INSTRUCTORS  Carleen Eaton Grant Fraser  Dr. Carleen Eaton

Cramer's Rule

Slide Duration:

Section 1: 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
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
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
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
Section 2: 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
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
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
Section 3: 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
Section 4: 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
0:18
Same Dimensions
0:25
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
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
Section 5: Quadratic Functions and Inequalities

31m 48s

Intro
0:00
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

27m 3s

Intro
0:00
0:16
Standard Form
0:18
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

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
10:12
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
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

22m 48s

Intro
0:00
0:21
Standard Form
0:29
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
13:50
16:03
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

27m 5s

Intro
0:00
0:11
Test Point
0:18
0:29
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
Section 6: 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
0:13
Like Terms and Like Monomials
0:23
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
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
9:23
Odd Degrees
12:51
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
8:22
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
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
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
Section 7: 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
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
12:23
13:29
16:07
18:18

41m 11s

Intro
0:00
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
3:24
3:47
6:33
7:16
Rationalizing Denominators
8:27
9:05
11:47
Conjugates
12:07
13:11
16:12
16:20
16:28
19:04
Distributive Property
19:10
19:20
24:11
28:43
32:00
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
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22

31m 27s

Intro
0:00
0:11
0:22
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
11:27
Restriction: Index is Even
11:53
12:29
15:41
17:44
20:24
24:34
Section 8: 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

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
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
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
Section 9: 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
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
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
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
16:54
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
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
Section 10: 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
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
11:51
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

47m 4s

Intro
0:00
0:22
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
12:48
13:09
21:42
29:13
35:02
40:29
Section 11: 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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• ## Related Books 0 answersPost by Jinjin Chai on February 21, 2021I just checked the work for example 1 at 3:34 because I was confused - I noticed that the denominator for x was -9, and 4 was the value of x. I solved for Dr.Eaton's determinant and got -18/4, or -4.5. However, when I plugged in the value for x into the first equation, I got that y was 2, not -4.5. Furthermore, if the denominator was -9, which was what the denominator was for x, the solution for the determinant for y would've been 2, which is correct. In short, Dr.Eaton made a small error for the first example at 3:34 - for any people confused. the answer is 2, and the equation for the determinant is|3     2||-3 -8|                 -18_______    =    __________     =    2!    -9                          -9 0 answersPost by Qing Xu on March 20, 2020THANKS SO MUCH! You can explain this to a person really well.  I'm a 5th grader, and I understood 1 answerLast reply by: Joshua ChenTue Jun 16, 2020 10:08 AMPost by DJ Sai on September 10, 2018Why are your outros different every time 1 answer Last reply by: Dr Carleen EatonSun Apr 8, 2018 2:48 PMPost by John Stedge on March 29, 2018Never mind you caught your mistake! Great job! 1 answerLast reply by: DJ SaiMon Sep 10, 2018 11:30 PMPost by John Stedge on March 29, 2018At about 22:24 you miss-multiplied the last group of #'s (1*2*-2) and you put (-2) in the denominator instead of (-4) as it should be therefore the final answer is incorrect. 1 answer Last reply by: Dr Carleen EatonSat Oct 3, 2015 7:15 PMPost by Peter Ke on September 30, 2015For example 4, I used the expansion b minors, and I used (2, -3 . 1) for my denominator. And when I solved it I got 12 + 18 - 18 which wasn't equal to 0. What did I do wrong? 3 answers Last reply by: Dr Carleen EatonWed Jan 1, 2014 12:52 AMPost by Rob Escalera on May 21, 2012Yes, I think the determinant (D) should be -9, and x = 4 while y = 2. 4 answersLast reply by: Miranda WintherTue Oct 29, 2013 9:19 PMPost by aloosh aloosh on February 17, 2011to solve for y isnt d=9 not 4 ???? 1 answer Last reply by: Dr Carleen EatonThu Feb 3, 2011 1:48 PMPost by Santhini Dheenathayalan on February 1, 2011Great video, Dr. Eaton! Thank you!

### Cramer's Rule

• Cramer’s rule allows us to solve a system of equations using only determinants. There is no graphing or algebraic work involved.
• Cramer’s rule indicates situations in which a system does not have a unique solution. It does not specify whether the system has no solution or an infinite number of solutions.

### Cramer's Rule

Solve the system of equations using Cramer's Rule
2x + y = − 2
2x − 3y = − 10
• To find the solution (x,y) for this problem, we're going to use Determinants, we need to compute three Determinants as follows
• x = [Dx/D] y = [Dy/D]
• Where D is the determinant of the coefficients of the linear equations.
• Where Dx is the determinant of D, but first column replaced with
 − 2
 − 10
• and Dy is the determinant of D, but the second column replaced with
 − 2
 − 10
• Find the three Determinants
• D = |
 2
 1
 2
 − 3
| Dx = |
 − 2
 1
 − 10
 − 3
| Dy = |
 2
 − 2
 2
 − 10
|
• D = − 8
• Dx = 16
• Dy = − 16
• Now Find x and y
• x = [Dx/D] y = [Dy/D]
• x = [16/( − 8)] = − 2
• y = [( − 16)/( − 8)] = 2
Solution = ( − 2,2)
Solve the system of equations using Cramer's Rule
2x − y = − 5
3x + y = 0
• To find the solution (x,y) for this problem, we're going to use Determinants, we need to compute three Determinants as follows
• x = [Dx/D] y = [Dy/D]
• Where D is the determinant of the coefficients of the linear equations.
• Where Dx is the determinant of D, but first column replaced with
 − 5
 0
• and Dy is the determinant of D, but the second column replaced with
 − 5
 0
• Find the three Determinants
• D = |
 2
 − 1
 3
 1
| Dx = |
 − 5
 − 1
 0
 1
| Dy = |
 2
 − 5
 3
 0
|
• D = 5
• Dx = − 5
• Dy = 15
• Now Find x and y
• x = [Dx/D] y = [Dy/D]
• x = [( − 5)/5] = − 1
• y = [15/5] = 3
Solution = ( − 1,3)
Solve the system of equations using Cramer's Rule
3x + y = − 6
2x + 2y = − 8
• To find the solution (x,y) for this problem, we're going to use Determinants, we need to compute three Determinants as follows
• x = [Dx/D] y = [Dy/D]
• Where D is the determinant of the coefficients of the linear equations.
• Where Dx is the determinant of D, but first column replaced with
 − 6
 − 8
• and Dy is the determinant of D, but the second column replaced with
 − 6
 − 8
• Find the three Determinants
• D = |
 3
 1
 2
 2
| Dx = |
 − 6
 1
 − 8
 2
| Dy = |
 3
 − 6
 2
 − 8
|
• D = 4
• Dx = − 4
• Dy = − 12
• Now Find x and y
• x = [Dx/D] y = [Dy/D]
• x = [( − 4)/4] = − 1
• y = [( − 12)/4] = − 3
Solution = ( − 1, − 3)
Solve the system of equations using Cramer's Rule
2x + 3y = 13
3x − 3y = − 3
• To find the solution (x,y) for this problem, we're going to use Determinants, we need to compute three Determinants as follows
• x = [Dx/D] y = [Dy/D]
• Where D is the determinant of the coefficients of the linear equations.
• Where Dx is the determinant of D, but first column replaced with
 13
 − 3
• and Dy is the determinant of D, but the second column replaced with
 13
 − 3
• Find the three Determinants
• D = |
 2
 3
 3
 − 3
| Dx = |
 13
 3
 − 3
 − 3
| Dy = |
 2
 13
 3
 − 3
|
• D = − 15
• Dx = − 30
• Dy = − 45
• Now Find x and y
• x = [Dx/D] y = [Dy/D]
• x = [( − 30)/( − 15)] = 2
• y = [( − 45)/( − 15)] = 3
Solution = (2,3)
Solve the system of equations using Cramer's Rule
− x + 2y = − 1
x + y = 4
• To find the solution (x,y) for this problem, we're going to use Determinants, we need to compute three Determinants as follows
• x = [Dx/D] y = [Dy/D]
• Where D is the determinant of the coefficients of the linear equations.
• Where Dx is the determinant of D, but first column replaced with
 − 1
 4
• and Dy is the determinant of D, but the second column replaced with
 1
 4
• Find the three Determinants
• D = |
 − 1
 2
 1
 1
| Dx = |
 − 1
 2
 4
 1
| Dy = |
 − 1
 − 1
 1
 4
|
• D = − 3
• Dx = − 9
• Dy = − 3
• Now Find x and y
• x = [Dx/D] y = [Dy/D]
• x = [( − 9)/( − 3)] = 3
• y = [( − 3)/( − 3)] = 1
Solution = (3,1)
Solve the system of equations using Cramer's Rule
x + y = − 6
− 2x − 2y = 12
• To find the solution (x,y) for this problem, we're going to use Determinants, we need to compute three Determinants as follows
• x = [Dx/D] y = [Dy/D]
• Where D is the determinant of the coefficients of the linear equations.
• Where Dx is the determinant of D, but first column replaced with
 12
• and Dy is the determinant of D, but the second column replaced with
 − 6
 12
• Find the three Determinants
• D = |
 1
 1
 − 2
 − 2
| Dx = |
 − 6
 1
 12
 − 2
| Dy = |
 1
 − 6
 − 2
 12
|
• D = 0
• Dx = 0
• Dy = 0
• Now Find x and y
• x = [Dx/D] y = [Dy/D]
• x = [0/0] = undefined
• y = [0/0] = undefined
No Unique Solution
Solve the system of equations using Cramer's Rule
− 2x − 3y = 2
x − 5y = − 14
• To find the solution (x,y) for this problem, we're going to use Determinants, we need to compute three Determinants as follows
• x = [Dx/D] y = [Dy/D]
• Where D is the determinant of the coefficients of the linear equations.
• Where Dx is the determinant of D, but first column replaced with
 2
 − 14
• and Dy is the determinant of D, but the second column replaced with
 2
 − 14
• Find the three Determinants
• D = |
 − 2
 − 3
 1
 − 5
| Dx = |
 2
 − 3
 − 14
 − 5
| Dy = |
 − 2
 2
 1
 − 14
|
• D = 13
• Dx = − 52
• Dy = 26
• Now Find x and y
• x = [Dx/D] y = [Dy/D]
• x = [( − 52)/13] = − 4
• y = [26/13] = 2
Solution = ( - 4,2)
Solve the system of equations using Cramer's Rule
5x + 2y − 4z = 9
− 6x + y + 5z = 2
− 6x − z = 22
• To find the solution (x,y) for this problem, we're going to use Determinants, we need to compute four Determinants as follows
• x = [Dx/D] y = [Dy/D] z = [Dz/D]
• Where D is the determinant of the coefficients of the linear equations.
• Where Dx is the determinant of D, but first column replaced with
 9
 2
 22
• Dy is the determinant of D, but the second column replaced with
 9
 2
 22
• Dz is the determinant of D, but the third column replaced with
 9
 2
 22
• Find the four Determinants. Choose between Expansion by Minors or Diagonal Method
• D = |
 5
 2
 − 4
 − 6
 1
 5
 − 6
 0
 − 1
| Dx = |
 9
 2
 − 4
 2
 1
 5
 22
 0
 − 1
| Dy = |
 5
 9
 − 4
 − 6
 2
 5
 − 6
 22
 − 1
| Dz = |
 5
 2
 9
 − 6
 1
 2
 − 6
 0
 22
|
• D = − 101
• Dx = 303
• Dy = − 404
• Dz = 404
• Now Find x, y and z
• x = [Dx/D] y = [Dy/D] z = [Dz/D]
• x = [303/( − 101)] = − 3
• y = [( − 404)/( − 101)] = 4
• z = [404/( − 101)] = − 4
Solution = ( − 3,4, − 4)
Solve the system of equations using Cramer's Rule
− 4x − 2y + z = − 7
− 5x − 2y + 2z = − 11
6x + 3y = 3
• To find the solution (x,y) for this problem, we're going to use Determinants, we need to compute four Determinants as follows
• x = [Dx/D] y = [Dy/D] z = [Dz/D]
• Where D is the determinant of the coefficients of the linear equations.
• Where Dx is the determinant of D, but first column replaced with
 − 7
 − 11
 3
• Dy is the determinant of D, but the second column replaced with
 − 7
 − 11
 3
• Dz is the determinant of D, but the third column replaced with
 − 7
 − 11
 3
• Find the four Determinants. Choose between Expansion by Minors or Diagonal Method
• D = |
 − 4
 − 2
 1
 − 5
 − 2
 2
 6
 3
 0
| Dx = |
 − 7
 − 2
 1
 − 11
 − 2
 2
 3
 3
 0
| Dy = |
 − 4
 − 7
 1
 − 5
 − 11
 2
 6
 3
 0
| Dz = |
 − 4
 − 2
 − 7
 − 5
 − 2
 − 11
 6
 3
 3
|
• D = − 3
• Dx = 3
• Dy = − 9
• Dz = 15
• Now Find x, y and z
• x = [Dx/D] y = [Dy/D] z = [Dz/D]
• x = [3/( − 3)] = − 1
• y = [( − 9)/( − 3)] = 3
• z = [15/( − 3)] = − 5
Solution = ( − 1,3, − 5)
Solve the system of equations using Cramer's Rule
− 4x − 2z = 4
− 4x + 6y + z = − 26
− x + 4y − 5z = 7
• To find the solution (x,y) for this problem, we're going to use Determinants, we need to compute four Determinants as follows
• x = [Dx/D] y = [Dy/D] z = [Dz/D]
• Where D is the determinant of the coefficients of the linear equations.
• Where Dx is the determinant of D, but first column replaced with
 4
 − 26
 7
• Dy is the determinant of D, but the second column replaced with
 4
 − 26
 7
• Dz is the determinant of D, but the third column replaced with
 4
 − 26
 7
• Find the four Determinants. Choose between Expansion by Minors or Diagonal Method
• D = |
 − 4
 0
 − 2
 − 4
 6
 1
 − 1
 4
 − 5
| Dx = |
 4
 0
 − 2
 − 26
 6
 1
 7
 4
 − 5
| Dy = |
 − 4
 4
 − 2
 − 4
 − 26
 1
 − 1
 7
 − 5
| Dz = |
 − 4
 0
 4
 − 4
 6
 − 26
 − 1
 4
 7
|
• D = 156
• Dx = 156
• Dy = − 468
• Dz = − 624
• Now Find x, y and z
• x = [Dx/D] y = [Dy/D] z = [Dz/D]
• x = [156/156] = 1
• y = [( − 468)/156] = − 3
• z = [( − 624)/156] = − 4
Solution = (1, − 3, − 4)

*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.

### Cramer's Rule

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
• System of Two Equations in Two Variables 0:16
• One Variable
• Determinant of Denominator
• Determinants of Numerators
• Example: System of Equations
• System of Three Equations in Three Variables 7:06
• Determinant of Denominator
• Determinants of Numerators
• Example 1: Two Equations 8:57
• Example 2: Two Equations 13:21
• Example 3: Three Equations 17:11
• Example 4: Three Equations 23:43

### Transcription: Cramer's Rule

Welcome to Educator.com.0000

In previous lessons, we talked about determinants; today, we are going to apply determinants0002

in order to solve systems of equations and solve for variables.0007

And we are going to do that using what is called Cramer's Rule.0011

Cramer's Rule shows us a way to solve for systems of equations; and we can do that in two variables or three variables.0017

Today, we are going to start out with two variables; and then we will go on in a minute and talk about systems of equations in three variables.0026

Now, one thing to notice, before we even delve into this, is that this is broken down into this formula for x and another one for y.0033

And what makes this rule helpful is that you may be in a situation where you just need to find one variable.0043

You have two or three variables in your system, but you only need one variable.0050

So, instead of solving the entire system, you can just hone in on the variable that you are looking for.0054

OK, so looking at this: if we have a system of equations ax + by = e, and the second equation cx + dy = f; we can find x by using determinants.0059

Let's first look at the determinant in the denominator; let's look at x, and I am going to call this determinant d.0074

Looking at y, it is actually the same determinant; and this is comprised of the coefficients of x in this row and the coefficients of y in this row.0085

So, whether you are looking for x or y, it doesn't matter; the denominator is going to be the same,0116

and it is going to be a determinant that is comprised of, in this first column, coefficients of x, and then here coefficients of y in the second column.0122

OK, now let's look at the numerator; and these are different; and I am going to call the numerator for x...0142

d is the determinant for x, and here I am going to have d and the subscript y to indicate that it is looking for the y; OK.0149

Now, if you look here, you will see that the second columns, where I had the y coefficients, are the same.0158

But if I am looking for x, I am actually going to replace the column where I had the x coefficients,0166

the coefficients of x, with the constants that are on the right side of the equation.0171

And look over here at y: this determinant for y--I kept the x coefficients the same; they are in their place.0179

But where I had the y coefficients, in that column, I replaced those with the constants.0190

So, I just think of this first column as the x column, and the second column as the y column.0195

And if I am looking for x, I replace the x column with the constants; if I am looking for y, I replace the y column with the constants.0200

And then, I just need to solve: OK, so using an example...let's say I have a system of equations, 3x - 5y = 2 and -3x + 2y = -8.0209

And from previous lessons, you know other ways to solve this.0224

You know substitution; you could use elimination; but this is introducing another way that is useful in certain situations.0227

So, looking at this: if I would like to find x, first I am going to find x.0235

Now, this determinant in the denominator is going to consist of the coefficients of x in this first column,0245

3 and -3, and the coefficients of y in the second column, which are -5 and 2.0260

OK, so I have the denominator, and now I need to find the numerator.0271

Recall that, in the numerator, if I am looking for x, I am going to replace the x column with constants; here, that is 2 and -8.0275

The y column--I am just going to keep it as it is down there.0288

OK, once I get this far, I just use my usual methods for finding a determinant.0292

And remember that that is...if a determinant is a, b, c, d--a second-order determinant--I just need to find ad - bc; OK.0298

So, I am going to do that: so x = 2(2) - -8(-5), over 3(2) - -3(-5).0312

OK, therefore, x equals...that is 4 minus...-8 times -5 is 40; in the denominator, 3 times 2 is 6; minus -3 times -5...that gives me 15.0340

This gives me x = (4 - 40), which is -36, over (6 - 15), which is -9; so my negatives cancel out, and -36/-9 is simply 4, so x equals 4.0359

If I wanted to solve for y, I would already have the determinant in the denominator.0379

So, if I wanted to solve for y, it would actually be even simpler, because I already know that the denominator is the same.0387

So, I would just put 4 in here; then I would find the determinant that I need in the numerator;0394

and I would look over here and say, "OK, my x column is the same as in the denominator; it is 3 and -3."0400

y I would replace with the constants, 2 and -8.0409

And then, I would just go ahead and solve, using ad - bc for the determinant, and divide by 4.0415

Cramer's Rule can also be used to solve systems of equations in three variables.0426

And although this looks pretty complicated, it is actually the same idea as we just covered.0432

Looking in the denominator: this first column is the coefficients of x, just like we talked about with systems of two equations.0438

This second column is coefficients of y, and the third is the coefficients of z.0452

And that is exactly the same, whether you are looking for the variable x, y, or z.0466

In the numerator, here you see that this is the same as below--coefficients of x.0474

Here, this is what I think of as my x column; I think of this as my y column; and then here, my z column--0484

I am going to replace that with the constants, since I am looking for z.0497

If I am looking for y, I am going to replace this second column with the constants, j, k, and l.0506

If I am looking for x, in the numerator, I replace that x column with the constants.0516

So, the denominator is always the same; in the numerator, these two columns are the same,0523

but for x this one is different; for y this one is different; and for z this one is different.0529

OK, for example, solving this system of equations using Cramer's Rule: this is in the form ax + by = some constant (I am going to call it e).0536

Here, I am going to call this cx + dy = f; and then, just to remind you that, if I am looking for x or y, what I am going to do...0557

In the denominator, I am going to have a determinant that consists of ac (coefficients of x)0573

in the first columns, and bd (coefficients of y) in the second column.0580

And that is going to be the same for both.0584

In the numerator, I am going to replace this first column with the constants; the second column will remain the same (coefficients of y).0587

Here, the x column still will have the coefficients of x, but the second column, where the y coefficients were--I am going to replace that with the constants, ef.0597

Therefore, if I am solving for x, let's start out with the denominator.0606

The denominator is going to be...the coefficient of x is 1; the other coefficient of x is 3.0614

The coefficient of y is -2, and the other is 1.0623

In the numerator, I am going to keep the y section the same; I am going to replace x with the constants 1 and 2, the x coefficients.0627

Now, I am just going to recall that, in order to find the second-order determinant here, it is going to be...this is a, b, c, d; it is going to be ad - bc.0635

Therefore, x equals...this is going to be 1 times 1, minus -2 times 2, over 1 times 1, minus -2 times 3.0647

Therefore, x equals 1 minus...-2 times 2 is -4, over 1 minus...-2 times 3 is -6; x equals 1...a negative and a negative gives me a positive;0670

the same here--a negative and a negative gives me a positive; so x equals 5/7.0688

Solving for y is actually going to be easier, because recall that these determinants are the same: the denominator is the same for x and y.0698

Well, I already did all this work, finding the denominator, which is 7.0709

So, I am not going to repeat that; I am just going to get rid of this and write 7 here.0713

In the numerator, in this first column, I am going to have my coefficients of x, 1 and 3, just like I did in the denominator.0717

However, in this second column, I am going to replace it with the constants 1 and 2.0728

Again, I am using my formula ad - bc to find this determinant; that is going to give me 1(2) - 3(1), all over 7.0737

This is y = (2 - 3)/7, so y = -1/7.0754

So, x = 5/7; y = -1/7; and I was able to figure that out using Cramer's rule, where the denominator is the determinant0763

formed by the coefficients of x and the coefficients of y, and the numerator is...0774

the first column consists of the constants; the second is the coefficients of y.0779

Once I did that, y was easier to find, because I already had the denominator.0785

I just plugged that in right here; for the numerator, I had a determinant consisting of the coefficients of x, and then (in the second column) the constants.0790

OK, again, we are solving a system of equations with two variables for x.0802

OK, recall that in the denominator, this first column is going to be coefficients of x; the second is going to be the coefficients of y.0815

In the numerator, I am going to leave the y's alone; but I am going to replace x with 8 and 4, the constants.0825

Now, I am going to use my rule, ad - bc, assuming that this is the set-up, to find these determinants.0837

x equals a times d, minus b times c; therefore, x equals...8 times 3 is 24, minus -16, or x = 24...a negative and a negative...that is + 16.0847

So, 24 and 16 is 40; x equals 40 in the numerator, over this denominator.0870

Let's find the denominator; the denominator determinant is 3 times 3, so that is the numerator for x.0886

Now, to find the denominator (this is just the numerator), I have 3 times 3, minus -4 times 2.0899

3 times 3 is 9, and in the denominator, I have -4 times 2, so that is -8; and then, this is going give me 9.0920

A negative and a negative is a positive; 9 plus 8 equals 17.0933

This is my numerator right here, and I have my x denominator right here; and this is going to give me x = 40/17.0937

I found my determinants; this consists of the x coefficients and the y coefficients; this one is the constants and the y coefficients.0951

And then, I found the numerator right here; I found the denominator right here.0960

And then, I went ahead put those together, and then x equals 40/17.0968

Now, I am finding y: finding y is going to be easier, because here I have my denominator,0975

which is 17--the same denominator, so that takes a lot of work out of it.0981

In the numerator, in that first column, go the x coefficients.0986

I replace the second column with the constants; and then, I go ahead and find this determinant,0992

which is going to be 3 times 4, minus 8 times 2.1002

This is going to give me 12 minus 16 over 17, and that is going to be...12 - 16 is -4, over 17.1010

So, here I have...x is 40/17 y is -4/17--using Cramer's Rule.1022

Now, looking at a system of equations with three variables, I am being asked only to solve for x.1032

And this shows you how this rule can be useful, because I only have to focus in on that, and not solve the entire system.1038

You use the same general approach as with three variables.1046

So, I am going to take x; and in the denominator, I am going to say, "OK, the first column is the x coefficients."1052

The coefficient here is 1; 2; 1; y column: -2, 1...there is no y here, so the coefficient is just 0.1060

z--there is no z at this first one, so that is going to be 0, 1, 1.1075

All right, now I am looking for x; the first column represents the x coefficients--I am going to hold on to that thought for a second.1081

And I am just going to copy over the other two columns, because I am not messing with those.1090

I am going to replace this first column with the constants, 3, 1, 3.1095

Now that I have set this up, I have to find a third-order determinant in the numerator and the denominator.1104

And you recall that we went over a couple of methods to do that: expansion of the minors and the diagonal method.1110

And now, I am going to use the diagonal method.1116

The first step of the diagonal method is to copy over the first two columns--so, up here, 3, 1, 3, -2, 1, 0.1120

Here, it is 1, 2, 1, -2, 1, 0.1130

Once I have done that, then I need to start at the upper left and go along the diagonals.1137

And I am going to find, actually, right along this way...the product of that and add1145

the product of this diagonal to the product of the next diagonal to the product of the next diagonal.1163

OK, so x equals...the first product...this is 3 times 1 times 1, plus -2 times 1 times 3, plus 0 times 1 times 0.1171

OK, I have taken care of those diagonals; now I start at the lower left and go along those diagonals.1200

And I am going to subtract those products: so minus 3 times 1 times 0.1206

Next, 0, 1, 3--minus 0 times 1 times 3--minus the last diagonal: and that is 1, times 1, times -2.1215

OK, that takes care of the numerator; in the denominator, I am going to start at the upper left.1232

My first diagonal is 1 times 1 times 1; and I am going to add that to the next diagonal, -2 times 1 times 1.1238

I am adding that to this diagonal, 0 times 2 times 0.1253

Now, subtracting: we are starting down here: 1 times 1 times 0; we are subtracting 0 times 1 times 1,1258

and then finally subtracting 1 times 2 times -2.1274

I'm running out of room; OK.1285

1 times 2 times -2; OK, I am working this out...1287

Therefore, x equals 3, and this is -2 times 1 times 3, which gives me -6, plus 0, minus 0, minus 0.1295

1 times 1 times -2...so that is minus -2.1308

In the denominator: the 1's multiply and give me 1; plus -2 times 1 is -2, times 1...-2 times 1 times 1 is -2;1313

plus...this is going to come out to 0; minus 1 times 1 times 0; minus 0 times 1 times 1 (that is another 0); minus 1 times 2 times -2.1326

OK, let's clean this up: this is 3 minus 6; get rid of all these 0's; and this is minus -2, and that is the same as plus 2.1340

In the denominator, I have 1 minus 2; get rid of all the 0's; and this is minus -2; actually, that is 2 times -2--this should be -4; minus -4 gives me plus 4.1352

Now, x equals...well, 3 minus 6 plus 2 is going to give me -1;1373

in the denominator, I have 1 minus 2 (that is -1), plus 4: that is going to give me 3.1382

So, x is -1/3; and as you can see, once you get it set up, then it is a matter of just keeping track of all of that multiplication and these signs.1388

So, I set this up by forming determinants with coefficients of x, y, and z in the denominator.1398

In the numerator, there are coefficients of y and z, and then the constants in the first row.1407

Then, I solve this by using the diagonal method; and I did that right here to come up with x = -1/3.1412

OK, again, we are asked to solve a system of three equations.1423

I am going to start out by looking for x, and looking at the denominator.1427

For the denominator, I am going to use the coefficients of x: 2, -6, and 4; the coefficients of y: -3, 9, -1; and the coefficients of z: 1, -3, 1.1437

OK, in the numerator, I am looking for x, so I am going to keep the y and z columns4 just as they were.1455

And I am going to go over here to the x area, and I am going to substitute the constants 4, 11, and 10.1465

Now, I am going to, again, use the diagonal method; and I am actually going to start with the denominator.1471

And you will see why in a minute--why you should always start with the denominator, and how it can save you work.1475

So, I need to find this third-order determinant; and I am going to use the diagonal method.1481

I am going to rewrite these first two columns: 2, -6, 4, -3, 9, -1.1486

After I have done that, I am just going ahead and rewriting this; I am not working with the numerator yet.1496

I am just going to leave that like this.1504

And I am only working at finding this determinant in the denominator.1510

Starting at the upper left, using the diagonal method, I am going to find the products of this first diagonal: that is 2 times 9 times 1,1513

plus this next number right here--next set of numbers: -3, -3, 4--that product, plus, coming down right here, 1, -6, -1.1531

Then, from that, I am going to start subtracting this other set of diagonals, starting down here, and going up:1558

the product of 4 times 9 times 1; continuing on: minus -1, -3, and 2, minus this last diagonal right here: 1, -6, -3.1566

So, x equals the same determinant; let's just work with the denominator right now.1591

2 times 9 times 1 is 18; plus -3 times -3 (is 9) times 4--that is 36; -6 and 1--that is 6.1598

OK, and now subtracting: 4 times 9 times 1--that is 36; minus...-1 and -3 is 3, times 2 is 6, minus 1 times -6 is -6, times -3 is 18.1610

Now, you can see what is happening here: x equals this determinant in the numerator.1628

But I have 18 and 18, so those cancel; that gives me 0.1634

I have 36 and 36, so those cancel: 0; 6 and 6--those cancel; so I have a 0 in the denominator, and that is not allowed.1640

Since that is not defined, what I have ended up with is a situation where Cramer's Rule does not work.1651

It doesn't mean that there is not necessarily a solution to this system of equations, but what it means that there is no unique solution.1657

It may be that there is an infinite number of solutions (it is a dependent system).1666

It may mean that there is no solution.1671

But I can't go any farther, and I can't use Cramer's Rule in this situation.1674

And that is why the first thing you should do, if you are using Cramer's Rule, is to check out the denominator and make sure that it is not 0.1678

If the determinant in the denominator, d, is 0, you can't use this method.1685

And that concludes this session of Educator.com on Cramer's Rule; I will see you next lesson.1699

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