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Solving Systems of Equations by Graphing

Parametric Representation of the Solution Set to a Linear Equation

Solving Systems of Equations by Graphing

Determining the Intersection of Two Graph on the TI83/84

Ex: Identify the Solution to a System of Equation Given a Graph, Then Verify

Ex 1: Solve a System of Equations by Graphing

Ex 2: Solve a System of Equations by Graphing

Ex: Solve a System of Equations by Graphing (Infinite Solutions)

Ex: Solve a System of Equations by Graphing (No Solution)

Regression and Systems of Equations: Application

Solving Systems of Equations Using Substitution

Solving Systems of Equations by Substitution

Ex 1: Solve a System of Equations Using Substitution

Ex 2: Solve a System of Equations Using Substitution

Ex 3: Solve a System of Equations Using Substitution

Ex 4: Solve a System of Equations Using Substitution

Ex: Solve a System of Equations Using Substitution - Infinite Solutions

Ex: Solve a System of Equations Using Substitution - No Solution

Solving Systems of Equations Using Elimination

Solving Systems of Equations by Elimination

Ex 1: Solve a System of Equations Using the Elimination Method

Ex 2: Solve a System of Equations Using the Elimination Method

Ex 3: Solve a System of Equations Using the Elimination Method

Ex: System of Equations Using Elimination (Infinite Solutions)

Ex: System of Equations Using Elimination (No Solution)

Applications of Systems of Equations

Applications Involving Systems of Equations

More Applications Involving Systems of Equations

Ex: System of Equations Application - Number Problem Using Substitution

Ex: System of Equations Application - Number Problem Using Elimination

Ex: System of Equations Application - Compare Phone Plans

Ex: System of Equations Application - Entrance Fees

Ex: System of Equations Application - Coin Problem

Ex: System of Equations Application - Plane and Wind Problem

Ex: System of Equations Application - Commission and Salary

Ex: System of Equations Application - Investment Accounts

Ex: System of Equations Application (Corral Perimeter)

Ex: System of Equations Application - Mixture Problem

Systems of Equations with Three Unknowns

System of Equations in Three Variables: Part 1, Part 2

Ex 1: System of Three Equations with Three Unknowns Using Elimination

Ex 2: System of Three Equations with Three Unknowns Using Elimination

Ex 3: System of Three Equations with Three Unknowns Using Elimination (Fractions)

Ex 4: System of Three Equations with Three Unknowns Using Elimination (No Solution)

Ex 5: System of Three Equations with Three Unknowns Using Elimination (Infinite Solutions)

Matrices

Dimensions of a Matrix

Ex: Determine the Dimensions (Size) of a Matrix

Ex: Determine the Dimensions and Elements of a Matrix

Introduction to Augmented Matrices

Diagonal Matrices

Ex: Find the Transpose of a Matrix

Transpose of a Matrix

Row Echelon Form, Pivot Positions, Basic and Free Variables

Echelon Form, Pivots, and Free Variables

Augmented Matrices

Augmented Matrices: Row Echelon Form

Ex: Write a System of Equations as an Augmented Matrix (3x3)

Ex: Write a System of Equations as a Matrix Equation (3x3)

Ex: Give the Solution From an Augmented Matrix in RREF (3x3)

Ex 1: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)

Ex 2: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)

Ex 3: Solve a System of Two Equations with Using an Augmented Matrix (Row Echelon Form)

Ex 1: Solve a System of Three Equations with Using an Augmented Matrix (Row Echelon Form)

Ex 2: Solve a System of Three Equations with Using an Augmented Matrix (Row Echelon Form)

Ex 3: Solve a System of Three Equations with Using an Augmented Matrix (Row Echelon Form)

Augmented Matrices: Reduced Row Echelon Form

Ex 1: Solve a System of Two Equations Using an Augmented Matrix (Reduced Row Echelon Form)

Ex 2: Solve a System of Two Equations Using an Augmented Matrix (Reduced Row Echelon Form)

Ex 3: Solve a System of Two Equations Using an Augmented Matrix (Reduced Row Echelon Form)

Ex: Solve a System of Three Equations Using an Augmented Matrix (Reduced Row Echelon Form)

Ex: Solve a System of Equation By Writing an Augmented Matrix in RREF (3x3 Infinite Sol)

Ex: Write a 3x3 Matrix in Reduced Row Echelon Form (Identity)

Ex: Write a 3x3 Matrix in Reduced Row Echelon Form (Not Identity)

Ex: Write a 4x4 Matrix in Reduced Row Echelon Form (Not Identity)

Augmented Matrices on the Graphing Calculator

Use Elementary Matrices to Perform Row Operations to Solve a System

Write a Matrix as a Product of Elementary Matrices

Matrix Operations

Use Elementary Matrices to Perform Row Operations to Solve a System

Write a Matrix as a Product of Elementary Matrices

Matrix Addition, Subtraction and Scalar Multiplication

Ex: Add Two 2x2 Matrices

Ex: Subtract Two 2x3 Matrices

Ex: Matrix Addition and Subtraction

Ex: Perform Matrix Scalar Multiplication

Ex: Perform Matrix Scalar Multiplication and Matrix Addition

Ex: Matrix Scalar Multiplication

Ex: Matrix Operations - Scalar Multiplication, Addition, and Subtraction

Ex: Matrix Addition Application - Translation

Ex 1: Matrix Application: Translation / Dilation

Ex 2: Matrix Application: Translation / Dilation

Ex: Use Matrices to Dilate and Translate a Triangle

Ex: Matrix Scalar Multiplication Application - Dilation

Matrix Multiplication

Ex: Determine if Matrix Multiplication is Possible

Ex: Solve a Basic Matrix Equation (No Inverse)

Ex: Matrix Multiplication (2x2)*(2x2)

Ex: Matrix Multiplication (2x3)*(3x4)

Ex: Square a 2x2 Matrix

Matrix Multiplication on the Graphing Calculator

Ex 1: Matrix Multiplication (Basic)

Ex 2: Matrix Multiplication (2x3)*(3x2)

Ex 3: Matrix Multiplication (3x2)*(2x3)

Ex Applicaton of Matrix Multiplication - Transformation

Animation: Matrix Multiplication

The Identity Matrix

Elementary Matrices

LU Decomposition Using Elementary Matrices

LU Decomposition - Shortcut

Inverse Matrices

2 x 2 Inverse Matrix Formula

Inverse Matrices Using Augmented Matrices

Ex: Find the Inverse of a 2x2 Matrix Using a Formula

Ex: Inverse of a 2x2 Matrix Using an Augmented Matrix

Ex 1: Inverse of a 3x3 Matrix Using an Augmented Matrix

Ex 2: Inverse of a 3x3 Matrix Using an Augmented Matrix

Inverse Matrices on the Graphing Calculator

Matrix Equations

Ex 1: Solve the Matrix Equation AX=B (2x2)

Ex 2: Solve the Matrix Equation AX=B (2x2)

Ex: Solve the Matrix Equation AX=B (3x3)

Matrix Equations

Ex 1: Solve a System of Two Equations Using a Matrix Equation

Ex 2: Solve a System of Two Equations Using a Matrix Equation

Ex: Solve a System of Three Equations Using a Matrix Equation

LU Decomposition Using Elementary Matrices

LU Decomposition - Shortcut

Solve a System of Linear Equations Using LU Decomposition

Perform Linear Regression Using Matrices

Determinants

Determinants

Ex: Determinant of a 2x2 Matrix

Ex: Solve an Equation Involving a Determinant

Ex 1: Determinant of 3x3 Matrix - Diagonal Method

Ex 2: Determinant of 3x3 Matrix - Diagonal Method

Ex 1: Determinant of 3x3 Matrix - Cofactor Method

Ex 2: Determinant of 3x3 Matrix - Cofactor Method

Ex: Find a 3x3 Determinant using Cofactor Expansion on Row 2

Ex: Find a 3x3 Determinant using Cofactor Expansion on Row 3

Ex: Find the Value of a 4x4 Determinant Using Cofactor Expansion (with Zeros)

Determinants on the Graphing Calculator

Cramer’s Rule

Ex: Solve a System of Two Equations Using Cramer's Rule

Ex: Solve a System of Three Equations Using Cramer's Rule

Area Using Determinants

Ex: Find the Area of a Triangle on the Coordinate Plane Using a Determinant

Ex: Determine if a 2x2 Matrix is Invertible (nonsingular) Using a Determinant

Ex: Determine if a 3x3 Matrix is Invertible (nonsingular) Using a Determinant

Ex: Determine a Value in a 2x2 Matrix To Make the Matrix Singular

Ex: Determine a Value in a 3x3 Matrix To Make the Matrix Singular

Vectors

Introduction to Vectors

Vector Operations

Unit Vector

Ex: Find the Sum of Two Vectors From a Graph (2 Dimensions)

Ex: 2D Vector Scalar Multiplication

Ex: Find the Unit Vector Given the Graph of a Vector in 2D

Find the Component Form of a Vector from the Graph of a Vector

Ex: Find the Direction and Magnitude of a Vector in Component Form

Find the Component Form of a Vector Given Magnitude and Direction

Ex: Find the Difference of Two Vectors in Component Form

Ex: Find the Sum of Two Vectors Given in Linear Combination Form

Ex: Find the Difference of Two Vector Given in Linear Combination Form

Ex: Find the Difference of Scalar Multiples of Vectors in 2D

Ex: Determine if Vectors are Linearly Independent or Linearly Dependent (Dependent)

Ex: Determine if Vectors are Linearly Independent or Linearly Dependent (Independent)

Ex: Dot Product of Vectors - 2D

Ex: Dot Product of Vectors From a Graph - 2D

Ex 1: Fine a Vector in Component Form Given an Angle and the Magnitude (30)

Ex 2: Fine a Vector in Component Form Given an Angle and the Magnitude (45)

Ex 3: Fine a Vector in Component Form Given an Angle and the Magnitude (60)

Ex 4: Fine a Vector in Component Form Given an Angle and the Magnitude (90)

Ex 5: Fine a Vector in Component Form Given an Angle and the Magnitude (180)

Ex: Find the Difference of Scalar Multiples of Vectors in 2D

Ex: Find the Magnitude of The Difference of Two Vectors and The Difference of Two Magnitudes

Ex: Write a Vector as a Combination of Two Vectors

Ex: Find the Net Force of Three Vectors and the Opposite Force

Ex: Find the Coordinates of a Rotated Point Using Vectors

Ex: Direction and Speed of a Plane in the Wind Using Vectors

Applications of Vectors

Applications of Vectors

Determining the Angle Between Two Vectors

Proof of the formula for the Angle Between Two Vectors

Vector Projection

Proof of the Vector Projection Formula

Ex: Vector Projection in Two Dimensions

Ex: Find the Angle of Intersection of Two Curves Using Vectors

Vector Applications: Force and Work

Vectors in Space

Plotting Points in 3D

The Equation of a Sphere

Ex: Find the Difference of Scalar Multiples of Two Vectors in 3D (Linear Combination Form)

Vectors in Space

Parallel Vectors

Ex: Dot Product of Vectors - 3D

Ex: Find the Component of a Vector so Two Vectors are Orthogonal (3D)

Ex: Find the Angle Between Two Vectors in Three Dimensions

Ex: Vector Projection in Three Dimensions

Ex: Find the Component Form of a Vector in Space Given the Initial and Terminal Point

Ex: Find a Unit Vector in the Direction of a Given Vector in 3D

Ex: Find the Magnitude of a Vector in 3D

Ex: Find the Sum of Scalar Multiples of Two Vectors in 3D (Component Form

Ex: Find the Difference of Scalar Multiples of Two Vectors in 3D (Linear Combination Form)

Vector Cross Products

Ex: Find the Cross Product of Two Vectors

Ex 1: Properties of Cross Products - Cross Product of a Sum and Difference

Ex 2: Properties of Cross Products - Cross Product of a Sum and Difference

Ex: Find the Area of a Triangle Using Vectors - 3D

Ex: Find the Distance Between Two Points In Space

An Application of Cross Products: Torque

The Triple Scalar Product: Volume of a Parallelepiped

Parametric Equations of Lines in 3D

The Equation of a Plane in 3D Using Vectors

Graphing a Plane in 3D

Determining the Angle Between Two Planes

Determining the Distance Between a Plane and a Point

Determining the Distance Between a Line and a Point

Ex: Find the Distance Between Two Parallel Planes

Ex: Find the Equation of the Plane Containing a Given Line and a Point Using Vectors

Ex: Find the Equation of a Plane Given Three Points in the Plane Using Vectors

Ex: Find the Equation of a Plane Given an Orthogonal Line (Parametric) and a Point

Ex: Find the Equation of Plane Containing a Line and Orthogonal to a Given Plane

Ex: Find the Equation of a Plane Given a Point in the Plane and a Parallel Plane

Ex 1: Find the Parametric Equations of the Line of Intersection of Two Planes Using Vectors

Ex 2: Find the Parametric Equations of the Line of Intersection of Two Planes Using Vectors

Ex: Find the Parametric Equations of a Line Perpendicular to a Plane Through a Point

Eigenvalues and Eigenvectors

Ex: Find a Value to Make 2x2 Matrix Singular (Eigenvalue Warm Up)

Ex: Find the Eigenvalues of a 2x2 Matrix

Ex: Find the Eigenvalues of a 2x2 Matrix (Complex)

Ex: Find the Eigenvalues of a 3x3 Matrix

Ex: Find the Eigenvalues of a 4x4 Matrix

Ex 1: Find the Eigenvalues and Corresponding Eigenvectors of a 2x2 Matrix

Ex 2: Find the Eigenvalues and Corresponding Unit Eigenvectors of a 2x2 Matrix

Ex: Find the Corresponding Eigenvectors Given an Eigenvalues (Complex)

Ex 1: Find the Eigenvalues and Corresponding Eigenvectors of a 3x3 Matrix

Ex 2: Find the Eigenvalues and Corresponding Unit Eigenvectors of a 3x3 Matrix

Ex: Find the Trace and Determinant of a 3x3 Matrix Using Eigenvalues