry in row 3 by -2 -Replace row 1 with the sum of row 1 and 4 times each entry in row 2

Example 1Perform the indicated row operations on matrix A.

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5PracticePerform the indicated row operations on matrix A.

HomeworkLesson 4.5 exercises 1, 3, 4, 9, 11, 12, 19-29 odd.Warm-UpPerform the indicated row operations on matrix A.

b) -3R2 + 4R1 R14.5 Using Matrix Row OperationsObjectives: Solve a system of linear equations by using elementary row operationsExample 1Solve the system of equations by using the row-reduction method. Then classify the system.

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x = 2; y = 7independentPracticeSolve the system of equations by using the row-reduction method. Then classify the system.

Example 2Solve the system of equations by using the row-reduction method. Then classify the system.

x 1.4z = 0y 0.2z = 00 = 1no solution, inconsistentExample 3Solve the system of equations by using the row-reduction method. Then classify the system.

x z = -1y + 2z = 30 = 0infinitely many solutions

dependentPracticeSolve the system of equations by using the row-reduction method. Then classify the system.

HomeworkLesson 4.5 exercises 31-36