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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

The Vocabulary of Linear Equations

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Table of Contents

I. Properties of Real Numbers
Basic Types of Numbers

30m 41s

Intro
0:00
Objectives
0:07
Basic Types of Numbers
0:36
Natural Numbers
1:02
Whole Numbers
1:29
Integers
2:04
Rational Numbers
2:38
Irrational Numbers
5:06
Imaginary Numbers
6:48
Basic Types of Numbers Cont.
8:09
The Big Picture
8:10
Real vs. Imaginary Numbers
8:30
Rational vs. Irrational Numbers
8:48
Basic Types of Numbers Cont.
10:55
Number Line
11:06
Absolute Value
11:44
Inequalities
12:39
Example 1
13:16
Example 2
17:30
Example 3
21:56
Example 4
24:27
Example 5
27:48
Operations on Numbers

19m 26s

Intro
0:00
Objectives
0:06
Operations on Numbers
0:25
Addition
0:53
Subtraction
1:33
Multiplication & Division
2:19
Exponents
3:24
Bases
4:04
Square Roots
4:59
Principle Square Roots
5:09
Perfect Squares
6:32
Simplifying and Combining Roots
6:52
Example 1
8:16
Example 2
12:30
Example 3
14:02
Example 4
16:27
Order of Operations

12m 6s

Intro
0:00
Objectives
0:06
The Order of Operations
0:25
Work Inside Parentheses
0:42
Simplify Exponents
0:52
Multiplication & Division from Left to Right
0:57
Addition & Subtraction from Left to Right
1:11
Remember PEMDAS
1:21
The Order of Operations Cont.
2:27
Example
2:43
Example 1
3:55
Example 2
5:36
Example 3
7:35
Example 4
8:56
Properties of Real Numbers

18m 52s

Intro
0:00
Objectives
0:07
The Properties of Real Numbers
0:23
Commutative Property of Addition and Multiplication
0:44
Associative Property of Addition and Multiplication
1:50
Distributive Property of Multiplication Over Addition
3:20
Division Property of Zero
4:46
Division Property of One
5:23
Multiplication Property of Zero
5:56
Multiplication Property of One
6:17
Addition Property of Zero
6:29
Why Are These Properties Important?
6:53
Example 1
9:16
Example 2
13:04
Example 3
14:30
Example 4
16:57
II. Linear Equations
The Vocabulary of Linear Equations

12m 22s

Intro
0:00
Objectives
0:09
The Vocabulary of Linear Equations
0:44
Variables
0:52
Terms
1:09
Coefficients
1:40
Like Terms
2:18
Examples of Like Terms
2:37
Expressions
4:01
Equations
4:26
Linear Equations
5:04
Solutions
5:55
Example 1
6:16
Example 2
7:16
Example 3
8:45
Example 4
10:20
Solving Linear Equations in One Variable

28m 52s

Intro
0:00
Objectives
0:08
Solving Linear Equations in One Variable
0:34
Conditional Cases
0:51
Identity Cases
1:09
Contradiction Cases
1:30
Solving Linear Equations in One Variable Cont.
2:00
Addition Property of Equality
2:10
Multiplication Property of Equality
2:43
Steps to Solve Linear Equations
3:14
Example 1
4:22
Example 2
8:21
Example 3
12:32
Example 4
14:19
Example 5
17:25
Example 6
22:17
Solving Formulas

12m 2s

Intro
0:00
Objectives
0:06
Solving Formulas
0:18
Formulas
0:26
Use the Same Properties as Solving Linear Equations
1:36
Addition Property of Equality
1:55
Multiplication Property of Equality
1:58
Steps to Solve Formulas
2:43
Example 1
3:56
Example 2
6:09
Example 3
8:39
Applications of Linear Equations

28m 41s

Intro
0:00
Objectives
0:10
Applications of Linear Equations
0:43
The Six-Step Method to Solving Word Problems
0:55
Common Terms
3:12
Example 1
5:03
Example 2
9:40
Example 3
13:48
Example 4
17:58
Example 5
23:28
Applications of Linear Equations, Motion & Mixtures

24m 26s

Intro
0:00
Objectives
0:21
Motion and Mixtures
0:46
Motion Problems: Distance, Rate, and Time
1:06
Mixture Problems: Amount, Percent, and Total
1:27
The Table Method
1:58
The Beaker Method
3:38
Example 1
5:05
Example 2
9:44
Example 3
14:20
Example 4
19:13
III. Graphing
Rectangular Coordinate System

22m 55s

Intro
0:00
Objectives
0:11
The Rectangular Coordinate System
0:39
The Cartesian Coordinate System
0:40
X-Axis
0:54
Y-Axis
1:04
Origin
1:11
Quadrants
1:26
Ordered Pairs
2:10
Example 1
2:55
The Rectangular Coordinate System Cont.
6:09
X-Intercept
6:45
Y-Intercept
6:55
Relation of X-Values and Y-Values
7:30
Example 2
11:03
Example 3
12:13
Example 4
14:10
Example 5
18:38
Slope & Graphing

27m 58s

Intro
0:00
Objectives
0:11
Slope and Graphing
0:48
Standard Form
1:14
Example 1
2:24
Slope and Graphing Cont.
4:58
Slope, m
5:07
Slope is Rise over Run
6:11
Don't Mix Up the Coordinates
8:20
Example 2
9:39
Slope and Graphing Cont.
14:26
Slope-Intercept Form
14:34
Example 3
16:55
Example 4
18:00
Slope and Graphing Cont.
19:00
Rewriting an Equation in Slope-Intercept Form
19:39
Rewriting an Equation in Standard Form
20:09
Slopes of Vertical & Horizontal Lines
20:56
Example 5
22:49
Example 6
24:09
Example 7
25:59
Example 8
26:57
Linear Equations in Two Variables

20m 36s

Intro
0:00
Objectives
0:13
Linear Equations in Two Variables
0:36
Point-Slope Form
1:07
Substitute in the Point and the Slope
2:21
Parallel Lines: Two Lines with the Same Slope
4:05
Perpendicular Lines: Slopes are Negative Reciprocals of Each Other
4:39
Perpendicular Lines: Product of Slopes is -1
5:24
Example 1
6:02
Example 2
7:50
Example 3
10:49
Example 4
13:26
Example 5
15:30
Example 6
17:43
IV. Functions
Introduction to Functions

21m 24s

Intro
0:00
Objectives
0:07
Introduction to Functions
0:58
Relations
1:03
Functions
1:37
Independent Variables
2:00
Dependent Variables
2:11
Function Notation
2:21
Function
3:43
Input and Output
3:53
Introduction to Functions Cont.
4:45
Domain
4:46
Range
4:55
Functions Represented by a Diagram
6:41
Natural Domain
9:11
Evaluating Functions
12:02
Example 1
13:13
Example 2
15:03
Example 3
16:18
Example 4
19:54
Graphing Functions

16m 12s

Intro
0:00
Objectives
0:09
Graphing Functions
0:54
Using Slope-Intercept Form
1:56
Vertical Line Test
2:58
Determining the Domain
4:20
Determining the Range
5:43
Example 1
6:06
Example 2
7:18
Example 3
8:31
Example 4
11:04
V. Systems of Linear Equations
Systems of Linear Equations

25m 54s

Intro
0:00
Objectives
0:13
Systems of Linear Equations
0:46
System of Equations
0:51
System of Linear Equations
1:15
Solutions
1:35
Points as Solutions
1:53
Finding Solutions Graphically
5:13
Example 1
6:37
Example 2
12:07
Systems of Linear Equations Cont.
17:01
One Solution, No Solution, or Infinite Solutions
17:10
Example 3
18:31
Example 4
22:37
Solving a System Using Substitution

20m 1s

Intro
0:00
Objectives
0:09
Solving a System Using Substitution
0:32
Substitution Method
1:24
Substitution Example
2:35
One Solution, No Solution, or Infinite Solutions
7:50
Example 1
9:45
Example 2
12:48
Example 3
15:01
Example 4
17:30
Solving a System Using Elimination

19m 40s

Intro
0:00
Objectives
0:09
Solving a System Using Elimination
0:27
Elimination Method
0:42
Elimination Example
2:01
One Solution, No Solution, or Infinite Solutions
7:05
Example 1
8:53
Example 2
11:46
Example 3
15:37
Example 4
17:45
Applications of Systems of Equations

24m 34s

Intro
0:00
Objectives
0:12
Applications of Systems of Equations
0:30
Word Problems
1:31
Example 1
2:17
Example 2
7:55
Example 3
13:07
Example 4
17:15
VI. Inequalities
Solving Linear Inequalities in One Variable

17m 13s

Intro
0:00
Objectives
0:08
Solving Linear Inequalities in One Variable
0:37
Inequality Expressions
0:46
Linear Inequality Solution Notations
3:40
Inequalities
3:51
Interval Notation
4:04
Number Lines
4:43
Set Builder Notation
5:24
Use Same Techniques as Solving Equations
6:59
'Flip' the Sign when Multiplying or Dividing by a Negative Number
7:12
'Flip' Example
7:50
Example 1
8:54
Example 2
11:40
Example 3
14:01
Compound Inequalities

16m 13s

Intro
0:00
Objectives
0:07
Compound Inequalities
0:37
'And' vs. 'Or'
0:44
'And'
3:24
'Or'
3:35
'And' Symbol, or Intersection
3:51
'Or' Symbol, or Union
4:13
Inequalities
4:41
Example 1
6:22
Example 2
9:30
Example 3
11:27
Example 4
13:49
Solving Equations with Absolute Values

14m 12s

Intro
0:00
Objectives
0:08
Solve Equations with Absolute Values
0:18
Solve Equations with Absolute Values Cont.
1:11
Steps to Solving Equations with Absolute Values
2:21
Example 1
3:23
Example 2
6:34
Example 3
10:12
Inequalities with Absolute Values

17m 7s

Intro
0:00
Objectives
0:07
Inequalities with Absolute Values
0:23
Recall…
2:08
Example 1
3:39
Example 2
6:06
Example 3
8:14
Example 4
10:29
Example 5
13:29
Graphing Inequalities in Two Variables

15m 33s

Intro
0:00
Objectives
0:07
Graphing Inequalities in Two Variables
0:32
Split Graph into Two Regions
1:53
Graphing Inequalities
5:44
Test Points
6:20
Example 1
7:11
Example 2
10:17
Example 3
13:06
Systems of Inequalities

21m 13s

Intro
0:00
Objectives
0:08
Systems of Inequalities
0:24
Test Points
1:10
Steps to Solve Systems of Inequalities
1:25
Example 1
2:23
Example 2
7:28
Example 3
12:51
VII. Polynomials
Integer Exponents

44m 51s

Intro
0:00
Objectives
0:09
Integer Exponents
0:42
Exponents 'Package' Multiplication
1:25
Example 1
2:00
Example 2
3:13
Integer Exponents Cont.
4:50
Product Rule for Exponents
4:51
Example 3
7:16
Example 4
10:15
Integer Exponents Cont.
13:13
Power Rule for Exponents
13:14
Power Rule with Multiplication and Division
15:33
Example 5
16:18
Integer Exponents Cont.
20:04
Example 6
20:41
Integer Exponents Cont.
25:52
Zero Exponent Rule
25:53
Quotient Rule
28:24
Negative Exponents
30:14
Negative Exponent Rule
32:27
Example 7
34:05
Example 8
36:15
Example 9
39:33
Example 10
43:16
Adding & Subtracting Polynomials

18m 33s

Intro
0:00
Objectives
0:07
Adding and Subtracting Polynomials
0:25
Terms
0:33
Coefficients
0:51
Leading Coefficients
1:13
Like Terms
1:29
Polynomials
2:21
Monomials, Binomials, Trinomials, and Polynomials
5:41
Degrees
7:00
Evaluating Polynomials
8:12
Adding and Subtracting Polynomials Cont.
9:25
Example 1
11:48
Example 2
13:00
Example 3
14:41
Example 4
16:15
Multiplying Polynomials

25m 7s

Intro
0:00
Objectives
0:06
Multiplying Polynomials
0:41
Distributive Property
1:00
Example 1
2:49
Multiplying Polynomials Cont.
8:22
Organize Terms with a Table
8:23
Example 2
13:40
Multiplying Polynomials Cont.
16:33
Multiplying Binomials with FOIL
16:48
Example 3
18:49
Example 4
20:04
Example 5
21:42
Dividing Polynomials

44m 56s

Intro
0:00
Objectives
0:07
Dividing Polynomials
0:29
Dividing Polynomials by Monomials
2:10
Dividing Polynomials by Polynomials
2:59
Dividing Numbers
4:09
Dividing Polynomials Example
8:39
Example 1
12:35
Example 2
14:40
Example 3
16:45
Example 4
21:13
Example 5
24:33
Example 6
29:02
Dividing Polynomials with Synthetic Division Method
33:36
Example 7
38:43
Example 8
42:24
VIII. Factoring Polynomials
Greatest Common Factor & Factor by Grouping

28m 27s

Intro
0:00
Objectives
0:09
Greatest Common Factor
0:31
Factoring
0:40
Greatest Common Factor (GCF)
1:48
GCF for Polynomials
3:28
Factoring Polynomials
6:45
Prime
8:21
Example 1
9:14
Factor by Grouping
14:30
Steps to Factor by Grouping
17:03
Example 2
17:43
Example 3
19:20
Example 4
20:41
Example 5
22:29
Example 6
26:11
Factoring Trinomials

21m 44s

Intro
0:00
Objectives
0:06
Factoring Trinomials
0:25
Recall FOIL
0:26
Factor a Trinomial by Reversing FOIL
1:52
Tips when Using Reverse FOIL
5:31
Example 1
7:04
Example 2
9:09
Example 3
11:15
Example 4
13:41
Factoring Trinomials Cont.
15:50
Example 5
18:42
Factoring Trinomials Using the AC Method

30m 9s

Intro
0:00
Objectives
0:08
Factoring Trinomials Using the AC Method
0:27
Factoring when Leading Term has Coefficient Other Than 1
1:07
Reversing FOIL
1:18
Example 1
1:46
Example 2
4:28
Factoring Trinomials Using the AC Method Cont.
7:45
The AC Method
8:03
Steps to Using the AC Method
8:19
Tips on Using the AC Method
9:29
Example 3
10:45
Example 4
16:50
Example 5
21:08
Example 6
24:58
Special Factoring Techniques

30m 14s

Intro
0:00
Objectives
0:07
Special Factoring Techniques
0:26
Difference of Squares
1:46
Perfect Square Trinomials
2:38
No Sum of Squares
3:32
Special Factoring Techniques Cont.
4:03
Difference of Squares Example
4:04
Perfect Square Trinomials Example
5:29
Example 1
7:31
Example 2
9:59
Example 3
11:47
Example 4
15:09
Special Factoring Techniques Cont.
19:07
Sum of Cubes and Difference of Cubes
19:08
Example 5
23:13
Example 6
26:12
IX. Quadratic Equations
Solving Quadratic Equations by Factoring

23m 38s

Intro
0:00
Objectives
0:08
Solving Quadratic Equations by Factoring
0:19
Quadratic Equations
0:20
Zero Factor Property
1:39
Zero Factor Property Example
2:34
Example 1
4:00
Solving Quadratic Equations by Factoring Cont.
5:54
Example 2
7:28
Example 3
11:09
Example 4
14:22
Solving Quadratic Equations by Factoring Cont.
18:17
Higher Degree Polynomial Equations
18:18
Example 5
20:22
Solving Quadratic Equations

29m 27s

Intro
0:00
Objectives
0:12
Solving Quadratic Equations
0:29
Linear Factors
0:38
Not All Quadratics Factor Easily
1:22
Principle of Square Roots
3:36
Completing the Square
4:50
Steps for Using Completing the Square
5:15
Completing the Square Works on All Quadratic Equations
6:41
The Quadratic Formula
7:28
Discriminants
8:25
Solving Quadratic Equations - Summary
10:11
Example 1
11:54
Example 2
13:03
Example 3
16:30
Example 4
21:29
Example 5
25:07
Equations in Quadratic Form

16m 47s

Intro
0:00
Objectives
0:08
Equations in Quadratic Form
0:24
Using a Substitution
0:53
U-Substitution
1:26
Example 1
2:07
Example 2
5:36
Example 3
8:31
Example 4
11:14
Quadratic Formulas & Applications

29m 4s

Intro
0:00
Objectives
0:09
Quadratic Formulas and Applications
0:35
Squared Variable
0:40
Principle of Square Roots
0:51
Example 1
1:09
Example 2
2:04
Quadratic Formulas and Applications Cont.
3:34
Example 3
4:42
Example 4
13:33
Example 5
20:50
Graphs of Quadratics

26m 53s

Intro
0:00
Objectives
0:06
Graphs of Quadratics
0:39
Axis of Symmetry
1:46
Vertex
2:12
Transformations
2:57
Graphing in Quadratic Standard Form
3:23
Example 1
5:06
Example 2
6:02
Example 3
9:07
Graphs of Quadratics Cont.
11:26
Completing the Square
12:02
Vertex Shortcut
12:16
Example 4
13:49
Example 5
17:25
Example 6
20:07
Example 7
23:43
Polynomial Inequalities

21m 42s

Intro
0:00
Objectives
0:07
Polynomial Inequalities
0:30
Solving Polynomial Inequalities
1:20
Example 1
2:45
Polynomial Inequalities Cont.
5:12
Larger Polynomials
5:13
Positive or Negative Intervals
7:16
Example 2
9:01
Example 3
13:53
X. Rational Equations
Multiply & Divide Rational Expressions

26m 41s

Intro
0:00
Objectives
0:09
Multiply and Divide Rational Expressions
0:44
Rational Numbers
0:55
Dividing by Zero
1:45
Canceling Extra Factors
2:43
Negative Signs in Fractions
4:52
Multiplying Fractions
6:26
Dividing Fractions
7:17
Example 1
8:04
Example 2
14:01
Example 3
16:23
Example 4
18:56
Example 5
22:43
Adding & Subtracting Rational Expressions

20m 24s

Intro
0:00
Objectives
0:07
Adding and Subtracting Rational Expressions
0:41
Common Denominators
0:52
Common Denominator Examples
1:14
Steps to Adding and Subtracting Rational Expressions
2:39
Example 1
3:34
Example 2
5:27
Adding and Subtracting Rational Expressions Cont.
6:57
Least Common Denominators
6:58
Transitioning from Fractions to Rational Expressions
9:08
Identifying Least Common Denominators for Rational Expressions
9:56
Subtracting vs. Adding
10:41
Example 3
11:19
Example 4
12:36
Example 5
15:08
Example 6
16:46
Complex Fractions

18m 23s

Intro
0:00
Objectives
0:09
Complex Fractions
0:37
Dividing to Simplify Complex Fractions
1:10
Example 1
2:03
Example 2
3:58
Complex Fractions Cont.
9:15
Using the Least Common Denominator to Simplify Complex Fractions
9:16
Both Methods Lead to the Same Answer
10:07
Example 3
10:42
Example 4
14:28
Solving Rational Equations

16m 24s

Intro
0:00
Objectives
0:07
Solving Rational Equations
0:23
Isolate the Specified Variable
1:23
Example 1
1:58
Example 2
5:00
Example 3
8:23
Example 4
13:25
Rational Inequalities

18m 54s

Intro
0:00
Objectives
0:06
Rational Inequalities
0:18
Testing Intervals for Rational Inequalities
0:38
Steps to Solving Rational Inequalities
1:05
Tips to Solving Rational Inequalities
2:27
Example 1
3:33
Example 2
12:21
Applications of Rational Expressions

20m 20s

Intro
0:00
Objectives
0:07
Applications of Rational Expressions
0:27
Work Problems
1:05
Example 1
2:58
Example 2
6:45
Example 3
13:17
Example 4
16:37
Variation & Proportion

27m 4s

Intro
0:00
Objectives
0:10
Variation and Proportion
0:34
Variation
0:35
Inverse Variation
1:01
Direct Variation
1:10
Setting Up Proportions
1:31
Example 1
2:27
Example 2
5:36
Variation and Proportion Cont.
8:29
Inverse Variation
8:30
Example 3
9:20
Variation and Proportion Cont.
12:41
Constant of Proportionality
12:42
Example 4
13:59
Variation and Proportion Cont.
16:17
Varies Directly as the nth Power
16:30
Varies Inversely as the nth Power
16:53
Varies Jointly
17:09
Combining Variation Models
17:36
Example 5
19:09
Example 6
22:10
XI. Radical Equations
Rational Exponents

14m 32s

Intro
0:00
Objectives
0:07
Rational Exponents
0:32
Power on Top, Root on Bottom
1:05
Example 1
1:37
Rational Exponents Cont.
4:04
Using Rules from Exponents for Radicals as Exponents
4:05
Combining Terms Under a Single Root
4:50
Example 2
5:21
Example 3
7:39
Example 4
11:23
Example 5
13:14
Simplify Rational Exponents

15m 12s

Intro
0:00
Objectives
0:07
Simplify Rational Exponents
0:25
Product Rule for Radicals
0:26
Product Rule to Simplify Square Roots
1:11
Quotient Rule for Radicals
1:42
Applications of Product and Quotient Rules
2:17
Higher Roots
2:48
Example 1
3:39
Example 2
6:35
Example 3
8:41
Example 4
11:09
Adding & Subtracting Radicals

17m 22s

Intro
0:00
Objectives
0:07
Adding and Subtracting Radicals
0:33
Like Terms
1:29
Bases and Exponents May be Different
2:02
Bases and Powers Must be Same when Adding and Subtracting
2:42
Add Radicals' Coefficients
3:55
Example 1
4:47
Example 2
6:00
Adding and Subtracting Radicals Cont.
7:10
Simplify the Bases to Look the Same
7:25
Example 3
8:23
Example 4
11:45
Example 5
15:10
Multiply & Divide Radicals

19m 24s

Intro
0:00
Objectives
0:08
Multiply and Divide Radicals
0:25
Rules for Working With Radicals
0:26
Using FOIL for Radicals
1:11
Don’t Distribute Powers
2:54
Dividing Radical Expressions
4:25
Rationalizing Denominators
6:40
Example 1
7:22
Example 2
8:32
Multiply and Divide Radicals Cont.
9:23
Rationalizing Denominators with Higher Roots
9:25
Example 3
10:51
Example 4
11:53
Multiply and Divide Radicals Cont.
13:13
Rationalizing Denominators with Conjugates
13:14
Example 5
15:52
Example 6
17:25
Solving Radical Equations

15m 5s

Intro
0:00
Objectives
0:07
Solving Radical Equations
0:17
Radical Equations
0:18
Isolate the Roots and Raise to Power
0:34
Example 1
1:13
Example 2
3:09
Solving Radical Equations Cont.
7:04
Solving Radical Equations with More than One Radical
7:05
Example 3
7:54
Example 4
13:07
Complex Numbers

29m 16s

Intro
0:00
Objectives
0:06
Complex Numbers
1:05
Imaginary Numbers
1:08
Complex Numbers
2:27
Real Parts
2:48
Imaginary Parts
2:51
Commutative, Associative, and Distributive Properties
3:35
Adding and Subtracting Complex Numbers
4:04
Multiplying Complex Numbers
6:16
Dividing Complex Numbers
8:59
Complex Conjugate
9:07
Simplifying Powers of i
14:34
Shortcut for Simplifying Powers of i
18:33
Example 1
21:14
Example 2
22:15
Example 3
23:38
Example 4
26:33
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Lecture Comments (12)

1 answer

Last reply by: Professor Eric Smith
Sun Dec 3, 2017 5:50 PM

Post by Juan Collado on November 28, 2017

You are truly a great teacher!  I have to refresh my math to pre calculus and then I need higher than that.  

Are you doing any new videos on ALgebra II , PreCal. and Calculus?

Also:  Could I skip Trigonometry and Geometry and go straight to precalculus?  I need to get my math to Calculus as soon as possible.

Thank you so much for these great videos.

3 answers

Last reply by: Juan Collado
Tue Nov 28, 2017 7:39 PM

Post by Timothy Davis on June 10, 2014

I am thoroughly enjoying your lectures!  I have always struggled with calculus and differential equations, and after watching your videos up this point, I realize it is because my algebra 1 skills are so weak!.   Thanks!

1 answer

Last reply by: Professor Eric Smith
Tue Oct 15, 2013 10:15 AM

Post by Emma Wright on October 14, 2013

So, When You had 20w-9+3 why did it end up as 20w-6?
Also, why when you subtract y-3y why is it 2, and when you add y2+6y2 it ends up as 7y2 ?

3 answers

Last reply by: Professor Eric Smith
Tue Oct 15, 2013 10:16 AM

Post by Ezuma Ngwu on August 29, 2013

How do you create and solve equations with tiles?

The Vocabulary of Linear Equations

  • A variable is a letter that is used to represent any unknown quantity. Usually we use the letter “x.”
  • A term is a number, variable, or product or quotient of numbers and variables raised to powers. Terms can be identified because they are connected to other terms using addition or subtraction.
  • Terms with exactly the same variables that have the same exponents are called like terms.
  • An equation is a statement that two algebraic expression are equal. A linear equation is a special type of equation that can be written in the form Ax + B = C where A, B, and C are real numbers with A not being zero.
  • A number is said to be a solution if it can be substituted for the variable, and it creates a true statement.

The Vocabulary of Linear Equations

Write the equation for:
Six times a number decreased by the square of that number is three more than five times the number.
  • x represents the unknown number
6xx2 =5x + 3
Write the equation for:
Eight times the sum of a number and the cube of another number is ten less than the difference of the second number and four times the first number.
  • x = 1st number
    y = 2nd number
8(x + y3) =(y4x)10
Write the equation for: Twelve times a number increased by the cube of that number is twenty more than eight times the number.
  • x represents the unknown number
12x + x3 =8x + 20
Write the equation for:
Sixteen times the difference of a number and half of another number is three less than the sum of the square of the second number and seven times the first number.
  • x = 1st number
    y = 2nd number
16(x − [(y)/(2)]) =(y2 + 7x)3
Write the equation for:
John's brother is ten years younger than he is. The product of their ages is seventeen less than the square of the difference of their ages.
  • j = John's age
    j - 10 = John's sister's age
  • j(j − 10) = [j − (j − 10)]2 − 17
  • j(j − 10) = (j − j + 10)2 − 17
  • j(j − 10) = 102 − 17
  • j(j − 10) = 100 − 17
j(j10) =83
Write the equation for:
Jason's friend is six years older than he is. The difference of their age is three times Jason's age.
  • j = Jason's age
    j + 6 = Jason's friend's age
j(j + 6) =3(j + 6)
Write the equation for: The square of a number decreased by ten times that number is two less than five times that number.
  • x represents the unknown number
x210x =5x2
Write the equation for:
Half of a number is seven more than four times the square of that number.
  • x represents the unknown number
[(x)/(2)] =4x2 + 7
Write the equation for:
Nine times the difference of a number and the cube of another number is seven more than the difference of two times the square of the second number and three times the first number.
  • x = 1st number
    y = 2nd number
9(xy3) =(2y23x) + 7
Write the equation for:
Sarah's sister is twice her age. The sum of their ages is three less than the square of the difference of their ages.
  • s = Sarah's age
    2s = Sarah's sister's age
s + 2s =(s2s)23

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

Answer

The Vocabulary of Linear Equations

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
  • Objectives 0:09
  • The Vocabulary of Linear Equations 0:44
    • Variables
    • Terms
    • Coefficients
    • Like Terms
    • Examples of Like Terms
    • Expressions
    • Equations
    • Linear Equations
    • Solutions
  • Example 1 6:16
  • Example 2 7:16
  • Example 3 8:45
  • Example 4 10:20

Transcription: The Vocabulary of Linear Equations

Welcome back to www.educator.com.0000

In this lesson we are going to start looking more at linear equation starting off with the vocabulary of linear equations.0003

There will be lots of new terms in here, it will definitely take some time to look at them all and what they mean and play around with them a little bit.0010

Some of the terms that will definitely get more familiar with are variable, term.0018

We will look at coefficients and we will definitely see how we can combine like terms.0026

We will be able to tell the difference between equations and expressions and get into a linear equation.0032

What we want to solve later on and solutions.0038

When looking at an equation, we often see lots of letters in there, those are our variables.0046

What they do is they represent our unknowns.0054

One favorite thing to use in a lot of equations is (x), but potentially we could use any letter.0058

You could use a, b, it does not really matter but most the time our unknown is (x).0064

A term is a little bit more than just that variable.0070

It is a number of variables or sometimes the product or quotient of those things put together.0074

To make it a little bit more clear, I have different examples of what I mean by a term.0079

All of these things down here are types of terms.0085

The first one is the product of an actual number and a variable.0088

Down here with the (k) it is simply just a variable all by itself.0093

A coefficient of a term is a number associated with that term.0102

If I'm looking at a term say 2m, the coefficient is the number right out front that is associated with that term.0109

I'm looking at another one like 5mq, but again the 5 would be the coefficient of that term.0121

Terms with the exactly the same variables that have the same exponents those are known as like terms.0133

There are two conditions in there you want to be familiar with.0140

It must have exactly the same variables and it must also have exactly the same exponents.0143

Let us say I have both of those and you can not consider them like terms.0150

Let us take a look at some real quick.0156

I'm looking at 5x and I'm looking at 5y, these are not like terms.0158

Why, you ask? They do not have the same variable.0169

One has (x) and the other one has (y).0172

How about 3x2 and 4x2, these are like terms.0176

These ones are definitely good because notice they have exactly the same variable and they have the same exponent.0188

They have both of those conditions.0203

This one is little bit trickier so be careful, they both have an (x), that looks good.0218

They both have a (y), that seems good but they have different exponents.0224

This one has the y2 and this one has nothing on its y.0230

I would say that these are not like terms.0238

An expression is the statement written using a combination of these numbers, operations, and variables.0244

This is when we start stringing things together so I might have a term 2x and then I decide use may be addition and put together a 4xy.0251

That entire thing would be my expression.0262

In the equation, we take a statement that two algebraic expressions are actually equal.0265

I can even borrow my previous expression to make an equation.0271

I simply have to set it equal to another expression, maybe 2x2.0276

What a lot of students like to recognize in these two cases is that in an equation there better be an equal sign somewhere in there.0285

With your expression there is not an equal sign because it is just a whole bunch of string of terms and coefficients, numbers, operations.0292

Since we are interested in linear equations and eventually getting solved for those, what exactly is a linear equation?0306

It is any equation that can be written in the form, ax + b = c.0314

There is some conditions on those a, b, and c.0321

Here we want a, b, and c to be real numbers, we would not have to deal with any of those imaginary guys.0324

We want to make sure that (a) is not 0.0330

The reason why we are throwing that condition in there is we do not want to get rid of our variable.0335

If (a) was 0, you would have 0 × x and then we would have a rare variable whatsoever.0340

It is an equation, not necessarily look like that but it almost can be written in that form, it is a linear equation.0347

A number is a solution of that equation if after substituting it in for the value the statement is true.0354

That means if you actually take out your variable and replace it with a number that is the solution.0361

Then it is going to balance out, it is going to be true with that number in there.0367

This first part we just want to identify the different parts of the equation so we can better feel of what we are looking at.0379

First of all I know that this is an equation because notice how we have an equal sign right there.0386

I have the expression 7x + 8 that is one expression one side and expression 15 on the other.0394

What else do I have here? I have my 7x and I have the 8 both of these are terms.0400

If I pick apart that one term on the left, I can say that the 7 is a coefficient and that the x is my variable.0410

There are many different parts of the equation and you want be able to keep track of it.0428

And probably terms are one of the most important for now.0432

Let us look at this one, let us see 30x = (4 × X) - 3 + (3 × 3) + (x + 2).0438

Again identify the parts of the equation, let us see.0444

It has the equal sign, I know it is an equation, it is important to recognize.0449

Let us see, over on this side I have a term 30x, I have this term, I have this term.0454

I have a bunch of different terms.0461

Terms are always combined using addition or subtraction.0465

That is how we can usually recognize them.0468

In my terms I have some coefficients but you know it might be easier to use my distributive properties to see even more of those coefficients.0471

Let us use that distributive property, let us take the 4 multiplied by the x and 3 and do the same thing with 3.0481

4x -12 + 3x + 6, not bad.0489

Looking at that I have even more terms, I got my 30x, I get 4x, 12, 3x, 6 and lots of different terms now.0497

Into those terms I can identify what its coefficient is and I can identify the variable, the x.0508

This one says simplify it by combining like terms.0528

Remember our like terms are terms that have exactly the same variable and they have exactly the same exponent.0532

We have to be careful on which things we can actually combine here.0539

Let us see I have 12w and 10w those are like terms, they both have a w to the first.0543

Over here is 8- 2w, all three of those are like terms, we only write them next to each other.0550

I know that I need to combine them.0558

The 9 and the 3 I would also consider those like terms because both of them do not have a variable associated with them.0567

I will combine those together.0574

Let us take care of everything with the (w), 12w + 10 would be 22w.0576

That would give me a 20w when combining those, now we will take care of 3 and 9, -3 + 9 = - 6.0591

It is important to recognize that you should not go any further than here because the 20w and 6, those are not like terms.0609

We are not going to put those together.0616

On to example 4, this one we want to simplify by combining like terms.0622

I have lots of grouping symbols in here so it is hard to pick out what my like terms are.0630

I think we can do it though but we might have to borrow our distributive property first.0637

I'm going to take this negative sign and distribute it inside my parentheses here.0641

Now that would give me (2y - 3y) - 4 + (y2 + 6y2), not bad.0648

I can see there is a few things I can combine, let us see.0665

Specifically I can put these y's together since both of them are single (y).0670

I can combine these ones together over here because both of them are y2.0675

Let us put those together, 1y - 3y would be -2y, I have a 4 hanging by itself, -4.0683

y2 + 6y2 would be a 7y2, that looks good.0695

Remember, I have not distributed my 2, it is not going to help me combine anymore like terms.0703

It will definitely help me see my final results, -4y - 8 + 14y2.0708

My final answer would be -4y - 8 + 14y2.0722

I would not combine those anymore together because none of those are like terms.0726

I have a single (y), I have an 8 that does not have any variables whatsoever.0731

I have that y2, definitely not like terms.0735

Alright, thanks for watching www.educator.com.0739

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