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INSTRUCTORS  Carleen Eaton Grant Fraser
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Solving Systems of Equations Algebraically

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

• ## Related Books

 0 answersPost by Zexin Li on January 3, 2019hmm 0 answersPost by edder villegas on March 2, 2014no questions 0 answersPost by Tiffany Huang on December 11, 2013GOOD

### Solving Systems of Equations Algebraically

• You can solve a system of equations by substitution or by elimination.
• Sometimes each equation must be multiplied before elimination can be used.
• If you obtain an equation that is always true, the system has an infinite number of solutions.
• If you obtain an equation that is never true, the system has no solution.

### Solving Systems of Equations Algebraically

Solve the system of equations algebraically
4x − y = − 4
− 4x + y = 0
• Whenever you are solving a system of equations algebraically you have two options, elimination method
• or substitution method. For elimination method, look out for variables with equal, opposite sign coefficients.
• With this method, one variable will drop out. For substitution method, look out for positive variables that have
• a leading coefficient of one.
• In this problem, the best method is elimination because the x's have equal, opposite sign coefficients.
• Step 1: Add the system of equations so x's drop out.
• (4x − y = − 4) + (− 4x + y = 0)=
• 0 + 0 = − 4
• 0 = − 4
This is never true, therefore, you have an inconsistent system. No solution.
Solve the system of equations algebraically
(a) 4x + y = 1
(b) − 3x − 3y = − 3
• Notice that y on equation (a) is positive with a coefficient of 1. Therefore solve by substitution.
• Step 1: solve (a) for y
• 4x + y = 1
• 4x+y− 4x = 1− 4x
• y = − 4x − 1
• Step 2:substitute y = 4x + 1 into equation (b) and solve for x.
• − 3x − 3y = − 3
• − 3x − 3( − 4x + 1) = − 3
• Simplify
• − 3x + 12x − 3 = − 3
• 9x − 3 = − 3
• 9x−3 + 3 = −3 + 3
• 9x = 0
• x = 0
• Step 3: Substitute x = 0 into y = − 4x + 1
• y = − 4x + 1 = − 4(0) + 1 = 1
Solution : (0,1)
Solve the system of equations algebraically
(a) 2x + 2y = 8
(b) x − 3y = 0
• Notice that x on equation (b) is positive with a coefficient of 1. Therefore solve by substitution.
• Step 1: solve (b) for x
• x − 3y = 0
• x−3y+3y=0+3y
• x = 3y
• Step 2:substitute x = 3y into equation (a) and solve for y.
• 2x + 2y = 8
• 2(3y) + 2y = 8
• Simplify
• 6y + 2y = 8
• 8y = 8
• y = 1
• Step 3: Substitute y = 1 into x = 3y
• x = 3y = 3(1) = 3
Solution : (3,1)
Solve the system of equations algebraically
(a) 8x − 6y = − 16
(b) − 8x + 6y = 18
• Notice that both the x and the y have equal, opposite sign coefficients. That means that both variables will drop out.
• (8x − 6y = − 16)+(−8x+6y=18)
• 0 + 0 = − 2
• 0 = − 2
Since 0 is never going to equal − 2, you have an Inconsistent System of equations. No Solution.
Solve the system of equations algebraically
(a) − 9x − 3y = 15
(b) 9x + 5y = − 25
• Notice that the variables x has equal, opposite sign coefficients. That means if we add the system of equations the x's will drop out.
• Step 1. Add the system
• (− 9x − 3y = 15)+(9x + 5y = − 25)
• 0 + 2y = − 10
• Step 2: Solve for y
• 2y = − 10
• y = − 5
• Step 3: Solve for x. You may use equation (a) or equation (b). Substitute y = − 5
• − 9x − 3y = 15
• − 9x − 3( − 5) = 15
• − 9x + 15 = 15
• −9x + 15 − 15 = 15 − 15
• − 9x = 0
• x = 0
Solution(0, − 5)
Solve the system of equations algebraically
(a) 6x − 10y = − 16
(b) − 2x + 10y = 12
• Notice that the variables y has equal, opposite sign coefficients. That means if we add the system of equations the y's will drop out.
• Step 1. Add the system
• (6x − 10y = − 16)+(−2x + 10y = 12)
• 4x + 0 = − 4
• Step 2: Solve for x
• 4x = − 4
• x = − 1
• Step 3: Solve for y. You may use equation (a) or equation (b). Substitute x = − 1
• − 2x + 10y = 12
• − 2( − 1) + 10y = 12
• 2 + 10y = 12
• 2+10y− 2=12 − 2
• 10y = 10
• y = 1
Solution ( − 1,1)
Solve the system of equations algebraically
(a) 9x − 4y = 10 (b) − 9x + 2y = 4
• Notice that the variables x has equal, opposite sign coefficients. That means if we add the system of equations
• the x's will drop out.
• Step 1. Add the system
• (9x − 4y = 10)+(− 9x + 2y = 4)
• 0 − 2y = 14
• Step 2: Solve for y
• − 2y = 14
• y = − 7
• Step 3: Solve for x. You may use equation (a) or equation (b). Substitute y = − 7
• − 9x + 2y = 4
• − 9x + 2( − 7) = 4
• − 9x − 14 = 4
• −9x−14 + 14=4 + 14
• − 9x = 18
• x = − 2
Solution ( − 2, − 7)
Solve the system of equations algebraically
(a) 5x − 2y = 16
(b) − 7x − 7y = 7
• Notice that we've run into a problem: none of the variables have coefficient = 1, and neither x's or y's have equal, different sign coefficients.
• That means we need to do extra work in order to create an equivalent system that can be solved by elimination.
• Since the x's already have opposite signs, now it's time to look for the Least Common Multiple of 5 and 7.
• The Least Common Multiple of 5 and 7 is 35. That means we'll multiply (a) by 7 and (b) by 5
• Step 1. Multiply (a) by 7, (b) by 5 to create an equivalent system.
• (a) 35x − 14y = 112
• (b) − 35x − 35y = 35
• Step 2: Add the System
• (35x − 14y = 112)+(− 35x − 35y = 35)
• 0 − 49y = 147
• − 49y = 147
• Step 3: Solve for y
• − 49y = 147
• y = − 3
• Step 4: Solve for x. You may use equation (a) or equation (b). Substitute y = − 3
• In this step you may use the original equations, or equations from the Equivalent System.
• 5x − 2y = 16
• 5x − 2( − 3) = 16
• 5x + 6 = 16
• 5x +6 − 6 = 16 − 6
• 5x = 10
• x=2
Solution (2, − 3)
Solve the system of equations algebraically
(a) − 15x − 15y = − 15 (b) 6x + 6y = 6
• Notice that we've run into a problem: none of the variables have coefficient = 1, and neither x's or y's have equal, different sign coefficients.
• That means we need to do extra work in order to create an equivalent system that can be solved by elimination.
• Since the x's and y's have opposite signs, now it's time to look for the Least Common Multiple of 6 and 16.
• The Least Common Multiple of 6 and 15 is 30. That means we'll multiply (a) by 2 and (b) by 5
• Step 1. Multiply (a) by 2, (b) by 5 to create an equivalent system.
• (a)− 30x − 30y = − 30
• (b)30x + 30y = 30
• Step 2: Add the System
• (− 30x − 30y = − 30)+(30x + 30y = 30)
• 0+ 0 = 0
• 0 = 0
In this case we have a Dependent System because 0 is always going to equal 0. Therefore, you will have infinite number of solutions.
Solve the system of equations algebraically
(a) − 5x + 6y = 19
(b) − 2x − 7y = 17
• Notice that we've run into a problem: none of the variables have coefficient = 1, and neither x's or y's have equal, different sign coefficients.
• That means we need to do extra work in order to create an equivalent system that can be solved by elimination.
• Since the easiest variable to find the Least Common Multiple is x, we'll Drop Out x even if that means (a) or (b) needs to be multiplied by a negative
• in order to have the required ( + )( − ) coefficients.
• The Least Common Multiple of 5 and 2 is 10. That means we'll multiply (a) by 2 and (b) by − 5 or (a) by − 2 and (b) 5
• Step 1. Multiply (a) by 2, (b) by − 5 to create an equivalent system.
• (a) − 10x + 12y = 38
• (b) 10x + 35y = − 85
• Step 2: Add the System
• (− 10x + 12y = 38) + (10x + 35y = − 85)=
• 0 + 47y = − 47
• Step 3: Solve for y
• 47y = − 47
• y = − 1
• Step 4: Solve for x. You may use equation (a) or equation (b). Substitute y = − 1.
• In this step you may use the original equations, or equations from the Equivalent System.
• − 2x − 7y = 17
• − 2x − 7( − 1) = 17
• − 2x + 7 = 17
• − 7 − 7
• − 2x = 10
• x = − 5
Solution ( − 5, − 1)

*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 Algebraically

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 by Substitution 0:08
• Example: System of Equations
• Solving by Multiplication 7:22
• Extra Step of Multiplying
• Example: System of Equations
• Inconsistent and Dependent Systems 11:14
• Variables Drop Out
• Inconsistent System (Never True)
• Constant Equals Constant
• Dependent System (Always True)
• Example 1: Solve Algebraically 13:58
• Example 2: Solve Algebraically 15:52
• Example 3: Solve Algebraically 17:54
• Example 4: Solve Algebraically 21:40

### Transcription: Solving Systems of Equations Algebraically

Welcome to Educator.com.0000

In today's lesson, we will be talking about solving systems of equations algebraically.0002

In the previous lesson, we talked about solving systems of equations by graphing.0008

However, that method has some limitations; therefore, we are going to talk about a few other methods of solving.0013

And this is some review from Algebra I; so again, if you need more detail--more review on these concepts--check out our Educator.com Algebra I series.0020

First, we are going to review solving by substitution, first jotting down a system of equations:0030

2x + 3y = 12, and the second equation in the system is x + 4y = 1.0036

In this method, you are going to solve one equation for one variable in terms of the other variable.0047

Then, substitute the expression for the variable in the other equation.0053

Let's look at what this means: first step: solve one equation for one variable in terms of the other one.0059

The easiest thing to do is to find a situation where you have a variable that has a coefficient of 1; and I have that right here.0065

So then, I can solve for x in terms of y pretty easily.0071

So, x + 4y = 1, so I am going to solve for x in terms of y; this gives me x = -4y + 1.0075

The second step is to substitute this expression for the variable in the other equation.0088

So, I am going to substitute -4y + 1 for x in the first equation; and that is going to create0094

an equation that has only one variable, and then I will be able to solve that.0101

It is important that you go back into the other equation to substitute in.0105

This is going to give me 2(-4y + 1) + 3y = 12; then, I can just solve for y.0111

2 times -4y; that is -8y, plus 2 times 1--that is just 2--plus 3y, equals 12.0124

Combine like terms: -8y and 3y gives me -5y, plus 2 equals 12; subtract 2 from both sides to get -5y = 10.0135

Dividing both sides by -5, I get y = -2.0150

Once I have one variable solved, I can easily solve for the other.0155

So, I just go back to either one of these (and this one is easier to work with), and I am going to substitute in...I am going to let y equal -2.0159

x + 4(-2) = 1; this is going to give me x + -8 = 1, or x - 8 = 1; adding 8 to both sides, I get x = 9.0170

Here, y equals -2 and x equals 9.0192

This method of substitution works really well when the coefficient of one of the variables is 1.0197

So, use substitution when a variable in one of the equations has a coefficient of 1.0203

So again, we looked at this system of equations, saw that this variable had a coefficient of 1,0232

then solved for this variable x in terms of the other variable y to get x = -4y + 1.0240

And then, I went ahead and substituted this 4x in the other equation; that gave me one equation with one variable, and I can solve for y.0247

Once you have y, you can substitute that value into either equation and then solve for x.0260

The second method is solving by elimination: in elimination, you add or subtract the two equations to eliminate one of the variables from the resulting equation.0267

And this system works well when you have variables, either the two x's or the two y's, that have either the same coefficient or opposite coefficients.0277

By "opposite coefficients," I mean the same number with opposite signs, such as 2 and -2 or 3 and -3.0286

For example, if you were asked to solve this system, you look and see that there is no variable with a coefficient of 1.0293

Therefore, substitution is not the ideal--it is not the easiest way to go.0306

But what you see is that you have one variable--you have y--that has the same coefficient.0312

So, adding is certainly not going to help me; if I add these together, I will get 6x + 10y = 8.0318

What I need to do is subtract, because my goal is to get one variable to drop out.0324

So, I am going to subtract the second equation from the first.0328

And just to help me keep my signs straight, I rewrite that as adding the opposite; so I am adding -4x, -5y, and -3.0333

OK, this is going to give me 2x - 4x is -2x; the y's are going to drop out: 5y - 5y gives me 0, so I can write + 0, but they just drop out.0352

And then, 5 minus 3 equals 2; this is -2x = 2; I can easily solve for x, and x equals -1.0365

Once I have one variable, I can go to either original equation, and then substitute in order to solve for the other variable.0376

I know that x equals -1, so I am going to substitute that up here.0385

Add 2 to both sides (that is going to give me 5y = 7), and divide both sides by 5 to get y = 7/5.0395

So, solving by elimination, I was able to come up with the solution that x equals -1 and y = 7/5, which will satisfy both of these equations.0403

Again, this works well; use elimination when the same variable (meaning both x's or both y's)0412

have the same or opposite coefficients (the same coefficient, but opposite signs).0426

Sometimes, you look, and you see that there is no variable with a coefficient of 1; and then you say, "OK, I will use elimination."0442

But then, you realize that none of the variables have the same or opposite coefficients.0449

In that case, you can still use elimination, but you are going to have to take an extra step before you do the elimination.0453

And the extra step is to multiply one or both equations so that one of the variables has the same or opposite coefficient in the two new equations.0459

So, your goal is to create a situation where one of the variables has the same or opposite coefficients.0469

Then, you just use elimination, as we did previously.0476

If you had a system of equations, 5x + 4y = 1, and 6x + 3y = 3, you see that neither of these0480

has a variable with a coefficient of 1, and neither set of variables has the same or opposite coefficients.0489

But if I look at these two, 4y and 3y, the least common multiple is 12; so what I want to do0497

is multiply each of these by something, in order to end up with a coefficient of 12 in front of the y.0505

Sometimes you will be lucky, and all you will have to do is multiply one of the equations by a number to get the opposite or same coefficient.0511

Other times, like this, you are going to have to multiply both.0519

So, for the first equation, I am going to multiply by 3; and this is going to give me 15x + 12y = 3.0522

The second equation I am going to multiply by 4: this is going to give me...4 times 6x is 24x + 12 y = 12.0542

OK, I now see that I have the same coefficient for y; so rewriting this over here, I need to subtract.0559

So, I am going to subtract the second equation from the first.0576

And to keep everything straight, I like to go about this by adding the opposite to make sure that I keep my signs straight.0581

So, I am going to add -24x - 12y, and then that is going to be a -12.0589

OK, -24x - 12y - 12: add these together: I get 15 - 24x--that is going to give me -9x; the y's drop out, and then I have 3 - 12; that is -9.0603

Divide both sides by -9 to get x = 1.0618

From here, I substitute back; I will choose this top equation, 5x + 4y = 1, and I know that x equals 1.0622

This is 5(1) + 4y = 1; so that is 5 + 4y = 1, or 4y = -4, so y = -1.0631

So, x equals 1; y equals -1.0645

Again, this is just the first step in elimination; and you use it in a situation like this,0648

where you want to use elimination, but you don't already have a set of variables with the same or opposite coefficients.0653

So, you figure out what the least common multiple is, and multiply one or both equations in order to achieve that.0660

From there, you just proceed as we did before, solving by elimination.0667

There are several different possibilities that can occur when you are solving systems of equations algebraically.0675

Usually, in the problems that you will see, what will happen is what just happened previously,0683

that I showed you, where you will end up with a value for x and a value for y that satisfy both equations.0688

However, there are times when that doesn't occur.0695

You can be going along, doing substitution, doing elimination, and things are going fine;0699

and then you end up with something like this: c = d.0705

Your variables drop out, and you end up with an equation that is saying that a constant is equal to a different constant--for example, 4 = 5.0710

Well, that is not true; so when you see this, this is an equation that is never true.0719

And what this tells me is that the system of equations is inconsistent.0727

So, this is a situation where there is no solution.0742

If you start seeing a situation where the constants drop out, and then you see something like 4 = 50752

or 9 = 10--something that is not true--then you know you have an inconsistent system.0758

The other possibility is that you could have a system that is dependent, or always true.0764

In that case, you are going along; you are doing elimination; you are doing substitution;0773

and then you see that you end up with variables dropping out, and a constant that equals a constant--the same constant--for example, 3 = 3.0778

Well, that is always true; so if you end up with an equation--the sum or difference of the two equations--0788

the system of equations--that is always true, this system is dependent, and it has an infinite number of solutions.0795

Recall, when we were solving systems of equations by graphing: this is analogous to the situation where you would end up with the same line.0812

So, if you have two equations (a system of equations), you graph both equations,0819

and you find out that they are the same line, well, then, there is an infinite number of solutions--all points along those lines.0823

Here, no solution would be like parallel lines, where they never intersect; there is no solution to that system.0829

OK, the first example is: Solve algebraically for the system of equations.0839

As soon as I see that I have variables (or even one variable) with a coefficient of 1, I recognize that substitution would be a really good method to use.0844

So, use substitution when a variable has a coefficient of 1.0853

So, I am going to solve for x in terms of y: x + y = 5, so x = 5 - y.0856

Now, I have solved for x in terms of y; and now I am going to substitute this value into the other equation.0867

2x + 3y = 13: I am going to substitute this expression for x, so 2 times (5 - y), plus 3y, equals 13.0875

Or, 10 - 2y + 3y = 13; this is 10 + y = 13; then subtract 10 from both sides to get y = 3.0889

Once I have this, I can plug y into this simple equation up here: x + y = 5.0906

Substitute in; let y equal 3; and I get x = 2.0916

And this is very easy to check; you can see that, if x is 2 and y is 3, this equation holds true.0924

And you could do the same--plug it back into that first equation, put these values in for x and y, and just verify that these solutions are correct.0929

OK, so again, we are solving by substitution because you had a situation where you had a variable (actually, two variables) with a coefficient of 1.0942

That is the easiest method to use.0950

Here, I am solving this system of equations algebraically; and I don't have any variables with a coefficient of 1, so I am not going to use substitution.0954

But if I look; I have variables (y) that have the opposite coefficient, -3 and 3.0963

What that means is that, if I add these equations, the y's are going to drop out; then, I can just solve for x.0970

Rewrite this right here and add: I am going to add these two.0977

2x + 4x: that gives me 6x; the y's drop out: -3y + 3y is 0; and then 0 + 0 is 9.0988

I am going to divide both sides by 6; this is going to give me 9/6, and that simplifies to 3/2.1004

I have x = 3/2; I am going to pick either one--I will go ahead and pick the top one: 2x -3y = 0.1011

And let x equal 3/2; substitute that for x.1018

The 2's cancel out; this gives me 3 - 3y = 0, or -3y = -3; if I divide both sides of the equation by -3, I will get y = 1.1028

So, the solution is x = 3/2, y = 1; again, I could always check these solutions1042

by substituting the values in here and making sure that this equation holds true.1051

And this was a perfect setup to use elimination, because I already had variables that had the opposite coefficients.1056

I simply added these equations together; the y dropped out, allowing me to solve for x, and then substitute 3/2 in for x.1063

OK, Solve algebraically: again, there are no variables with a coefficient of 1, so I am going to go to elimination.1075

But these do not have the same or opposite coefficients, and the y's do not have the same or opposite coefficients.1083

So, this time I am going to need to use that extra step; I am going to need to use multiplication1090

in order to create a situation where I have variables with the same or opposite coefficients.1095

And I see that, if I multiply the top equation by 3, I will get 6x; and if I multiply the bottom equation by 2, I will get 6x.1102

So, I am going to go ahead and do that--multiply the top equation by 3: that is going to give me 3 times 2x - 3y = -16.1112

And this comes out to 3(2x) is 6x; 3 times -3...that is -9y; and 3 times -16 is -48.1129

The second equation I am going to multiply by 2.1144

So, 2 times 3x, plus 5y, equals 14; OK, 2 times 3x gives me 6x; 2 times 5y is 10y; and 2 times 14 is 28.1147

I have the same coefficient here; so what I need to do is subtract.1165

I am going to subtract, and again, I am going to convert this so that I am adding the opposite.1171

Rewrite the top equation the same, and the bottom as adding the opposite (adding -6x - 10y - 28).1185

OK, when I do that, the 6x's drop out, and that is going to give me -19y equals...-48 and -28 comes out to -76.1196

Divide both sides by -19, and -76 divided by -19 actually equals 4, so it divided very nicely and evenly; so I ended up with y = 4.1215

Now that I have that, I am going to substitute y = 4 into either equation; I am going to choose the top one.1227

So, 2x - 3y = 16; let y equal 4; so, 2x - 3(4) = -16, or 2x - 12 = 16.1233

Add 12 to both sides to get 2x = -4, and divide both sides by 2 to get x = -2.1251

So, the solution is that x equals -2 and y equals 4.1259

So, when I looked at this, I saw that I could use elimination, if I got the x's to have the opposite coefficient1264

by multiplying the top equation by 3 to give me 6x, and the bottom equation by 2 to give me 6x here, as well.1274

So, I did that; and then, these have the same coefficient, so I subtracted and solved for y.1283

Once I had a value for y, I substituted that value into one of the equations and then solved for x to get the solution.1291

Here, I have 2x + 3y = 8, and -4x - 6y = -16.1303

Since there is no variable with a coefficient of 1, I am going to use elimination.1311

But again, I am going to have to do a little work to get the opposite coefficients.1314

And I see here that all I have to do is multiply the top by 2, and that will give me 4x; that is opposite coefficients.1319

I don't have to do anything to the bottom equation.1329

So, that is 2 times 2x, plus 3y, equals 8, and this is going to give me 4x + 6y = 16.1330

That is the opposite coefficient, so I am going to add this new equation and this equation: + -4x - 6y - 16.1343

And you may have already realized that each one of these terms is opposite.1354

And so, here is what is going to happen: 4x - 4x is 0; 6y - 6y is 0; 16 - 16 is 0; so I end up with 0 = 0.1360

And I didn't make any mistakes--I did everything correctly--but then all my variables went away.1373

And what this tells me is that I have a dependent system, and it has an infinite number of solutions.1377

If I were to graph this, I would see that I have an infinite number of solutions--that these are intersecting at every point along the line.1386

So, we call this a dependent system.1399

If I had been going along, and all of the variables dropped out, and then I got something that wasn't true,1405

like 4 = 2 or 4 = 0, then I would have a situation where it is an inconsistent system and there are just no solutions.1410

But here, this is always true; so I have an infinite number of solutions.1418

So today, we covered solving systems of equations algebraically.1425

And that concludes today's lesson for Educator.com; I will see you again.1430

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