INSTRUCTORS  Carleen Eaton Grant Fraser  Dr. Carleen Eaton

Solving Systems of Equations by Graphing

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
Bookmark & Share Embed

## Copy & Paste this embed code into your website’s HTML

Please ensure that your website editor is in text mode when you paste the code.
(In Wordpress, the mode button is on the top right corner.)
×
• - Allow users to view the embedded video in full-size.
Since this lesson is not free, only the preview will appear on your website.

• ## Related Books 0 answersPost by Alex Leung on August 25, 2020At 4:02, I don't understand why the function is called a "dependent system." Is it because the points depend on the function? 1 answerLast reply by: Ak LiuTue Jul 28, 2020 2:58 PMPost by Ak Liu on July 28, 2020On example 1, isn't the answer (1,2)? because I did the math, and the y-intercept is 1... But in your graph, it's -1.

### Solving Systems of Equations by Graphing

• You can solve a system of equations by graphing the equations and finding the point of intersection of the two graphs.
• If the graphs are parallel lines, there is no point of intersection and no solution to the system.
• If the graphs are the same lines, the system has an infinite number of solutions.
• If the graphs intersect at one point, the system has a unique solution.

### Solving Systems of Equations by Graphing

Solve by graphing
y = − 3x − 4
y = [1/2]x + 3
• There are three ways to solve system of equations by graphing. Choose the method that best fits the system of equation given.
• Slope Intercept Form:
y = mx + b y2 = m2x + b

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

This is the best method to use when the system of equations is in this format.
1. Graph the y - intercept b, @ (0,b)
2. Use the slope m = [rise/run] to graph additional points from the y - intercept.
3. Draw the line with ruler, find the intersection

You would have to re - write the equations into standard form. Use this method if you don't like graphing using y - intercept and slope.
• Standard Form:

ax + by = c
a2x + b2y = c2

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

In order to use this method when the system of equations is in Standard Form, you must solve for y. In other words, re - write the system into slope - intercept form.

This is the best method to use if and only if you do not get fractions when finding the x - and y - intercepts. If you get fractions, the best way is to solve for y and use the method on the left - Y - Intercept + Slope
When using this method, you will get four points (x,0) and (0,y) from equation 1 and (x,0), (0,y) from equation 2. Carefully draw the lines through the x - and y - intercept and locate the point of intersection.
• Given that the system is given in slope - intercept form, use Y - intercept + slope method.
• Step 1: Find the y - intercep for equation 1 and plot it
• b =
• b = − 4, the point is (0, − 4)
• Step 2: Find the slope for equation 1 and plot the next point starting from the y - intercept (0, − 4)
• m = [rise/run] =
• m = [rise/run] = [( − 3)/1] or [3/( − 1)]
• Step 3: Draw the line. Make line extends your entire graph.
• Step 4: Find the y - intercep for equation 2 and plot it
• b =
• b = 3, the point is (0,3)
• Step 5: Find the slope for equation 2 and plot the next point starting from the y - intercept (0,3)
• m = [rise/run] =
• m = [rise/run] = [1/2] or [( − 1)/( − 2)]
• Step 6: Draw the line. Make line extends your entire graph.
• Step 7: Find the point of intersection.
• Solution = ( , )
• Solution = ( − 2,2) Solve by graphing
y = − [1/3]x + 1
y = [2/3]x + 4
• There are three ways to solve system of equations by graphing. Choose the method that best fits the system of equation given.
• Slope Intercept Form:
y = mx + b y2 = m2x + b

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

This is the best method to use when the system of equations is in this format.
1. Graph the y - intercept b, @ (0,b)
2. Use the slope m = [rise/run] to graph additional points from the y - intercept
3. Draw the line with ruler, find the intersection

You would have to re - write the equations into standard form. Use this method if you don't like graphing using y - intercept and slope.
• Standard Form :
ax + by = c
a2x + b2y = c2

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

In order to use this method when the system of equations is in Standard Form, you must solve for y. In other words, re - write the system into slope - intercept form.

This is the best method to use if and only if you do not get fractions when finding the x - and y - intercepts. If you get fractions, the best way is to solve for y and use the method on the left - Y - Intercept + Slope
When using this method, you will get four points (x,0) and (0,y) from equation 1 and (x,0), (0,y) from equation 2. Carefully draw the lines through the x - and y - intercept and locate the point of intersection.
• Given that the system is given in slope - intercept form, use Y - intercept + slope method.
• Step 1: Find the y - intercep for equation 1 and plot it
• b =
• b = 1, the point is (0,1)
• Step 2: Find the slope for equation 1 and plot the next point starting from the y - intercept (0,1)
• m = [rise/run] =
• m = [rise/run] = [( − 1)/3] or [1/( − 3)]
• Step 3: Draw the line. Make line extends your entire graph.
• Step 4: Find the y - intercep for equation 2 and plot it
• b =
• b = 4, the point is (0,4)
• Step 5: Find the slope for equation 2 and plot the next point starting from the y - intercept (0,3)
• m = [rise/run] =
• m = [rise/run] = [2/3] or [( − 2)/( − 3)]
• Step 6: Draw the line. Make line extends your entire graph.
• Step 7: Find the point of intersection.
• Solution = ( , )
• Solution = ( − 3,2) Solve by graphing
y = − [3/2]x − 2
y = x + 3
• There are three ways to solve system of equations by graphing. Choose the method that best fits the system of equation given.
• Slope Intercept Form:
y = mx + b
y2 = m2x + b

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

This is the best method to use when the system of equations is in this format.
1. Graph the y - intercept b, @ (0,b)
2. Use the slope m = [rise/run] to graph additional points from the y - intercept
3. Draw the line with ruler, find the intersection

You would have to re-write the equations into standard form. Use this method if you don't like graphing using y - intercept and slope.
• Standard Form:
ax + by = c
a2x + b2y = c2
Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

In order to use this method when the system of equations is in Standard Form, you must solve for y. In other words, re - write the system into slope - intercept form.

This is the best method to use if and only if you do not get fractions when finding the x - and y - intercepts. If you get fractions, the best way is to solve for y and use the method on the left - " Y - Intercept + Slope"
When using this method, you will get four points (x,0) and (0,y) from equation 1 and (x,0), (0,y) from equation 2. Carefully draw the lines through the x - and y - intercept and locate the point of intersection.
• Given that the system is given in slope - intercept form, use Y - intercept + slope method.
• Step 1: Find the y - intercep for equation 1 and plot it
• b =
• b = − 2, the point is (0, − 2)
• Step 2: Find the slope for equation 1 and plot the next point starting from the y - intercept (0, − 2)
• m = [rise/run] =
• m = [rise/run] = [( − 3)/2] or [3/( − 2)]
• Step 3: Draw the line. Make line extends your entire graph.
• Step 4: Find the y - intercep for equation 2 and plot it
• b =
• b = 3, the point is (0,3)
• Step 5: Find the slope for equation 2 and plot the next point starting from the y - intercept (0,3)
• m = [rise/run] =
• m = [rise/run] = [1/1] or [( − 1)/( − 1)]
• Step 6: Draw the line. Make line extends your entire graph.
• Step 7: Find the point of intersection.
• Solution = ( , )
• Solution = ( − 2,1) Solve by graphing
y = − [5/4]x − 2
y = − [5/4] + 2
• There are three ways to solve system of equations by graphing. Choose the method that best fits the system of equation given.
• Slope Intercept Form:
y = mx + b
y2 = m2x + b
Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

This is the best method to use when the system of equations is in this format.
1. Graph the y - intercept b, @ (0,b)
2. Use the slope m = [rise/run] to graph additional points from the y - intercept.
3. Draw the line with ruler, find the intersection

You would have to re - write the equations into standard form. Use this method if you don't like graphing using y - intercept and slope. ax + by = c
a2x + b2y = c2

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

In order to use this method when the system of equations is in Standard Form, you must solve for y. In other words, re - write the system into slope - intercept form.

This is the best method to use if and only if you do not get fractions when finding the x - and y - intercepts. If you get fractions, the best way is to solve for y and use the method on the left - "Y - Intercept + Slope"
When using this method, you will get four points (x,0) and (0,y) from equation 1 and (x,0), (0,y) from equation 2. Carefully draw the lines through the x - and y - intercept and locate the point of intersection.
• Given that the system is given in slope - intercept form, use Y - intercept + slope method.
• Step 1: Find the y - intercep for equation 1 and plot it
• b =
• b = − 2, the point is (0, − 2)
• Step 2: Find the slope for equation 1 and plot the next point starting from the y - intercept (0, − 2)
• m = [rise/run] =
• m = [rise/run] = [( − 5)/4] or [5/( − 4)]
• Step 3: Draw the line. Make line extends your entire graph.
• Step 4: Find the y - intercep for equation 2 and plot it
• b =
• b = 2, the point is (0,2)
• Step 5: Find the slope for equation 2 and plot the next point starting from the y - intercept (0,2)
• m = [rise/run] =
• m = [rise/run] = [( − 5)/4] or [5/( − 4)]
• Step 6: Draw the line. Make line extends your entire graph.
• Step 7: Find the point of intersection.
• Solution = ( , )
• Solution = Parallel Lines do not intersect, therefore, no solution. Solve by graphing
y = [7/2]x + 3
y = [1/2]x − 3
• There are three ways to solve system of equations by graphing. Choose the method that best fits the system of equation given.
• Slope Intercept Form:
y = mx + b
y2 = m2x + b

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

This is the best method to use when the system of equations is in this format.
1. Graph the y - intercept b, @ (0,b)
2. Use the slope m = [rise/run] to graph additional points from the y - intercept
3. Draw the line with ruler, find the intersection

You would have to re - write the equations into standard form. Use this method if you don't like graphing using y - intercept and slope.
• Standard Form:
ax + by = c
a2x + b2y = c2

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

In order to use this method when the system of equations is in Standard Form, you must solve for ÿ". In other words, re - write the system into slope - intercept form.

This is the best method to use if and only if you do not get fractions when finding the x - and y - intercepts. If you get fractions, the best way is to solve for y and use the method on the left - "Y-Intercept + Slope"
When using this method, you will get four points (x,0) and (0,y) from equation 1 and (x,0), (0,y) from equation 2. Carefully draw the lines through the x - and y - intercept and locate the point of intersection.
• Given that the system is given in slope - intercept form, use Y - intercept + slope method.
• Step 1: Find the y - intercep for equation 1 and plot it
• b =
• b = 3, the point is (0,3)
• Step 2: Find the slope for equation 1 and plot the next point starting from the y - intercept (0,3)
• m = [rise/run] =
• m = [rise/run] = [7/2] or [( − 7)/( − 2)]
• Step 3: Draw the line. Make line extends your entire graph.
• Step 4: Find the y - intercep for equation 2 and plot it
• b =
• b = − 3, the point is (0, − 3)
• Step 5: Find the slope for equation 2 and plot the next point starting from the y - intercept (0, − 3)
• m = [rise/run] =
• m = [rise/run] = [1/2] or [( − 1)/( − 2)]
• Step 6: Draw the line. Make line extends your entire graph.
• Step 7: Find the point of intersection.
• Solution = ( , )
• Solution = ( − 2, − 4) Solve by graphing
x − 2y = − 2
x + 2y = − 6
• There are three ways to solve system of equations by graphing. Choose the method that best fits the system of equation given.
• Slope Intercept Form:
y = mx + b
y2 = m2x + b

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

This is the best method to use when the system of equations is in this format.
1. Graph the y - intercept b, @ (0,b)
2. Use the slope m = [rise/run] to graph additional points from the y - intercept.
3. Draw the line with ruler, find the intersection

You would have to re - write the equations into standard form. Use this method if you don't like graphing using y - intercept and slope.
• Standard Form:
ax + by = c
a2x + b2y = c2

In order to use this method when the system of equations is in Standard Form, you must solve for y. In other words, re - write the system into slope - intercept form.

This is the best method to use if and only if you do not get fractions when finding the x - and y - intercepts. If you get fractions, the best way is to solve for y and use the method on the left - "Y-Intercept + Slope"

When using this method, you will get four points (x,0) and (0,y) from equation 1 and (x,0), (0,y) from equation 2. Carefully draw the lines through the x - and y - intercept and locate the point of intersection.
• Given that the system is given in Standard Form, use X - intercept Y - Intercept method.
• Step 1: Find the x - intercep for equation 1 by eliminating the y term, solve for x if necessary. Plot it.
• x − 2y = − 2x = − 2
• x - intercpet: ( − 2,0)
• Step 2: Find the y - intercept for equation 1 by eliminating the x - term, solve for y if necessary. Plot it.
• x − 2y = − 2 − 2y = − 2
• y = 1
• y - Intercept:(0,1)
• Step 3: Draw the line through the x - and y - intercepts.
• Step 4: Find the x - intercep for equation 2 by eliminating the y term, solve for x if necessary. Plot it.
• x + 2y = − 6x = − 6
• x - intercpet: ( − 6,0)
• Step 5: Find the y - intercept for equation 2 by eliminating the x - term, solve for y if necessary. Plot it.
• x + 2y = − 62y = − 6
• y = − 3
• y - Intercept:(0, − 3)
• Step 6: Draw the line through the x - and y - intercepts.
• Step 7: Find the point of intersection
• Solution:( − 4, − 1) Solve by graphing
x + y = 2
x + y = − 3
• There are three ways to solve system of equations by graphing. Choose the method that best fits the system of equation given.
• Slope Intercept Form :
y = mx + b
y2 = m2x + b

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

This is the best method to use when the system of equations is in this format.
1. Graph the y - intercept b, @ (0,b)
2. Use the slope m = [rise/run] to graph additional points from the y - intercept.
3. Draw the line with ruler, find the intersection

You would have to re - write the equations into standard form. Use this method if you don't like graphing using y - intercept and slope.
• Standard Form:
ax + by = c
a2x + b2y = c2

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

In order to use this method when the system of equations is in Standard Form, you must solve for y. In other words, re - write the system into slope - intercept form.

This is the best method to use if and only if you do not get fractions when finding the x - and y - intercepts. If you get fractions, the best way is to solve for y and use the method on the left - "Y-Intercept + Slope"

When using this method, you will get four points (x,0) and (0,y) from equation 1 and (x,0), (0,y) from equation 2. Carefully draw the lines through the x - and y - intercept and locate the point of intersection.
• Given that the system is given in Standard Form, use X - intercept Y - Intercept method.
• Step 1: Find the x - intercep for equation 1 by eliminating the y term, solve for x if necessary. Plot it.
• x + y = 2x = 2
• x - intercpet: (2,0)
• Step 2: Find the y - intercept for equation 1 by eliminating the x - term, solve for y if necessary. Plot it.
• x + y = 2y = 2
• y = 2
• y - Intercept:(0,2)
• Step 3: Draw the line through the x - and y - intercepts.
• Step 4: Find the x - intercep for equation 2 by eliminating the y term, solve for x if necessary. Plot it.
• x + y = − 3x = − 3
• x - intercpet: ( − 3,0)
• Step 5: Find the y - intercept for equation 2 by eliminating the x - term, solve for y if necessary. Plot it.
• x + y = − 3y = − 3
• y = − 3
• y - Intercept:(0, − 3)
• Step 6: Draw the line through the x - and y - intercepts.
• Step 7: Find the point of intersection
• Since the lines are parallel, there is no solution. Solve by graphing
x − 4y = 4
x + 2y = − 8
• There are three ways to solve system of equations by graphing. Choose the method that best fits the system of equation given.
• Slope Intercept Form:
y = mx + b
y2 = m2x + b

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

This is the best method to use when the system of equations is in this format.
1. Graph the y - intercept b, @ (0,b)
2. Use the slope m = [rise/run] to graph additional points from the y - intercept.
3. Draw the line with ruler, find the intersection

You would have to re - write the equations into standard form. Use this method if you don't like graphing using y - intercept and slope.
• Standard Form:
ax + by = c
a2x + b2y = c2

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

In order to use this method when the system of equations is in Standard Form, you must solve for y. In other words, re - write the system into slope - intercept form.

This is the best method to use if and only if you do not get fractions when finding the x - and y - intercepts. If you get fractions, the best way is to solve for y and use the method on the left-"Y-Intercept + Slope"

When using this method, you will get four points (x,0) and (0,y) from equation 1 and (x,0), (0,y) from equation 2. Carefully draw the lines through the x- and y-intercept and locate the point of intersection.
• Given that the system is given in Standard Form, use X - intercept Y - Intercept method.
• Step 1: Find the x - intercep for equation 1 by eliminating the y term, solve for x if necessary. Plot it.
• x − 4y = 4x = 4
• x - intercpet: (4,0)
• Step 2: Find the y - intercept for equation 1 by eliminating the x - term, solve for y if necessary. Plot it.
• x − 4y = 4 − 4y = 4
• y = − 1
• y - Intercept:(0, − 1)
• Step 3: Draw the line through the x - and y - intercepts.
• Step 4: Find the x - intercep for equation 2 by eliminating the y term, solve for x if necessary. Plot it.
• x + 2y = − 8x = − 8
• x - intercpet: ( − 8,0)
• Step 5: Find the y - intercept for equation 2 by eliminating the x - term, solve for y if necessary. Plot it.
• x + 2y = − 82y = − 8
• y = − 4
• y - Intercept:(0, − 4)
• Step 6: Draw the line through the x - and y - intercepts.
• Step 7: Find the point of intersection
• Solution ( − 4, − 2) Solve by graphing
x + 2y = 2
x + 2y = 4
• There are three ways to solve system of equations by graphing. Choose the method that best fits the system of equation given.
• Slope Intercept Form:
y = mx + b
y2 = m2x + b

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

This is the best method to use when the system of equations is in this format.
1. Graph the y - intercept b, @ (0,b)
2. Use the slope m = [rise/run] to graph additional points from the y - intercept.
3. Draw the line with ruler, find the intersection

You would have to re - write the equations into standard form. Use this method if you don't like graphing using y - intercept and slope.
• Standard Form:
ax + by = c
a2x + b2y = c2

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

In order to use this method when the system of equations is in Standard Form, you must solve for y. In other words, re - write the system into slope - intercept form.

This is the best method to use if and only if you do not get fractions when finding the x - and y - intercepts. If you get fractions, the best way is to solve for y and use the method on the left - "Y-Intercept + Slope"

When using this method, you will get four points (x,0) and (0,y) from equation 1 and (x,0), (0,y) from equation 2. Carefully draw the lines through the x-and y-intercept and locate the point of intersection.
• Given that the system is given in Standard Form, use X - intercept Y - Intercept method.
• Step 1: Find the x - intercep for equation 1 by eliminating the y term, solve for x if necessary. Plot it.
• x + 2y = 2x = 2
• x - intercpet: (2,0)
• Step 2: Find the y - intercept for equation 1 by eliminating the x - term, solve for y if necessary. Plot it.
• x + 2y = 22y = 2
• y = 1
• y - Intercept:(0,1)
• Step 3: Draw the line through the x - and y - intercepts.
• Step 4: Find the x - intercep for equation 2 by eliminating the y term, solve for x if necessary. Plot it.
• x + 2y = 4x = 4
• x - intercpet: (4,0)
• Step 5: Find the y - intercept for equation 2 by eliminating the x - term, solve for y if necessary. Plot it.
• x + 2y = 42y = 4
• y = 2
• y - Intercept:(0,2)
• Step 6: Draw the line through the x - and y - intercepts.
• Step 7: Find the point of intersection
• Since the lines are parallel, there is no solution. Solve by graphing
− 2x − 2y = 6
6x + y = 2
• There are three ways to solve system of equations by graphing. Choose the method that best fits the system of equation given.
• Slope Intercept Form:
y = mx + b
y2 = m2x + b

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

This is the best method to use when the system of equations is in this format.
1. Graph the y - intercept b, @ (0,b)
2. Use the slope m = [rise/run] to graph additional points from the y - intercept.
3. Draw the line with ruler, find the intersection

You would have to re - write the equations into standard form. Use this method if you don't like graphing using y - intercept and slope.
• Standard Form:
ax + by = c
a2x + b2y = c2

Use the same values of x for both equations. Graph it. With a ruler, connect the points and find the solution if there's any.

In order to use this method when the system of equations is in Standard Form, you must solve for y. In other words, re - write the system into slope - intercept form.

This is the best method to use if and only if you do not get fractions when finding the x - and y - intercepts. If you get fractions, the best way is to solve for y and use the method on the left - "Y-Intercept + Slope"

When using this method, you will get four points (x,0) and (0,y) from equation 1 and (x,0), (0,y) from equation 2. Carefully draw the lines through the x - and y - intercept and locate the point of intersection.
• Eventhough this system of equations is in standard form, you should notice right away that there will be a fraction on equation two when solving for the x - intercept.
• To avoid fractions, solve this system of equations using a table of values.
• Step 1:Create two tables of values. Choose 4 points to solve for y.
•  x 2x-2y=6 -1 0 1 2
•  x 2x-2y=6 -1 -2 0 -3 1 -4 2 -5
•  x 6x+y=2 -1 0 1 2
•  x 6x+y=2 -1 8 0 2 1 -4 2 -10
• Step 2: Plot the points. Draw the lines through the points.
• Step 3: Locate the point of intersection
• Solution (1, − 4) *These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Solving Systems of Equations by Graphing

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
• Systems of Equations 0:09
• Example: Two Equations
• Solving by Graphing 0:53
• Point of Intersection
• Types of Systems 2:29
• Independent (Single Solution)
• Dependent (Infinite Solutions)
• Inconsistent (No Solution)
• 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

### Transcription: Solving Systems of Equations by Graphing

Welcome to Educator.com.0000

In today's Algebra II lesson, we will be discussing solving systems of equations by graphing.0002

Recall that a system of equations, for our purposes, is defined as two equations and two variables.0009

Later on, we will be looking at systems of equations involving more than two variables.0016

But right now, we are just going to stick to this definition.0021

For example, a system of equations could be something such as 2x - y = 7, when it is considered along with another equation, such as x + 3y = -4.0025

And a solution to a system of equations would be a value for x and a value for y--a set of values that satisfies both of the equations.0040

The first technique we are going to discuss in solving systems of equations is solving by graphing.0053

In addition, later on, we will be talking about how these systems can be solved algebraically.0059

So, one way to solve is to graph each equation: the solution for the system is the point of intersection of the two graphs.0065

And we will do some examples in a minute; but for now, imagine that you are given an equation,0075

and you figure out some points, or one of the points in the slope, and it turns out that this is the graph of the line described by the equation.0081

And you are given another equation as part of that system; and you go ahead and graph that one out.0098

And it comes out to a line that looks like this.0105

The solution for the system is the point of intersection, so this is the solution.0110

The x- and y-coordinate for this point are the solution.0117

Now, here you can see a weakness in this method: and that is that, if the solution doesn't land right on an integer, it is difficult to get an exact value.0121

And that is one advantage to the algebraic methods.0136

But if you graph carefully, and the solution does land on, say, (3,4), then you could get an exact solution.0138

There are several types of systems of equations; we call a system independent if it has exactly one solution.0150

For example, just as I showed you in the previous slide, sometimes you will graph out a system, and usually,0159

in the problems you will be doing, you will see a single point of intersection, which is the solution--the single solution.0167

And this set of lines describes an independent system.0175

Other times, you will go along, and you will graph out the system of equations.0185

You have graphed the first equation, and perhaps the graph looks like this.0191

Then, you go to graph the second equation, and it turns out that it describes the same line.0196

So, if the system of equations--both equations within that system describe the same line, then there would be an infinite number of solutions.0202

And the reason is that every single point along this line, every single set of (x,y) coordinates is an intersection of the two systems.0216

The line...every single point along this line (and there is an infinite number of points along the line)0226

is the set of solutions, the set of (x,y) values where the two lines intersect, since they are the same line.0233

Here, we call this a dependent system; here we have one solution; here an infinite number of solutions; and this is a dependent system.0239

The third possibility is that there are no solutions; and we call this an inconsistent system.0254

So, thinking about a situation where there would not be any solution, where there is no point of intersection: it would be a set of parallel lines.0263

Drawing this over here for clarity: here is my x and y axis; and if I graphed the first line, the first linear equation,0273

and then I went ahead and graphed the second one, and those two turned out to have the same slope, those would be parallel lines.0284

They are never going to intersect; therefore, we can see on the graph: there is no solution.0292

So, no solution--that is an inconsistent system.0298

So, there are three possibilities: one solution--intersection at one point; an infinite number of solutions--0306

all the points along the line; or no solution, because it is a set of parallel lines that do not intersect.0312

The first example gives us a system of equations with two variables: x + y = 3 and x - y = 1.0322

So, graphing this first line, I am going to start with x + y = 3; and I find a few x and y values, some ordered pairs, so I can do the graphing.0331

First, I am going to let x equal 0; well, if x is 0, I can see easily that y is 3.0343

So, that gives me the y-intercept; next, I am going to let y equal 0 to find the x-intercept.0351

So, this would be 0; x would be 3; and just one more point to make it a better graph0361

(because it is especially important when you are not just graphing one line--you are actually looking0367

for a solution to a system of equations--to have a really good graph, so you can find a point of intersection accurately):0372

so, when x is 1, x plus y equals 3; so when x is 1, y equals 2.0378

So, I am going to graph this line: first, I have the y-intercept at 3; I have the x-intercept at 3; and another point in between those--x is 1; y is 2.0387

And this gives me a better sense of the slope of the line than if I had just done two points.0399

OK, the second equation--graphing that: x - y =1: again, I am finding a few values for x and y--a few sets of values.0406

When x is 0, that would give me 0 - y = 1, so - y equals 1; therefore, y is -1; the y-intercept is -1.0422

Now, finding the x-intercept, let y equal 0; if y is 0, x is 1.0434

One final point: I am going to just let x equal 2; so that would be 2 - y = 1; and that is going to give me -y = -1, so y = 1.0442

Again, I have three sets of points to graph.0457

So, the y-intercept is at -1; the x-intercept is at 1; and one more point: when x is 2, y is 1.0460

OK, drawing this line, I can immediately see that I have a single point of intersection, right here.0476

This is the solution; and this occurs at (2,1).0487

So, the solution for this is x = 2, y = 1.0493

And therefore, this is an independent system, since it has one solution.0501

OK, and you can always check your work by substituting x and y values back into these equations.0515

I have x + y = 3, so that is 2 + 1 = 3; and that checks out.0528

I have x - y = 1, so 2 - 1 = 1, and that checks out; so you can easily check and make sure that both of these are valid solutions for both of the equations.0534

The second example is a little bit more complicated, but using the same system.0551

And this time, instead of just finding (x,y) values, I am going to graph these out by putting both equations into slope-intercept form.0557

So, the first one is 2x - 4y = 12, so I am going to subtract 2x from both sides, and then I am going to divide both sides by -4.0565

And that is going to give me y = 1/2x - 3.0577

Slope-intercept form is very helpful when you are graphing: this is -2, -4, -6, -8, -10, 2, 4, 6, 8, 10.0582

OK, here I have a y-intercept at -3; and I know the slope--the slope is 1/2.0597

That tells me, when I increase y by 1, I am going to increase x by 2; if I increase y by another 1, I am going to increase x by 2.0606

So, you can see the slope of this line right here; that gives me enough to work with.0620

I am going to do the same with the second equation, 4x - y = 10, putting this into slope-intercept form, y = mx + b,0638

subtracting 4x from both sides, and then dividing both sides by -1 to give me y = 4x - 10.0646

So here, I have a y-intercept at -10 and a slope of 4; I am going to increase y by 4 (that is 2, 4), and increase x by 1.0656

Increase y by 2, 4; increase x by 1; increase y by 4 (2, 4); increase x by 1; OK.0669

So, this is a much steeper line; the slope is much steeper--the slope of 4; so I am going to go ahead and draw it like that.0687

And the point of intersection is right here, and again, we have an independent system with a single solution.0695

And it is at (2,-2), so that is my point of intersection: x is 2; y is -2.0708

Again, you could always check your answers by substituting either or both equations with these x and y values, and ensuring that the solutions are valid.0723

So again, I have an independent system; it has one solution--one set of valid solutions.0733

OK, this next example: again, solve by graphing; and my approach is going to be to put these into y-intercept form and use that to graph.0748

First, 2x - 3y = 6, so -3y = -2x + 6; divide both sides by -3; that is going to give me...-2/-3 is 2/3x, and then 6/-3 is -2.0758

So, this line has a y-intercept of -2 and a slope of 2/3; so increase y by 2; increase x by 3.0781

OK, that is my first equation; the second equation--again, putting it into the slope-intercept form, which is an easy way of graphing:0799

This gives me 6y = 4x - 12; dividing both sides by 6 gives me 4/6x - 2.0808

And I am going to go ahead and simplify this to 2/3x - 2.0823

So, you may already see that these are the same equation--these are going to describe the same line.0828

Even if you didn't notice it right away, when you start graphing, you are going to see: the y-intercept is -2; the slope is 2/3.0835

You are going to end up with the same line.0842

Because this is the same line, they intersect at every single point; these two equations--their graphs intersect at every single point,0844

which is an infinite number of points, along the line; they don't intersect everywhere, but everywhere on this line.0852

Therefore, this is a dependent system, and there is an infinite number of solutions; all points along this line are solutions.0857

And this is known as a dependent system.0871

So initially, you looked at this; you might not have recognized that these are actually describing the same line.0877

But once you started to plot it, either by putting it in the y-intercept form or by finding points along these lines,0882

you would have quickly realized that this is the same line.0890

In this last example, again, there is a system of equations that we need to solve by graphing.0896

Again, using the method of slope-intercept form for graphing: 4x - 2y = 8; subtract 4x from both sides, and then divide both sides by -2 to give me y = 2x - 4.0905

OK, the y-intercept is -4; the slope is 2; increase y by 2; increase x by 1; increase y by 2; increase x by 1; and so on.0925

I have three points; that is plenty to graph this line...so that is my first line.0942

The second line: again, slope-intercept form: add 6x to both sides--it gives you 3y = 6x + 12.0947

Now, divide both sides by 3 to get y = 2x + 4.0961

So here, I have a y-intercept of 4 and a slope of 2; so increase x by 2; increase y by 1.0968

So, I am going to get a line looking like this; and what you may quickly realize is that these are parallel lines.0978

One thing you could have noted is that the slope, m, equals 2 for this line, and it equals 2 for this line.0991

Since these are parallel lines, they are never going to intersect; and this is an inconsistent system, and there is no solution.0998

And what we say here is: you could say the solution is just the empty set; there is no solution.1008

This is described as an inconsistent system.1016

OK, that concludes this lesson of Educator.com, describing graphing to solve systems of equations.1024

And I will see you next time.1032

OR

### Start Learning Now

Our free lessons will get you started (Adobe Flash® required).