INSTRUCTORS Carleen Eaton Grant Fraser

Dr. Carleen Eaton

Solving Systems of Equations in Three Variables

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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 1 answerLast reply by: Dr Carleen EatonSun Apr 8, 2018 2:47 PMPost by John Stedge on March 27, 2018At approx. 4 mins why don't you just multiply the equation by a negative two; this gets rid of having to carry the negative sign and the possibility of adding the wrong number on accident. Just a suggestion :)! 1 answerLast reply by: Dr Carleen EatonSat Feb 23, 2013 6:35 PMPost by bo young lee on February 19, 2013i dont get the example 1 of the slove for y. can you explain more easy? 1 answerLast reply by: Dr Carleen EatonSun Jan 29, 2012 4:41 PMPost by Ken Mullin on January 25, 2012Very concise explanations accompanying video---good review of each step.I sometimes accompany an explanation of this type systems of equations--triples--with a quick solution using matrices on the TI-84.Students become skillful in obtaining a quick answer for x, y, and z when the identity matrix appears.

Solving Systems of Equations in Three Variables

• Use substitution and elimination to solve a system in three variables.
• A system in 3 variables can have a unique solution, infinitely many solutions, or no solution.

Solving Systems of Equations in Three Variables

Solve:
(a) − 2x + 4y + 4z = − 4
(b) − 2x − 5y − 3z = − 12
(c) 2x − 2y + z = − 8
• The strategy to solve a system of equations in three variables is to divide and conquer.
• Break down 3x3 system into 2x2 systems, solve for one variable, then work backwards to solve for all three variables. This is done in several steps.
• Choose the easiest variable to eliminate.
• The Easiest variable to eliminate is x because we can add (a) + (c) to eliminate x, and we can add (b) and (c) to eliminate x
• (a) + (c):
(− 2x + 4y + 4z = − 4) + (2x − 2y + z = − 8) = (2y + 5z = − 12)
• (b) + (c):
(− 2x − 5y − 3z = − 12) + (2x − 2y + z = − 8) = (− 7y − 2z = − 20)
• Combine and solve for one of the variables. The easiest variable to eliminate is y.
• (2y + 5z = − 12) + (− 7y − 2z = − 20)
7*(2y + 5z = − 12) + 2*(− 7y − 2z = − 20)
(14y + 35z = − 84) + (− 14y − 4z = − 40)
31z = −124
z = −4
• Use z = − 4 to solve for y
• 2y + 5z = − 12
2y + 5( − 4) = − 12
2y − 20 = − 12
2y = 8
y = 4
• Using y = 4 and z = − 4, solve for x using (a), (b), or (c) (c) 2x − 2y + z = − 8
2x − 2(4) − 4 = − 8
2x − 12 = − 8
2x = 4
x = 2
Solution (x,y,z) = (2,4, − 4)
Solve:
(a)2x + 5y + 3z = − 9
(b)6x + 2y − 3z = − 18
(c) − x − 12y − 3z = − 18
• The strategy to solve a system of equations in three variables is to divide and conquer.
• Break down 3x3 system into 2x2 systems, solve for one variable, then work backwards to solve for all three variables. This is done in several steps.
• Choose the easiest variable to eliminate.
• The Easiest variable to eliminate is z because we can add (a) + (b) to eliminate z, and we can add (a) and (c) to eliminate z
• (a) + (b):
(2x + 5y + 3z = − 9) + (6x + 2y − 3z = − 18) = (8x + 7y = − 27)
• (a) + (c):
(2x + 5y + 3z = − 9) + (− x − 12y − 3z = − 18) = (x − 7y = − 27)
• Combine and solve for one of the variables. The easiest variable to eliminate is y.
• (8x + 7y = − 27) + (x − 7y = − 27)
9x = − 54
x = − 6
• Use x = − 6 to solve for y
• x − 7y = − 27
− 6 − 7y = − 27
− 7y = − 21
y = 3
• Using y = 3 and x = − 6, solve for z using (a), (b), or (c)
• (a) 2x + 5y + 3z = − 9
2( − 6) + 5(3) + 3z = − 9
− 12 + 15 + 3z = − 9
3 + 3z = − 9
3z = − 12
z = − 4
Solution (x,y,z) = ( − 6,3, − 4)
Solve:
(a)− 4x + 5y − 5z = − 18
(b)− 6x − 5y − 4z = − 11
(c)− 6x + 5y + 13z = − 4
• The strategy to solve a system of equations in three variables is to divide and conquer.
• Break down 3x3 system into 2x2 systems, solve for one variable, then work backwards to solve for all three variables. This is done in several steps.
• Choose the easiest variable to eliminate.
• The Easiest variable to eliminate is y because we can add (a) + (b) to eliminate y, and we can add (b) and (c) to eliminate y
• (a) + (b):
(− 4x + 5y − 5z = − 18 )+(− 6x − 5y − 4z = − 11)=( 10x − 9z = − 29)
• (b) + (c):
( − 6x − 5y − 4z = − 11 ) + (− 6x + 5y + 13z = − 4 ) = (− 12x + 9z = − 15)
• Combine and solve for one of the variables. The easiest variable to eliminate is z.
• (− 10x − 9z = − 29) + (− 12x + 9z = − 15)
− 22x = − 44
x = 2
• Use x = 2 to solve for y
• − 12x + 9z = − 15
− 12(2) + 9z = − 15
− 24 + 9y = − 15
y = − 1
• Using y = − 1 and x = 2, solve for z using (a), (b), or (c)
• (a) − 4x + 5y − 5z = − 18
− 4(2) + 5( − 1) − 5z = − 18
z = 1
Solution (x,y,z) = (2, − 1,1)
Solve:
(a)3x − y + 4z = 18
(b) − 4x − y − 2z = 10
(c)2x − 2y + z = 10
• The strategy to solve a system of equations in three variables is to divide and conquer.
• Break down 3x3 system into 2x2 systems, solve for one variable, then work backwards to solve for all three variables. This is done in several steps.
• Choose the easiest variable to eliminate.
• The Easiest variable to eliminate is y because we can add (a) + (b) to eliminate y, and we can add (b) and (c) to eliminate y, however, (a) and (b) need to be multiplied by 2 and − 2 in order to eliminate y.
• (a)2*(3x − y + 4z = 18)
(b)2*(− 4x − y − 2z = 10)
(c)2x − 2y + z = 10
• (a)6x−2y+8z=36
(b)8x+2y+4z=−20
(c)2x − 2y + z = 10
• (a) + (b):
(6x − 2y + 8z = 36) + (8x + 2y + 4z = − 20 ) = (14x + 12z = 16)
• (b) + (c):
(8x + 2y + 4z = − 20 ) + (2x − 2y + z = 10) = (10x + 5z = − 10)
• Combine and solve for one of the variables. The easiest variable to eliminate is z.
• (14x + 12z = 16) + (10x + 5z = − 10) =
5*(14x + 12z = 16) + −12*(10x + 5z = − 10) =
(70x + 60z = 80) + (− 120x − 60z = 120) =
− 50x = 200
x = − 4
• Use x = − 4 to solve for z
• 10x + 5z = − 10
10( − 4) + 5z = − 10(
− 40 + 5z = − 10
z = 6
• Using x = − 4 and z = 6, solve for y using (a), (b), or (c)
• 3x − y + 4z = 18
3( − 4) − y + 4(6) = 18
y = − 6
Solution (x,y,z) = ( − 4, − 6,6)
Solve:
(a) − 3x + 2y + 3z = − 2
(b)3x + y − 3z = − 1
(c)x + 2y − 3z = − 6
• The strategy to solve a system of equations in three variables is to divide and conquer.
• Break down 3x3 system into 2x2 systems, solve for one variable, then work backwards to solve for all three variables. This is done in several steps.
• Choose the easiest variable to eliminate.
• The Easiest variable to eliminate is x because we can add (a) + (b) to eliminate x, and we can add (a) and (c) to eliminate x, however, (c) need to be multiplied by 3 in order to eliminate x.
• (a) − 3x + 2y + 3z = − 2
(b)3x + y − 3z = − 1
(c)3*(x + 2y − 3z = − 6)
• (a) − 3x + 2y + 3z = − 2
(b)3x + y − 3z = − 1
(c)3x + 6y − 9z = − 18
• (a) + (b):
( − 3x + 2y + 3z = − 2 ) + (3x + y − 3z = − 1) = (3y = − 3)
• (a) + (c):
(− 3x + 2y + 3z = − 2) + (3x + 6y − 9z = − 18 ) = (8y − 6z = − 20)
• Solve for y using (3y = − 3)
• y = − 1
• Use y = − 1 to solve for z using (8y − 6z = − 20)
• 8y − 6z = − 20
8( − 1) − 6z = − 20
z = 2
• Using y = − 1 and z = 2, solve for x using (a), (b), or (c)
• − 3x + 2y + 3z = − 2
− 3x + 2( − 1) + 3(2) = − 2
x = 2
Solution (x,y,z) = (2, − 1,2)
Solve:
(a)x + y + z = − 3
(b)3x − 4y = 5
(c)2x + 3y − z = − 8
• The strategy to solve a system of equations in three variables is to divide and conquer.
• Break down 3x3 system into 2x2 systems, solve for one variable, then work backwards to solve for all three variables. This is done in several steps.
• Choose the easiest variable to eliminate.
• The Easiest variable to eliminate is z because we can add (a) + (c) to eliminate z,
• (a) + (c):
(x + y + z = − 3) + (2x + 3y − z = − 8) = (3x + 4y = − 11)
• Combine (3x + 4y = - 11) and equation (b) and solve for one of the variables.
• (3x + 4y = − 11) + (3x − 4y = 5) =
6x = − 6
x = − 1
• Use x = − 1 to solve for y using (3x + 4y = − 11)
• 3x + 4y = − 11
3( − 1) + 4y = − 11
− 3 + 4y = − 11
y = − 2
• Using x = − 1 and y = − 2, solve for z using (a), (b), or (c)
• x + y + z = − 3
− 1 + − 2 + z = − 3
z = 0
Solution (x,y,z) = ( − 1, − 2,0)
Solve:
(a)2x − 2y + 3z = − 2
(b) − y + 3z = 0
(c) − 2x + y − 3z = 2
• The strategy to solve a system of equations in three variables is to divide and conquer.
• Break down 3x3 system into 2x2 systems, solve for one variable, then work backwards to solve for all three variables. This is done in several steps.
• Choose the easiest variable to eliminate.
• The Easiest variable to eliminate is x because we can add (a) + (c) to eliminate x,
• (a) + (c):
(2x − 2y + 3z = − 2) + ( −2x + y − 3z = 2 ) = (− y = 0)
• y = 0
• Solve for z using y=0 and (b)
• − y + 3z = 0
− 0 + 3z = 0
z = 0
• Use y = 0 and z = 0 to solve for x using (a) or (b) or (c)
• 2x − 2y + 3z = − 2
2x − 2(0) + 3(0) = − 2
x = − 1
solution (x,y,z) = ( − 1,0,0)
Solve:
(a)x + y + 2z = 4
(b) − 2y − z = − 3
(c) − x + y − z = − 4
• The strategy to solve a system of equations in three variables is to divide and conquer.
• Break down 3x3 system into 2x2 systems, solve for one variable, then work backwards to solve for all three variables. This is done in several steps.
• Choose the easiest variable to eliminate.
• The Easiest variable to eliminate is x because we can add (a) + (c) to eliminate x,
• (a) + (c):
(x + y + 2z = 4) + (− x + y − z = − 4) = (2y + z = 0)
• Combine (2y + z = 0) and equation (b) and solve for one of the variables.
• (2y + z = 0) + (− 2y − z = − 3)
0 = −3
This is never true, therefore the system of eq. has no solution.
Solve:
(a)x + 4y + 2z = 6
(b)3x + 4y − 6z = − 12
(c) − 2x − 4y + 2z = − 18
• The strategy to solve a system of equations in three variables is to divide and conquer.
• Break down 3x3 system into 2x2 systems, solve for one variable, then work backwards to solve for all three variables. This is done in several steps.
• Choose the easiest variable to eliminate.
• The Easiest variable to eliminate is y because we can add (a) + (c) and (b) and (c),
• Derive the two Systems of Equations Sys.
• (a) + (c):
(x + 4y + 2z = 6) + ( − 2x − 4y + 2z = − 18 ) = ( − x + 4z = − 12)
• (b) + (c):
(3x + 4y − 6z = − 12) + (− 2x − 4y + 2z = − 18 ) = (x − 4z = − 30 )
• Combine and solve for one of the variables.
• (− x + 4z = − 12) + (x − 4z = − 30)
0 = −42
This is never true, therefore the system of eq. has no solution.

*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.

Solving Systems of Equations in Three Variables

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
• Solving Systems in Three Variables 0:17
• Triple of Values
• Example: Three Variables
• Number of Solutions 5:55
• One Solution
• No Solution
• Infinite Solutions
• 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

Transcription: Solving Systems of Equations in Three Variables

Welcome to Educator.com.0000

Today we are going to be going on to talk about solving systems of equations in three variables.0002

In previous lessons, we talked about how to solve systems with two variables.0007

So now, we are going to go up to systems involving three variables.0012

In order to solve these systems, you are actually going to use the same strategies you used for solving systems with two variables.0017

Recall those techniques: substitution, elimination, and multiplication.0027

This time, though, a solution is an ordered triple of values.0031

So, you will end up having three variables: for example, (x,y,z).0035

And the solution would be something like (5,3,-2), where x is 5, y is 3, and z is -2.0040

Now, just to work out an example to show you how to approach these using the same techniques that you already know:0056

looking at this system of equations, I have three equations, and I have three total variables.0082

And the idea is to work with the three equations so that you get one of the variables to drop out.0088

Once you get one of the variables to drop out, you will be left with a system of two equations0096

with two variables, and you already know how to work with that.0100

So, the technique would be to first just consider two of the equations together; I am going to call these equations 1, 2, and 3, so I can keep track of them.0103

I am going to first consider equations 1 and 2.0113

And when I look at these, I see that the two z's have opposite coefficients.0116

And you will recall that elimination works really well in that situation.0125

So, I am going to take 2x + 3y - z = 5 (that is equation 1) and equation 2: 3x - 2y + z = 4, and I am going to add those.0129

This will give me 5x, and then 3y - 2y is going to give me y; the z's drop out; 5 and 4 is 9.0146

So now, I have a new equation, and I will just mark this out so I can keep track of it.0158

So, once you have gotten a variable to drop out, work with two different equations to get the same variable to drop out.0167

So, I got z to drop out; and what I want to do is work with two different equations--I worked with 1 and 2.0175

I could work with 1 and 2, or I could work with 2 and 3.0182

And I am actually going to work with 2 and 3; and I want to get z to drop out.0187

Equations 2 and 3: looking at this, how am I going to get z to drop out?0193

Well, in order to do that, I could use elimination; but I am first going to have to multiply this second equation by 2.0197

So, this is equation 2; and it is 3x - 2y + z = 4; and I am going to multiply that by 2 to give me 6x - 4y + 2z = 8.0207

Now, I am going to take equation 3 (this is equation 2, and I am going to take equation 3): I want to make sure that I am working with two different equations.0229

So, I have 1 and 2, and 2 and 3 (or I could have done 1 and 3--either way).0237

-4x + y + 2z = 3: my goal is to get the z's to drop out.0243

In order to do that, I am going to have to subtract: I need to subtract 3 from 2.0256

I want to be very careful with my signs here, so I am going to change this to adding the opposite.0267

This is going to give me 6x + 4x, which is 10x; -4y and -y is -5y; 2z and -2z drops out 8 minus 3 is 5; OK.0276

At this point, what I am left with is a system of two equations with two unknowns.0297

Once you get that far, you proceed using the techniques that we learned previously (again, substitution, elimination, and multiplication).0303

But since you are only working with two equations with two unknowns, you are on familiar territory.0312

And you can solve for one of the variables and then find that value; substitute in for the other variable0316

and find that value; and then you can go back and find z.0325

And we are going to work more examples on this; but the basic technique is to work with two equations0329

to eliminate a variable, using either elimination or substitution, then work with two other equations0335

to eliminate that same variable, resulting in two equations with two variables,0342

allowing you to solve for one variable, then the other, and then the third.0349

Just as in systems with two variables, a system with three variables may have one solution, no solutions, or an infinite number of solutions.0357

So, recall: the solution set here, if I had three variables (x, y, and z) would be a value for x0368

(such as 2), a value for y (such as -4), and a value for z (such as 1) that would satisfy all three equations.0375

The other possibility is that there may be no solutions; recall from working with systems of equations with two variables--0384

you know that you are in this situation when you are using elimination, or you are using substitution;0393

you are going along; and then you see variables start to drop out, and you end up with an equation0399

where you have a constant equaling another constant, which is never true.0406

So, if you start seeing variables drop out, and you end up with something such as 4 = 7 (which is never true),0413

this tells you that there is no solution to this system of equations.0420

There can be an infinite number of solutions; when you are working with your equations;0426

you are eliminating; you are substituting; you are using your techniques; you are being careful;0434

you are doing everything right, and then you see variables drop out, and you get a constant equaling a constant, like 2 = 2.0438

Well, that is always true; and this means that that system of equations has an infinite number of solutions.0447

So, typically, you will get one solution: a value for x, y, and z that is the set of values that makes the equations true--that satisfies the equations.0457

You may end up, though, with no solutions (there are no solutions to this equation) or an infinite number of solutions for this system of equations.0467

OK, in the first example, we are given a system of three equations with three variables.0479

So again, I am going to work with two of the equations; I will number them 1, 2, and 3.0486

And I will work with two of them, and then a different two.0493

So, first, I am going to look at the second two equations; and they are very easy to work with, because I have opposite coefficients.0500

I could also use substitution, because I have coefficients of 1; but I am just going to use elimination.0509

So first, I am going to work with 2 and 3: y - z = 2, x (let me move that, so it doesn't create confusion) + 2y + z = 2.0515

OK, here I end up with x; all I am doing is adding these together using elimination.0533

x + 3y; the z's drop out; 2 + 2 is 4.0543

OK, so I eliminated z from this first set of equations; now, I need to work with a different set of equations.0550

And there are actually multiple different ways to approach this, and I am going to work with 1 and 2.0567

I already worked with 2 and 3, so I eliminated z; and I want to...actually, I already have...this does not have z in it, so I don't even need to proceed.0579

That way, I already have two equations with two unknowns; this is a particularly easy situation, compared with when all three have 3 variables.0598

OK, so I look up here, and I have my new equation; I can call it equation 4.0607

And then, I have equation 1; and these just have x and y.0613

So, I am just going to proceed, like I usually do, with two equations with two unknowns.0616

Let me rewrite these right here: 2x + y = 3, and this is x + 3y = 4.0620

I can use substitution; that would be fine, because I have a coefficient of 1.0634

So here, I am going to solve for y; and this will give me y = -2x + 3.0640

And then, I am going to substitute into this equation; so I have x, and I am going to substitute for y: plus 3, times -2x, plus 3, equals 4.0651

Working this out: x...3 times -2x is -6x; 3 times 3 is 9; it equals 4.0668

Here, now, I just have an equation with a single variable, so I can solve that.0679

First, I am combining like terms: x - 6x is -5x; plus 9 equals 4.0684

Subtract 9 from both sides: -5x = -5; x = 1.0692

OK, so the first thing I wanted to do is just get rid of one of the variables; I am just working with two variables.0700

And I did that by just adding these two; in the second and third equations, the z dropped out.0705

I was lucky, because the first equation already didn't have a z; so I had two equations with two unknowns.0711

And then, I just used those two; I solved by substitution, and I came up with x = 1.0717

Since I know that x equals 1, I can go ahead and substitute this into this equation to find y.0726

So, looking at equation 1: 2x + y = 3; I know that x equals 1, so that is 2(1) + y = 3, or 2 + y = 3.0734

Subtracting 2 from both sides, I get y = 1; so now, I have x, and I have y.0754

I need to find z: well, this will easily tell me what z is.0759

That is y - z = 2, and I know y: y equals 1, so 1 - z = 2; -z = 1, therefore z = -1.0765

So, the solution to this set of equations is that x equals 1, y equals 1, and z equals -1.0780

The hardest step was just getting rid of that third unknown (the third variable).0791

I did that by adding these two together: then, working with two equations with two variables, I was able to solve for x.0797

Once I am there, all I have to do is start substituting.0805

Here, I substituted x into the first equation and solved for y.0807

Once I got y, then I was able to substitute y into the second equation to solve for z.0813

So, 1, 1, -1 is the solution for this system of equations with three variables.0822

OK, Example 2: again, my goal is going to be to get a variable to drop out, so I am just left with two variables.0833

Looking at this first and second equation, considering these together, y and -y...if I add those together,0843

the y's will drop out, because they have opposite coefficients (1 and -1).0852

So, I am going to start off by adding the first two equations: this is 1 and 2.0857

x + y + z = 2; and then, I am going to add x - y + 2z = -1; and I am going to come up with a new equation.0861

This is 2x; the y's drop out; z + 2z is 3z; 2 - 1 is 1; OK, I have this.0874

And I worked with these first two; I now need to work with two different equations to get the same variable to drop out.0886

This time, I am going to pick equations 1 and 3, and I want y to drop out.0894

So, I have equations 1 and 3: that is x + y + z = 2, and (equation 3) that is 2x + y + 2z = 2.0900

Now, I need to subtract in order to get the y to drop out.0919

To keep everything straight, as far as my signs go, I am going to keep the first equation the same;0925

but for the second one, I am going to change it to adding the opposite: add -2x, -y, -2z, and -2.0931

OK, x - 2x is -x; the y's drop out, which is just what I wanted; z - 2z is -z; 2 - 2 is 0.0944

Now, I have two equations and two variables; I am rewriting these two up here to see what I have to work with.0956

2x + 3z = 1; now, I just use my usual methods of solving a system of equations with two variables.0965

Since I have coefficients here of -1, it is pretty easy to use substitution, so I am going to solve for x in this second equation,0976

and then substitute that value up in the first equation.0984

I have -x - z = 0, which would give me -x = z, or x = -z.0989

So, I am going to take this -z and substitute it in right here; OK, that gives me 2x + 3z = 1, and let x equal -z.0996

So, 2 times -z, plus 3z, equals 1; that is -2z + 3z = 1.1014

Combine these two like terms to get z = 1; now, I have my first value.1025

OK, so I know that z equals 1, so I am on my way.1034

And I look up here, and I see, "Well, I know that x equals -z, so that makes it very easy to solve for x."1038

If x equals -z, and z equals 1, then x equals -1.1047

So now, I have x = -1, z = 1; I am just missing y.1059

Well, look at that first equation: it tells me that x + y + z = 2.1063

The x is -1; I don't know y; and I know that z is 1; these two cancel, and that gives me y = 2.1074

So, -1 + 1 is 0, so I end up with y = 2.1088

Putting all this together up here as my solution, I end up with x = -1, y = 2, and z = 1.1092

That was a lot of steps; it is really important to keep track of what you are working with--especially, in the beginning,1106

that you work with two equations (I worked with 1 and 2) to get a variable to drop out.1112

I added those, and the y's dropped out; then I want to work with either 1 and 3 or 2 and 3 (two different equations) to get the y to drop out.1118

I chose 1 and 3; and I saw that I could get the y to drop out of 1 and 3 if I just subtracted 3 from 1; that is what I did right here.1128

At that point, I clearly mark out what I ended up with, which is two equations with two variables.1140

We wrote those up here; and I decided I was going to use substitution.1146

I solved for x in this second equation: x equals -z; I substituted that in right here, into the first equation.1151

That allowed me to have one equation with one variable, z; and I determined that z equals 1.1162

From there, it was much easier, because I saw that x equals -z, and I knew z; so x equals -1.1170

but I picked the easiest one to substitute in x and z and solve for y to get my set of solutions.1184

Again, this is a set of three equations with three variables that I need to approach systematically.1198

And my first goal is to eliminate the same variable, so I am working with two equations with two variables.1204

And I see several possibilities; you could approach it differently, and you will come up with the same answer, as long as you follow the rules and the steps.1216

I am seeing that -y and y are opposite in terms of coefficients (-1 and 1), so I am going to add those two.1227

I am going to add 1 and 3; that is going to give me 5x; the y's drop out, so 5x - z; 1 - 3 is -2.1235

I just marked that, so I can keep track of it, because I am going to need to use it in a minute,1260

once I generate another equation in which y has been eliminated.1265

OK, I worked with the first and the third; now, I need to work with two different equations;1270

and I am going to work with the first and the second, and I want to eliminate y.1276

In order to eliminate y, I need to multiply the first equation by 2; so I will do that up here.1282

This is going to give me 4x - 2y + 2z = 2; that is the first equation.1294

Now, I am going to add it to the second equation: + x + 2y - z = 0.1305

Adding these together, I am going to get 5x; the y's drop out; 2z - z is z; 2 and 0 is 2.1319

OK, I first worked with the first and the third, and then I worked with the first and the second, to get the y's to drop out.1330

So now, I have two equations and two variables: x and z.1338

So, put these together so I can see what is going on with them: 5x - 2 = -z; 5x + z = 2.1345

Well, I can see that, if I add these, z will drop out; and I am just back to my usual two equations with two variables--usual techniques.1355

5x and 5x is 10x; the z's drop out; -2 and 2 is 0.1365

Divide both sides by 10; it gives me x = 0.1373

So, I have my first value, which makes things much easier.1377

Since I know that x equals 0, I can substitute into either of these to solve for z.1384

So, 5x - z = -2; so 5(0) - z = -2; 0 - z = -2; -z = -2; divide both sides by -1 to get z = 2.1389

I have my second value; OK, so I know x; I know z; I just need y.1411

I am going to solve for y; I could use any of these--I am going to pick the top one and solve for y, knowing that x equals 0 and z equals 2.1417

OK, that is 0 - y + 2 = 1, which gives me -y + 2 = 1.1437

Subtract 2 from both sides to get -y; if I say -1 - 2, that is going to give me -1.1447

Multiply all of that by -1 to give me y = 1.1455

OK, so the solution here is x = 0, y = 1, and z = 2.1461

And I approached that by seeing that I could add 1 and 3 because of -y and y, and those would drop out.1470

I could have added the first two and had the z drop out, and had that be my variable to eliminate; I happened to choose y.1479

I added those together and got this equation.1486

Then, I worked with the first and the second equation--a little more complicated, because to get opposite coefficients,1490

I had to multiply the first equation by 2.1496

I did that to generate this equation, which I added to the second; this is the resulting equation.1500

I then had two equations with two variables; I looked at those two and saw I had opposite coefficients with z.1507

So, when I added them together, z dropped out, and I could solve for x.1514

Once I determined that x is 0, I substituted 0 into this top equation for x to solve for z, and determined that z equals 2.1519

At that point, I just needed to solve for y, so I took this equation and substituted my value for x and my value for z, and determined that y equals 1.1529

So again, we are using the same techniques that we used previously, only you are working with more equations, and there is more to keep track of.1543

OK, in this system of equations (three equations with three unknowns), it is a little bit more complicated.1551

I do have one equation that has a coefficient of 1, but the rest of them have larger coefficients.1559

So, I am going to work with the first and second equations; and what I want to do is eliminate the z.1566

So, in order to do that, I am going to need to multiply this first equation by 2.1577

I want to work with the first and second equations; I want to get the z to drop out.1583

But I need to multiply this by 2 first; so let me do that right over here.1589

That is going to give me 6x - 4y + 2z = 8; so I am going to rewrite that right here: 6x - 4y + 2z = 8.1600

And that came from that first equation, multiplied by 2; and I am going to add it to the second equation.1615

So, + -6x + 4y - 2z = 2: now, you might have already seen what happened.1621

I was really just focusing on "OK, I want to get the z's to be the same or opposite coefficients," so that they would cancel out.1635

But what happened is: everything ended up with opposite coefficients: 6 and -6; -4 and 4; 2 and -2; OK.1640

So, I have 6x - 6x; that is 0; -4y and 4y--0; 2z and -2z--0; 8 and 2--10.1654

0 = 10: well, we know that 0 does not equal 10, so this is not true; it is never true that 0 equals 10.1669

So, there is no solution to this set of equations.1678

I could have had a situation where I got a solution.1682

I could have also had a situation where maybe I got 10 = 10 with a different system of equations.1686

If I had come up with something like 0 = 0 (that is always true) or 10 = 10, then I would have had an infinite number of solutions.1694

But instead, what happened is that my variables drop out; I got c = d, which is saying1701

that I have a constant that is equal to another constant, which is never true; so there is no solution.1706

So, this one turned out to actually be less work than the others.1712

But when this happens, you just want to be really careful that you were doing everything correctly;1714

you didn't make a mistake; but actually, it can end up that there simply is no solution to the system of equations.1718

That concludes this lesson on Educator.com on solving systems of equations with three variables; and I will see you next lesson!1725

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