Math 102/Chapter 8

Chapter 8: Systems of Equations and Matrices

Chapter 8: Systems of Equations and Inequalities

• Systems of linear equations: substitution, elimination
• Systems of linear equations: matrices
• Systems of linear equations: determinants
• Matrix algebra
• Partial fractions decomposition
• Systems of nonlinear equations
• Systems of inequalities
• Linear programming

Section 8.1: Systems of Lin Eqns Substitution and Elimination

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Examples of systems of equations

Graphing a system of linear equations

Solve systems of equations by substitution

How to solve a system of linear equations by substitution

How to solve a system of linear equations by elimination

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Inconsistent system of linear equations

Solving system of dependent equations

Solving system of 3 linear equations with 3 variables

Identify inconsistent system of linear equations

Solving system of dependent equations

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Section 8.2: Systems of Lin Eqns Matrices

Writing the augmented matrix for system of linear equations

Writing systems of linear equations from augmented matrix

Applying row operation to augmented matrix

Finding particular row operation

Matrix in row echelon form

How to solve system of linear equations using matrices

Steps: Matrix method for solving a system of linear equations (row echelon form)

Solving a system of linear equations using matrices (row echelon form)

Solving a dependent system of linear equations using matrices

Solving inconsistent system of linear equations using matrices

SOlving system of linear equations using matrices

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Section 8.3: Systems of Lin Eqns Determinants

Evaluating a 2x2 determinant

Cramer's Rule for two equations containing two variables

Solving a system of linear equations using determinants

Evaluate 3 x 3 determinants

Finding minors of a 3 x 3 determinant

Cofactor of a matrix

Evaluating a 3 x 3 determinant

Cramer's Rule for three equations and three variables

Using Cramer's Rule

Demonstrating theorems of matrix manipulation

Section 8.4: Matrix Algebra

Definition of a matrix

Arranging data in a matrix

Examples of matrices

Sum and difference of two matrices and equality

Commutative property of matrix addition

Associative property of matrix addition

Demonstrating commutative property

Operations on matrices

Properties of scalar multiplication

Finding product of two matrices

Product of row vector and column vector

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Multiplying two matrices

Multiplying two matrices

Multiplying two square matrices

Multiplication with identity matrix

Multiplying matrix by its inverse

Steps: Procedure for finding inverse of non-singular matrix

Finding inverse of matrix

Showing that a matrix has no inverse

Using the inverse matrix to solve a system of linear equations

Section 8.5: Partial Fraction Decomposition

Case 1: Decompose P/Q where Q has only non-repeated linear factors

Non-repeated linear factors

Case 2: Decompose P/Q where Q has repeated linear factors

Repeated linear factors

Repeated linear factors

Case 3: Decompose P/Q where Q has irreducible quadratic factor

Non-repeated irreducible quadratic factor

Case 4: Decompose P/Q where Q has repeated irreducible quadratic factor

Repeated irreducible quadratic factor

Section 8.6: Systems of Nonlinear Equations

Solving system of non-linear equations using substitution

Solving system of non-linear equations using elimination

Solving system of non-linear equations

Solving a system of non-linear equations using elimination

Solving a system of non-linear equations

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Section 8.7: Sections of Inequalities (SKIP)

Inequalities with two variables

Graphing inequalities

Graphing system of linear equalities

Section 8.8: Linear Programming (SKIP)

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Linear programming problem with multiple constraints

Chapter 8 Objectives

Section 8.1:

• Solve systems of equations by substitution
• Solve systems of equations by elimination
• Identify inconsistent/impossible linear systems
• Express solution of system of dependent equations containing two variables
• Solve systems of three equations containing three variables
• Identify incosnistent ssytems of equations with 3 variables
• Express solution of system of dependent equations containing three variables

Section 8.2:

• Write augmented matrix of system of linear equations
• Write the system of equations from augmented matrix
• Perform row operations on matrix
• Solve system of linear equations using matrix

Section 8.3

• Evaluate 2x2 determinant
• Use Cramer's rule to solve system of equations with two vars
• Evaluate 3x3 determinant
• Use Cramer's rule to solve a 3x3 system
• Know the properties of determinants

Section 8.4

• Find sum and diff of two matrices
• Find scalar multiples of a matrix
• Find product of 2 matrices
• Find inverse of a matrix
• Solve system of linear equations using an inverse matrix