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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Solving Formulas

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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 (8)

1 answer

Last reply by: Professor Eric Smith
Tue Apr 22, 2014 8:11 PM

Post by Maria Camila Bernal on April 16, 2014

One of the practice questions says:

2(13x-5) = 2-(6-5x)

26x-10 = -4+5x (How is that the -5x above turned to a +5x?)

1 answer

Last reply by: Mohamed Elnaklawi
Tue Mar 18, 2014 1:35 PM

Post by Mohamed Elnaklawi on March 18, 2014

in example two, why are you done with the problem if you still have a FRACTION in the end?

3 answers

Last reply by: Rose A
Tue Feb 13, 2018 10:59 AM

Post by Paul Cassidy on February 23, 2014

does anyone know if these questions go to those who monitor this web site or know how to let them know that there test questions are not
correct.  Here is another one.  Equation shown:   Q. 2( 13x − 5 ) = 2 − ( 6 − 5x ) the answer that is give is 2/3  that is not correct.  the answer is 2/7 and proofs out correctly at that.  the error is in the fact that they show -4+10 as a positive 14 instead of a 6.  I love this site for the lectures but am continuously finding errors with the practice questions.  I'm concerned for those home schooling as they need the right answers as well as the fact that they are paying for it.

Solving Formulas

  • A formula is a type of equation that usually conveys some fundamental principle. They usually have many variables that stand in for unknown quantities.
  • To solve a formula, we isolate the desired variable on one side of the equal sign. To do this we use the same process we saw with solving linear equations.
  • With formulas, the solution may still contain many variables. It can still be substituted into the original to see if it creates a true statement.

Solving Formulas

3n − 6 = 5n + 9
  • 3n − 6 − 5n = 5n + 9 − 5n
  • − 2n − 6 = 9
  • − 2n − 6 + 6 = 9 + 6
  • − 2n = 15
  • [( − 2n)/( − 2)] = [15/( − 2)]
n = − 7.5
5x + 8 = 13x − 9
  • 5x + 8 − 13x = 13x − 9 − 13x
  • − 8x + 8 = − 9
  • − 8x + 8 − 8 = − 9 − 8
  • − 8x = − 17
x = [17/8]
7( 3x + 5 ) = 2( 12x − 8 ) + 14
  • 21x + 35 = 24x − 16 + 14
  • 21x + 35 = 24x − 2
  • 21x + 35 − 24x = 24x − 2 − 24x
  • − 3x + 35 = − 2
  • − 3x + 35 − 35 = − 2 − 35
  • − 3x = − 37
x = [37/3]
2( 13x − 5 ) = 2 − ( 6 − 5x )
  • 26x − 10 = 2 − 6 + 5x
  • 26x − 10 = − 4 + 5x
  • 26x − 10 − 5x = − 4 + 5x − 5x
  • 21x − 10 = − 4
  • 21x − 10 + 10 = − 4 + 10
  • 21x = 14
  • x = [14/21] = [2/3]
x = [2/3]
2( 8y + 11 ) / 4 = 6( y − 5 )
  • 16y + 22 / 4 = 6y − 30
  • 4( [(16y + 22)/4] ) = ( 6y − 30 )4
  • 16y + 22 = 24y − 120
  • 16y + 22 − 24y = 24y − 120 − 24y
  • − 8y + 22 = − 120
  • − 8y + 22 − 22 = − 120 − 22
  • − 8y = − 142
  • y = [142/8] = [71/4]
y = [71/4]
4( 6s + 2 ) = 8( 3 − 2s )
  • 24s + 8 = 24 − 16s
  • 24s + 8 + 16s = 24 − 16s + 16s
  • 40s + 8 = 24
  • 40s + 8 − 8 = 24 − 8
  • 40s = 16
  • s = [16/40] = [2/5]
s = [2/5]
8n − 14 = 5n − 20
  • 8n − 14 − 5n = 5n − 20 − 5n
  • 3n − 14 = − 20
  • 3n − 14 + 14 = − 20 + 14
  • 3n = − 6
n = − 2
3(5f − 6) = 2(12f − 4) + 13
  • 15f − 18 = 24f − 8 + 13
  • 15f − 18 = 24f + 5
  • 15f − 18 − 24f = 24f + 5 − 24f
  • − 9f − 18 = 5
  • − 9f − 18 + 18 = 5 + 18
  • − 9f = 23
  • [( − 9f)/( − 9)] = [23/( − 9)]
f = − [23/9]
[(5(4x − 8))/10] = 2(15x + 5)
  • [(20x − 40)/10] = 30x + 10
  • 10( [(20x − 40)/10] ) = (30x + 10)10
  • 20x − 40 = 300x + 100
  • 20x − 40 − 300x = 300x + 100 − 300x
  • − 280x − 40 = 100
  • − 280x − 40 + 40 = 100 + 40
  • − 280x = 140
  • x = − [140/280] = − [1/2]
x = − [1/2]
4(4k − 8) = 4(3k − 6) − 18 − 2k
  • 16k − 32 = 12k − 24 − 18 − 2k
  • 16k − 32 = 10k − 42
  • 16k − 32 − 10k = 10k − 42 − 10k
  • 6k − 32 = − 42
  • 6k − 32 + 32 = − 42 + 32
  • 6k = − 10
  • k = − [10/6]
k = − [5/3]
Solve for y:4x − 11y = 15
  • 4x − 11y − 4x = 15 − 4x
  • − 11y = 15 − 4x
  • [( − 11y)/( − 11)] = [(15 − 4x)/( − 11)]
y = − [(15 − 4x)/11]
Solve for a:11a + 12b − 6 = 14
  • 11a + 12b − 6 + 6 = 14 + 6
  • 11a + 12b = 20
  • 11a + 12b − 12b = 20 − 12b
  • 11a = 20 − 12b
  • [11a/11] = [(20 − 12b)/11]
a = [(20 − 12b)/11]
Solve for t:s − 8st = 3t − 7
  • s − 8st − 3t = 3t − 7 − 3t
  • s − 8st − 3t = − 7
  • s − 8st − 3t − s = − 7 − s
  • − 8st − 3t = − 7 − s
  • t( − 8s − 3) = − 7 − s
  • [(t( − 8s − 3))/(( − 8s − 3))] = [( − 7 − s)/(( − 8s − 3))]
t = [( − 7 − s)/(( − 8s − 3))]
Solve for v:4u + 3uv = v − 1
  • 4u + 3uv − v = v − 1 − v
  • 4u + 3uv − v = − 1
  • 4u + 3uv − v − 4u = − 1 − 4u
  • 3uv − v = − 1 − 4u
  • v(3u − 1) = − 1 − 4u
  • [(v(3u − 1))/((3u − 1))] = [( − 1 − 4u)/(3u − 1)]
v = [( − 1 − 4u)/(3u − 1)]
Solve for n:[(n + m)/(n − m)] = 4m
  • (n − m) ×[(n + m)/(n − m)] = 4m(n − m)
  • n + m = 4m(n − m)
  • n + m = 4mn − 4m2
  • n + m − 4mn = 4mn − 4m2 − 4mn
  • n + m − 4mn = − 42
  • n + m − 4mn − m = − 4m2 − m
  • n − 4mn = − 4m2 − m
  • n(1 − 4m) = − 4m2 − m
  • [(n(1 − 4m))/((1 − 4m))] = [( − 4m2 − m)/((1 − 4m))]
n = [( − 4m2 − m)/((1 − 4m))]
Solve for c:
[(c − d)/2c] = 5d
  • 2c( [(c − d)/2c] ) = 5d(2c)
  • c − d = 5d(2c)
  • c − d = 10dc
  • c − d + d = 10cd + d
c = 10cd + d
Solve for k:
[(j − k)/(j + k)] = 3j
  • (j + k)( [(j − k)/(j + k)] ) = 3j(j + k)
  • j - k = 3j(j + k)
  • j − k = 3j2 + 3k
  • j − k − j = 3j2 + 3k − j
  • − k = 3j2 + 3k − j
  • k = − (3j2 + 3k − j)
k = − 3j2 − 3k + j
Solve for r:
16r − 12s = 48
  • 16r − 12s + 12s = 48 + 12s
  • 16r = 48 + 12s
  • [16r/16] = [(48 + 12s)/16]
  • r = [(48 + 12s)/16] = [(12 + 3s)/4]
r = [(12 + 3s)/4]
Solve for u:2u − 7uv = 5v + 9
  • u(2 − 7v) = 5v + 9
  • [(u(2 − 7v))/((2 − 7v))] = [(5v + 9)/((2 − 7v))]
u = [(5v + 9)/((2 − 7v))]
Solve for x:6xy + y = 3x − 2y
  • 6xy + y − 3x = 3x − 2y − 3x
  • 6xy + y − 3x = − 2y
  • 6xy + y − 3x − y = − 2y − y
  • 6xy − 3x = − 2y − y
  • 6xy − 3x = − 3y
  • x(6y − 3) = − 3y
  • [(x(6y − 3))/((6y − 3))] = [( − 3y)/((6y − 3))]
x = [( − 3y)/((6y − 3))]

*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

Solving Formulas

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 0:06
  • Solving Formulas 0:18
    • Formulas
    • Use the Same Properties as Solving Linear Equations
    • Addition Property of Equality
    • Multiplication Property of Equality
    • Steps to Solve Formulas
  • Example 1 3:56
  • Example 2 6:09
  • Example 3 8:39

Transcription: Solving Formulas

Welcome back to www.educator.com.0000

In this lesson we will take a look at solving formulas and that is our only objective for our list of things to do.0002

But one thing I want you to know while going through this one is how similar formulas to solve the equations.0011

If you are curious what exactly is a formula and how they differ from those linear equations that we saw earlier.0020

A formula is a type of the equation and usually conveys some sort of fundamental principle.0025

Below, I have examples of all kinds of different formulas, D = RT and I = P × R × T.0031

What these represents are usually something else.0039

For example, D = RT we will look at that a little bit later on because it stands for distance = rate × time.0042

I = PRT, that is an interesting formula and it stands for interest = principal, rate, time.0058

It is not that these equations are too unusual but they actually have lots of good applications behind them.0075

One thing that is a little bit scarier with these formulas is you will notice how they have a lot more variables than what we saw earlier.0081

They have a lot of L, W and other could be multiple variables for just a single formula.0089

You know we have those multiple variables present in there and usually we are only interested in solving for a single variable in the entire thing.0097

The way we go about solving for that variable is we use exactly the same tools that we used for solving several linear equations.0105

You have the addition property for equality and you have the multiplication property for equality.0114

Both of these say exactly the same thing that they did before.0120

One, you can add the same amount to both sides of an equation, keep it nice and balanced.0124

And that you can multiply both sides of an equation by the same value and again keep it nice and balanced.0129

The only difference here is that we will be usually adding and subtracting or multiplying, dividing by a variable of some sort.0135

One thing that we usually thrown in as an assumption is that when we do multiply by something then we make sure that it is not 0.0142

If I do multiply it by a variable, multiply by both sides by an X, then I will make the assumption that X is not 0.0151

Just to make sure I do not violate the multiplication property of 0.0158

You have lots of variables and the very first thing is probably identify what variable you are solving for.0167

I usually like to underline it, highlight it in some sort of way just to keep my eye on it.0174

If you have multiple copies of this variable, try to work on getting them together so that you can eventually isolate it.0180

Some of these formulas do involve some fractions.0186

Use the common denominator techniques so you can clear them out and not have to worry about them.0189

Simplify each side of your equation as much as possible then isolate that variable that you are looking for.0196

Try and get rid of all the rest of the things that are around you by using the addition and multiplication property.0203

You can check to make sure that your solution works out by taking your solution for that variable and playing it back into the original.0212

When you do this for your formula, what you often see is that a lot of stuff will cancel out and you will be left with a very simple statement.0221

Usually we do not worry about that one too much unless we have some more numbers present in there but you can do it.0229

Let us actually look at a formula and watch the solving process.0238

You will see that it actually flows pretty easily.0242

In this first one I have P=2L + 2W.0245

This formula stands for the perimeter of a rectangle.0248

It goes through and adds up both of the lengths and both of the widths.0251

What we want to do with this one is we want to solve it for L.0257

Let us identify where L is in our formula right there.0261

What I’m going to do is I’m trying to get rid of everything else around it.0265

The 2W is not part of what I'm interested in so I will subtract that from both sides.0271

P and W are not like terms so even though it will end up on the other side of the equation, I will not be able to combine them any further.0280

Now you have 2 - 2 or P - 2W = 2L, continuing on.0291

I almost have that L completely isolated, let us get rid of that 2 by dividing.0298

Notice how we are dividing the entire left side of this equation by 2 so that we can get that L isolated.0310

What I have here is (P -2W)/2 = L.0322

One weird part about solving a formula is sometimes it is tough to tell when you are done0328

because usually with an equation when you are done you will have a number like L = 5 or L = 2.0334

Since you have many variables in these formulas that when we get to the end what we developed is another formula right here.0340

It is just in a different order or different way of looking at things.0348

This formula that we have created could be useful if we were looking for L many times in a row and we had information about the perimeter and W.0351

It is solved, it is a good the way that it is and the way we know it is solved is because L is completely isolated.0362

Let us look at another one, in this one we want to solve for R.0371

I have Q= 76R + 37, a lots of odd ball numbers but let us go ahead and underline what we would be looking for.0375

We want to know about that R.0382

I think we can get rid of a few fractions and we would have to multiply everything through by 6.0387

Let me do that first, I will multiply the left side by 6, multiply the right side by 6.0395

Q = 76R + 37, on the right side we better distribute that 6 and then we will see that it actually does take care of our fraction like it should.0403

6Q = 7R, I know I have to take care of 37 × 6.0422

I guess I better do some scratch work.0434

37 × 6 I have 42, 3 × 6 is 18 + 4 = 22, a lot of 2.0436

6Q = 7R + 222, I do not have to deal with any more fractions at this point so let us continue isolating the R and get it all by itself.0452

I will subtract 222 from both sides, awesome.0463

One final step to get R all by itself, we will divide it by 7.0482

(6Q – 222)/7 = R and I will consider this one as solved.0490

The reason why we can consider this one done is because we have isolated R completely.0503

Even though we do have a Q still floating around in there, it is solved.0508

Do not get too comfortable with that, it is not equal to just a single number, very nice.0513

This one is a little different, we want to solve the following for V.0521

The reason why this one is a little bit different is I have a V over here but I also have another V sitting over there.0525

When you have more than one copy of the variable like this, you have to work on getting them together before you can get into the isolating process.0534

Let us take care of our fractions and see if we can actually get those V’s together and work on isolating it.0543

We only isolate one of them then it is not solved, we still have a V in there.0551

To take care of our fraction I will multiply both sides of this one by W, it is my common denominator.0556

(RV + Q) / W = W × 5V, on the left side those W’s would take care of each other.0569

I will be left with RV + Q = W × 5V.0585

In this point I do not have to deal with the fractions but notice we have not got those V’s any closer together so let us keep working on that.0594

If I'm going to get them together, I at least better get them on the same side of the equation.0602

I'm going to subtract say an RV from both sides right.0609

Now comes the fun part, I still have two V’s, I’m going to make them into one.0627

The way I’m going to do this is I’m going to think of my distributive property but I’m going to think of it in the other direction.0633

If both of these have a V then I will pull it out front and I will be left with W5 – R.0641

This step is a little bit tough when you see it at first but notice how it does work is that I'm taking both of these and moving them out front into a single V.0653

I'm sure if it is valid, go ahead and take the V and put it back in using our distributive property and you will see that you be right back at this step.0664

It is valid.0673

The important part of why we are using it though is now we only have a single V and we can work further by isolating it.0675

How do we get it all by itself, these V’s be multiplied by 5W – R, that entire thing inside the parentheses.0682

We will divide both sides by that and then it should be all alone 5W – R = V.0690

I literally just took this entire thing right here and divided it on both sides.0702

To your left, now I can call this one done because V is completely isolated, it is all alone.0709

There is no other V’s running around in there.0717

I have worked hard to get them together and I know it is completely solved.0719

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