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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Solving Systems of Equations Using Matrices

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Table of Contents

I. Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
II. Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
III. Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
IV. Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
V. Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
VI. Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
VII. Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
VIII. Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
IX. Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
X. Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
XI. Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (12)

1 answer

Last reply by: Dr Carleen Eaton
Fri May 24, 2013 9:29 PM

Post by Kavita Agrawal on May 23, 2013

Which method would you say is faster - Cramer's rule or matrix multiplication (as shown above)?

1 answer

Last reply by: richard lawlor
Tue Sep 4, 2012 6:55 AM

Post by richard lawlor on September 4, 2012

What about dimension requirements for matrix multiplication i.e. number of rows must equal number of columns?This is not the case in the above example?

1 answer

Last reply by: Dr Carleen Eaton
Sun Jan 29, 2012 4:42 PM

Post by Edmund Mercado on January 27, 2012

Dr. Eaton, thank you for a very detailed explanation of a very difficult procedure.

3 answers

Last reply by: Carroll Fields
Sat Sep 14, 2013 2:20 PM

Post by Lemel Covington on November 10, 2011

example 3 was doe incorrectly as well. you did not flip the -7 and 1 in the matrix. you flipped the a and c.

1 answer

Last reply by: Dr Carleen Eaton
Mon Nov 14, 2011 10:59 PM

Post by Jeff Mitchell on March 7, 2011

In example 2 (approx. 10:22 into lecture) the equation for A inverse was incorrectly written as 1 / ab-bc and should be 1 / ad-bc.

Jeff

Solving Systems of Equations Using Matrices

  • We can write a system of equations in matrix form. The result is a matrix equation.
  • A system of equations can be solved using matrices. The solution is the product of the inverse of the matrix of coefficients and the matrix of constants.
  • The system has a unique solution if and only if the matrix of coefficients has an inverse.
  • If the matrix of coefficients does not have an inverse, then either there is no solution or there are an infinite number of solutions.

Solving Systems of Equations Using Matrices

Write the system as a matrix equation.
x + 6y = − 12
− 7x − 3y = 6
  • Recall that a matrix equation looks like this Ax = B
  • Find A, x, and B
  • A = [
    1
    6
    − 7
    − 3
    ]
  • x = [
    x
    y
    ]
  • B = [
    − 12
    6
    ]
  • Ax = B
  • [
    1
    6
    − 7
    − 3
    ]*[
    x
    y
    ] = [
    − 12
    6
    ]
[
1
6
− 7
− 3
]*[
x
y
] = [
− 12
6
]
Write the system as a matrix equation.
x − 3y = − 20
5x − y = 12
  • Recall that a matrix equation looks like this Ax = B
  • Find A, x, and B
  • A = [
    1
    − 3
    5
    − 1
    ]
  • x = [
    x
    y
    ]
  • B = [
    − 20
    12
    ]
  • Ax = B
  • [
    1
    − 3
    5
    − 1
    ]*[
    x
    y
    ] = [
    − 20
    12
    ]
[
1
− 3
5
− 1
]*[
x
y
] = [
− 20
12
]
Write the system as a matrix equation.
5x − 4y = − 2
x + 2y = 22
  • Recall that a matrix equation looks like this Ax = B
  • Find A, x, and B
  • A = [
    5
    − 4
    1
    2
    ]
  • x = [
    x
    y
    ]
  • B = [
    − 2
    22
    ]
  • Ax = B
  • [
    5
    − 4
    1
    2
    ]*[
    x
    y
    ] = [
    − 2
    22
    ]
[
5
− 4
1
2
]*[
x
y
] = [
− 2
22
]
Write the system as a matrix equation.
x + 3y = − 8
− 3x − 6y = 15
  • Recall that a matrix equation looks like this Ax = B
  • Find A, x, and B
  • A = [
    1
    3
    − 3
    − 6
    ]
  • x = [
    x
    y
    ]
  • B = [
    − 8
    15
    ]
  • Ax = B
  • [
    1
    3
    − 3
    − 6
    ]*[
    x
    y
    ] = [
    − 8
    15
    ]
[
1
3
− 3
− 6
]*[
x
y
] = [
− 8
15
]
Solve using a matrix equation
5x − 3y = 19
− 3x + y = − 17
  • In order to solve this system of equations using matrices, it is assumed that you know how to
  • 1) Find Determinant of a 2x2 matrix
  • 2) Find the Inverse of a matrix
  • 3) Multiply Matrices
  • Recall that to find the solution, you must isolate
  • x = A − 1B
  • A = [
    5
    − 3
    − 3
    1
    ]
  • B = [
    19
    − 17
    ]
  • x = [
    x
    y
    ]
  • Step 1: Find the Inverse A − 1
  • A − 1 = [1/detA][
    1
    3
    3
    5
    ] = [
    − [1/4]
    − [3/4]
    − [3/4]
    − [5/4]
    ]
  • Step 2: Multiply A − 1B
  • [
    x
    y
    ] = [
    − [1/4]
    − [3/4]
    − [3/4]
    − [5/4]
    ]*[
    19
    − 17
    ] = [
    8
    7
    ]
Solution (8,7)
Solve using a matrix equation
x − 3y = − 20
5x − y = 12
  • In order to solve this system of equations using matrices, it is assumed that you know how to
  • 1) Find Determinant of a 2x2 matrix
  • 2) Find the Inverse of a matrix
  • 3) Multiply Matrices
  • Recall that to find the solution, you must isolate
  • x = A − 1B
  • A = [
    1
    − 3
    5
    − 1
    ]
  • B = [
    − 20
    12
    ]
  • x = [
    x
    y
    ]
  • Step 1: Find the Inverse A − 1
  • A − 1 = [1/detA][
    − 1
    3
    − 5
    1
    ] = [
    − [1/14]
    [3/14]
    − [5/14]
    [1/14]
    ]
  • Step 2: Multiply A − 1B
  • [
    x
    y
    ] = [
    − [1/14]
    [3/14]
    − [5/14]
    [1/14]
    ]*[
    − 20
    12
    ] = [
    4
    8
    ]
Solution (4,8)
Solve using a matrix equation
5x − 4y = − 2
x + 2y = 22
  • In order to solve this system of equations using matrices, it is assumed that you know how to
  • 1) Find Determinant of a 2x2 matrix
  • 2) Find the Inverse of a matrix
  • 3) Multiply Matrices
  • Recall that to find the solution, you must isolate
  • x = A − 1B
  • A = [
    5
    − 4
    1
    2
    ]
  • B = [
    − 2
    22
    ]
  • x = [
    x
    y
    ]
  • Step 1: Find the Inverse A − 1
  • A − 1 = [1/detA][
    2
    4
    − 1
    5
    ] = [
    [1/7]
    [4/14]
    − [1/14]
    [5/14]
    ]
  • Step 2: Multiply A − 1B
  • [
    x
    y
    ] = [
    [1/7]
    [4/14]
    − [1/14]
    [5/14]
    ]*[
    − 2
    22
    ] = [
    6
    8
    ]
Solution (6,8)
Solve using a matrix equation
x + 3y = − 8
− 3x − 6y = 15
  • In order to solve this system of equations using matrices, it is assumed that you know how to
  • 1) Find Determinant of a 2x2 matrix
  • 2) Find the Inverse of a matrix
  • 3) Multiply Matrices
  • Recall that to find the solution, you must isolate
  • x = A − 1B
  • A = [
    1
    3
    − 3
    − 6
    ]
  • B = [
    − 8
    15
    ]
  • x = [
    x
    y
    ]
  • Step 1: Find the Inverse A − 1
  • A − 1 = [1/detA][
    − 6
    − 3
    3
    1
    ] = [
    − 2
    − 1
    1
    [1/3]
    ]
  • Step 2: Multiply A − 1B
  • [
    x
    y
    ] = [
    − 2
    − 1
    1
    [1/3]
    ]*[
    − 8
    15
    ] = [
    1
    − 3
    ]
Solution (1, − 3)
Solve using a matrix equation
− 4x + 6y = 0
7x + y = 23
  • In order to solve this system of equations using matrices, it is assumed that you know how to
  • 1) Find Determinant of a 2x2 matrix
  • 2) Find the Inverse of a matrix
  • 3) Multiply Matrices
  • Recall that to find the solution, you must isolate
  • x = A − 1B
  • A = [
    − 4
    6
    7
    1
    ]
  • B = [
    0
    23
    ]
  • x = [
    x
    y
    ]
  • Step 1: Find the Inverse A − 1
  • A − 1 = [1/detA][
    1
    − 6
    − 7
    − 4
    ] = [
    − [1/46]
    [3/23]
    [7/47]
    [2/23]
    ]
  • Step 2: Multiply A − 1B
  • [
    x
    y
    ] = [
    − [1/46]
    [3/23]
    [7/47]
    [2/23]
    ]*[
    0
    23
    ] = [
    3
    2
    ]
Solution (3,2)
Solve using a matrix equation
4x − 3y = 6
x − 8y = 16
  • In order to solve this system of equations using matrices, it is assumed that you know how to
  • 1) Find Determinant of a 2x2 matrix
  • 2) Find the Inverse of a matrix
  • 3) Multiply Matrices
  • Recall that to find the solution, you must isolate
  • x = A − 1B
  • A = [
    4
    − 3
    1
    − 8
    ] B = [
    6
    16
    ] x = [
    x
    y
    ]
  • Step 1: Find the Inverse A − 1
  • A − 1 = [1/detA][
    − 8
    3
    − 1
    4
    ] = [
    [8/29]
    − [3/29]
    [1/29]
    − [4/29]
    ]
  • Step 2: Multiply A − 1B
  • [
    x
    y
    ] = [
    [8/29]
    − [3/29]
    [1/29]
    − [4/29]
    ]*[
    6
    16
    ] = [
    0
    − 2
    ]
Solution (0, − 2)

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Solving Systems of Equations Using Matrices

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Matrix Equations 0:11
    • Example: System of Equations
  • Solving Systems of Equations 4:01
    • Isolate x
    • Example: Using Numbers
    • Multiplicative Inverse
  • Example 1: Write as Matrix Equation 7:18
  • Example 2: Use Matrix Equations 9:12
  • Example 3: Use Matrix Equations 15:06
  • Example 4: Use Matrix Equations 19:35

Transcription: Solving Systems of Equations Using Matrices

Welcome to Educator.com.0000

In today's lesson, we are going to apply what we have learned with matrices so far,0002

in order to solve systems of equations using matrices.0006

OK, first: a system of equations can be written as a single matrix equation, and that is the first step to solving the system of equations.0011

So, let's use an example here: now, recall that, in previous lessons, we talked about a lot of other methods to solve equations,0022

such as substitution and elimination; and it depends on the situation which is the easiest method.0034

But this gives you another option, and it can be the best method in certain situations.0040

So, first, you are going to write one matrix using the coefficients of x and the coefficients of y; and I am going to call that matrix A.0044

In the first column is the coefficients of x; in the second column, I am going to have coefficients of y.0056

OK, that is the first matrix: the second matrix I will call B, and for this matrix,0080

you are going to use the constants that are on the right side of the equation: 1 and 3.0088

Finally, you are going to write a third matrix that I will call X; and that is going to consist of variables in the system of equations.0095

OK, so now I have these three matrices; but I want to write them as an equation.0106

And how I could write these is that I could say that matrix A times matrix X equals matrix B.0112

And let's look at why: if I have matrix A, that is 5, 6, 4, 3 for the elements, times matrix X (that is my variables), it is going to give me my constants.0124

Well, let's just look at what this is saying by using matrix multiplication.0140

This is my matrix equation for this system of equations: I have written this matrix equation right here.0147

All right, let's multiply this out and see what happens.0156

Well, with matrix multiplication, I am going to multiply this first row times the column (I only have one column).0159

And it is going to give me, for my product matrix, 5x + 4y; I can't go any farther.0165

OK, now for this second row, I am going to also multiply it by the column; and then, that is going to give me 6x + 3y =...0177

Recall the definition of equal matrices: equal matrices have corresponding elements that are equal.0193

So, this element here (row 1, column 1) and this element here (row 2, column 2) correspond to row 1, column 1, row 2, column 1, here and here.0200

OK, so that gives me 5x + 4y =...I look at the corresponding element in row 1, column 1.0213

Here, I look at row 2, column 1, and my corresponding element here; and I get my original equations back.0222

So, that shows that this system of equations is equivalent to this matrix equation.0229

And that is what allows us to write this system of equations as a matrix equation.0235

OK, now let's talk about solving systems of equations.0242

We found this matrix equation AX = B: however, if I want to solve, what I really want to find is...0246

I want to isolate X, because recall that the matrix X, we defined as containing the variables.0257

And I want to isolate that, the same way as, if I am solving a regular system of equations, I want to isolate the variable.0265

Well, we already talked about how this matrix equation, AX = B, can be written for a system of equations.0272

So now, let's talk about isolating X.0279

Recall that what we want to do is get rid of the A; so we can do that by multiplying it by its inverse.0284

I am thinking for a second about why we are doing that: we are taking it, and we are multiplying it by its multiplicative inverse--matrix inverse--A-1.0296

Well, think about when we are working with just regular numbers.0308

If I have 3x + 4, what I am going to do is multiply 3 by its multiplicative inverse, 1/3.0310

If I have 3x = 4, and I want to isolate the x, I am going to divide by 3; or I could say I am going to multiply by 1/3.0319

So, if I multiply 3 by its multiplicative inverse, 1/3, this is going to cancel out, and I am going to end up with 1.0330

Well, what is going to happen here? Recall that, when we talked about identity matrices,0344

we said they function somewhat like the number 1 in the world of matrices.0348

So, if I am multiplying a matrix by its multiplicative inverse, then I am going to get the identity matrix.0353

In the same, when I multiplied a number by its inverse, I got the number 1.0362

So, A times A-1, which we discussed in previous lessons on identity matrices, is the identity matrix.0367

So, this is going to give me the identity matrix, times X, equals A-1 times B.0377

OK, now also recall that, if you multiply a matrix by its identity matrix, you get the original matrix back.0384

What that tells me is that, if I multiply X by its identity matrix, that is the same as the matrix X.0396

So, I can just write this as X = A-1B.0405

OK, so this is what is important to know; but this shows you how we got there.0409

Now, in the last slide, we talked about writing the matrix equation AX = B.0414

Here, we talked about isolating X; so what you really want to work with is this, where X equals A-1 times B,0419

because that will allow you to solve for this matrix, and therefore to solve for the variables in the system of equations.0429

For example, write the system as a matrix equation.0441

First, all this is asking is to write it as a matrix equation; we don't even need to solve.0446

So, I have 2x - 3y = -7; this is 3x + 2y = 8; and we talked earlier about writing a matrix equation, AX = B.0457

And that is going to consist of three matrices.0472

The first matrix, A, is going to have, in the first column, the coefficients of x.0474

In the second column, it is going to have the coefficients of y.0482

Then, I am going to write a second matrix, B, which is going to have elements that consist of the constants, -7 and 8.0487

Then, I am going to have a third matrix that contains the variables.0499

The next thing to do is to put these together into an equation: that is going to give me A, times X, equals B.0506

The matrix equation for this system of equations is shown here: and this is AX = B.0532

Again, I wasn't asked to solve it, so I have done what I have been asked, which is to write the system as a matrix equation.0540

And we can use that as the basis of solving systems of equations, just as we are going to do in this problem.0547

The first thing to do is to write the matrix equation, so I need my three matrices: the first one is A;0563

and the coefficient of x is 1; and the other is 2; the coefficients of y are -2 and 1.0574

The second matrix, B, contains the constants for elements.0581

And then, finally, we have X, with my variables x and y.0587

Now, from this I could write AX = B; and we saw, earlier on, how, in order to solve it, I need to rearrange things0590

and do some manipulation and end up with this, because I really want to solve for the matrix X, because it contains the variable.0601

So, this is the equation I want; and I have A, B, and X, so what I need is A-1.0609

Recall from an earlier lesson that A-1 is 1 over the determinant, times this matrix.0615

OK, so all I am doing right now is finding A-1, if it exists.0630

This is ad, which is 1 times 1, minus -2 times 2.0636

And just looking over at this, this is going to be -4, so it is 1 minus -4, so that is not 0.0643

So, I am OK--the determinant does exist--so I am going to continue.0650

I am going to switch these two, but they are the same number, so it ends up 1 and 1.0654

Then, I am going to take -2 and reverse its sign; so that is going to be 2.0658

Here, I am going to take 2 and reverse its sign (because that is c), and I am going to make it -2.0664

This is going to give me 1 over 1 - -4, times this matrix.0672

A-1 is 1 over 1 + 4, which is going to give me 1/5.0682

So, I don't even have to multiply this out, because I am going to be doing some multiplication over here.0692

So, leaving this as it is: now I want to have X = A-1 times B.0696

So, X equals...here is A-1, 1/5 times its matrix, 1, 2, -2, 1, times B; B is 3, 7.0703

Now, I am going to go ahead and do my matrix multiplication to find the product matrix.0718

So, the product matrix would be: for row 1, column 1, this is 1, 2, times 3, 7--so 1 times 3, plus 2 times 7, equals 3 + 14; that is 17.0727

So, 17 goes right here; now, in row 1, column 2, right here, this is going to be 1...actually, it only has one row,0747

so it is just row 2, column 1; that is all I need to find; row 2 right here is the next position, because I have no other column here.0764

So, it is going to be row 2, column 1; that is going to give me -2 times 3, plus 1 times 7; that is -6 plus 7, which is going to give me 1.0774

OK, I found that I have x = 1/5 times 17/1, which equals...I am going to multiply 1/5 by 17--that is 17/5.0792

And multiplying 1/5 by 1, that is 1/5.0814

Now, the matrix X equals this; recall that matrix X also equals xy; so these two are equal;0820

therefore, the corresponding elements are equal; so I can say that x = 17/5 and y = 1/5.0830

It is a little bit complicated, but you just have to take it one step at a time.0840

I was asked to solve this using a matrix equation.0843

I wanted to find this equation, because then I would have the matrix with the variables in it isolated.0847

I started out by writing three matrices--one containing the x and y coefficients, one containing the constants, and one containing the variables.0854

Then, I had A, but I needed to find A-1; and I used my formula to find that.0864

I went through and found that this is A-1.0870

With that, I could then insert it into this formula: A-1, right here, times B.0874

I went ahead and did matrix multiplication to find that I got this matrix as the product.0884

And then, I multiplied it by 1/5, which I still had out here, to get the matrix X equals 17/5 and 1/5, which equals x and y.0892

So, that is my solution to this system of equations.0901

Example 3: again, solve using a matrix equation.0908

So, I am going to go ahead and find my three matrices that I will need to have,0913

because I eventually want to end up with X = A-1B.0918

So, A is a matrix with coefficients of x 3 and -1, and coefficients of y 7 and -2.0923

Matrix B has elements that are the constants 4 and 3; and then, matrix x has the variables.0934

The next thing is: I actually don't need A; I need A-1.0943

And I am going to recall my formula, that this equals 1 over the determinant, ad - bc, times this matrix.0948

So, A-1 is 1 over the determinant, which is 3 times -2, minus 7 times -1,0962

times this matrix: a and d switch positions, so it is going to be -2, 3; b reverses its sign; and c reverses its sign.0976

Proceeding to find the determinant: 3 times -2 is -6...that is minus -7, times this matrix...0989

that is going to give me A-1 is 1 over...well, this is -6 minus -7, or -6 + 7, so that is just 1.1006

OK, I have A-1: I found A, B, and C matrices; now I have A-1.1022

And this is what I want: I want that X = A-1B.1027

So, I am going to write A-1 up here; and I am just going to write this as 1/1, times B; B is right here.1033

I am doing some matrix multiplication: row 1, column 1 is going to be -2 times 4, plus 7 times 3.1050

That is going to be -8; let me see, this should actually be a -7.1069

This is going to be -2 times 4; and this is going to be 3, -7, times 3.1087

So, row 1, column 1: -2 times 4 and -7 times 3.1097

That is going to give me -8, minus 21, which is going to be -29 for row 1, column 1.1101

Row 2, column 1: row 2 here, column 1 here: that is going to be 1 times 4, plus 3 times 3, equals 4 + 9, equals 13.1112

It is actually -29; OK, so row 1, column 1 gave me -29; row 2, column 1 gave me 13.1130

So, x equals 1 times this; well, since this is just 1, I am going to end up with x equals -29, 13 as elements.1142

And I also know that x is this; corresponding elements in equal matrices are equal, so -29 equals x, and 13 equals y.1153

So, this is my solution to this system of equations, using matrices.1168

Again, solve using a matrix equation: and I am going to keep in mind that I want to find x = A-1B,1177

where A is a matrix with the coefficients of x and the coefficients of y;1186

B is a matrix containing the constants 8 and 9; and x is a matrix containing variables--the variables in the system of equations.1196

I need A-1: A-1 is 1 over the determinant of A, ad - bc, times the matrix d, -b, -c, a.1209

Therefore, A-1 is 4 times -6, minus -3 times 8, times this matrix.1225

We are reversing a and d; so I take a, and I put it in the d position; d goes in the a position.1238

For b, I am going to reverse the sign; I am going to make it 3.1245

For c, I am going to reverse the sign; and that is -8.1250

OK, here I have 4 times -6; that is -24, minus -3 times 8; that is -24; and you can probably already see what the issue is.1255

That A-1 would equal 1 over -24 minus -24, times its matrix; and we have a problem,1271

because taking this one step further, this gives me 1/0.1284

And it really doesn't matter what this matrix is over here, because this is undefined.1289

A-1 does not exist; I cannot use this method.1296

So, the situation is that there is no unique solution; there may be no solution at all; there may be an infinite number of solutions.1304

But this method only works if there is a unique solution--a unique solution for x, and a unique value for y, that satisfies the system of equations.1314

So, I started out trying to use my method and writing this matrix equation.1324

And I got my matrices OK; but once I got to this step, finding A-1, then I discovered1328

that the determinant for this matrix A is 0, and in that case, A-1 does not exist; I can't use this method.1335

OK, that concludes this lesson on solving systems of equations using matrices.1344

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