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INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith
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Eric Smith

Eric Smith

Linear Equations in Two Variables

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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)

0 answers

Post by Rose A on February 15 at 04:48:35 PM

Thanks for explaining while your writing, it's very helpful...

Will there be another lexture on motion and mixtures? I believe it was deleted?

1 answer

Last reply by: sherman boey
Tue Aug 12, 2014 11:13 AM

Post by sherman boey on August 12, 2014

example 5 is wrong.. y =8 not -8

1 answer

Last reply by: Professor Eric Smith
Sun Jul 6, 2014 2:18 PM

Post by patrick guerin on June 24, 2014

Thank you for the lecture!

0 answers

Post by Farhat Muruwat on March 21, 2014

can someone please fix the answers to the questions on the practice questions.

0 answers

Post by Professor Eric Smith on December 10, 2013

The key you want to recognize for many of these problems is that to build a line using point-slope form we need to know a slope, and a point on the line.  In the problems we are usually given just enough information to find these things, but may have to use other formulas to do it.  Let me know if that helps out.  :^D

0 answers

Post by Araksya Fernandes on December 10, 2013

is there a little simple way to explain similar examples.

Linear Equations in Two Variables

  • To build the equation of a line you can use point-slope form ( y – y1 = m(x-x1) ). To use this you must know the slope of the line and one point on the line.
  • Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of one another.
  • You can test if two lines are perpendicular by multiplying their slopes together. If you get -1 then you know they are perpendicular.
  • You may have to use other formulas such as the slope formula, before using the point-slope form.

Linear Equations in Two Variables

A line has a slope of - 4 and passes through the point ( - 4, - 3).
Find the equation of this line in point slope form.
  • y − y1 = m(x − x1)
  • y − ( − 3) = m[x − ( − 3)]
  • y + 3 = m(x + 3)
y + 3 = − 4(x + 3)
A line has a slope of - 7 and passes through the point (11, - 1).
Find the equation of this line in point slope form.
  • y − y1 = m(x − x1)
  • y − ( − 1) = − 7(x − 11)
y + 1 = − 7(x − 11)
A line has a slope of - 8 and passes through the point ( - 14,10).
Find the equation of this line in point slope form.
  • y − y1 = m(x − x1)
  • y − 10 = − 8[x − ( − 14)]
y − 10 = − 8(x + 14)
A horizontal line passes through the point (5,3). Find the equation of this line in point slope form.
  • slope = 0 for a horizontal line
    m = 0
  • y − y1 = m(x − x1)
y − 3 = 0(x − 5)
A horizontal line passes through the point ( - 12, - 13). Find the equation of this line in point slope form.
  • slope = 0 for a horizontal line
    m = 0
  • y − y1 = m(x − x1)
  • y − ( − 13) = 0[x − ( − 12)]
y + 13 = 0(x + 12)
The equation of a line is
y − 5 = [1/5](x + 3)
Find the equation of this line in slope intercept form.
  • y − 5 = [1/5](x + 3)
  • 5(y − 5) = [ [1/5](x + 3) ]5
  • 5y − 25 = 1(x + 3)
  • 5y − 25 = x + 3
  • 5y − 25 + 25 = x + 3 + 25
  • 5y = x + 28
  • y = [(x + 28)/5]
y = [1/5]x + [28/5]
The equation of a line is
y + 10 = [7/30](x − 2)
Find the equation of this line in slope intercept form.
  • y + 10 = [7/30](x − 2)
  • 30(y + 10) = [ [7/30](x − 2) ]30
  • 30y + 300 = 7(x − 2)
  • 30y + 300 = 7x − 14
  • 30y = 7x − 314
  • y = [(7x − 314)/30]
  • y = [7/30]x − [314/30]
y = [7/30]x − [157/15]
The equation of a line is y + 12 = [5/6](x + 9)
Find the equation of this line in slope intercept form.
  • y + 12 = [5/6](x + 9)
  • 6(y + 12) = [ [5/6](x + 9)]6
  • 6y + 72 = 5(x + 9)
  • 6y + 72 = 5x + 45
  • 6y = 5x − 27
y = [(5x − 27)/6]
A line passes through the points (7,9) and (2,5). Find the equation of this line in point slope form.
  • m = [(y2 − y1)/(x2 − x1)]
  • m = [(5 − 9)/(2 − 7)]
  • m = [( − 4)/( − 5)]
  • m = [4/5]
  • y − y1 = m(x − x1)
y − 9 = [4/5](x − 7)
A line passes through the points ( - 1,3) and ( - 7, - 11). Find the equation of this line in point slope form.
  • m = [(y2 − y1)/(x2 − x1)]
  • m = [( − 11 − 3)/( − 7 − ( − 1))]
  • m = [( − 14)/( − 6)]
  • m = [14/6] = [7/3]
  • y − y1 = m(x − x1)
  • y − 3 = [7/3][x − ( − 1)]
y − 3 = [7/3](x + 1)
A line is parallel to the line whose equation is
y = − [1/4]x − 3
This line also passes through the point ( - 5,7)
Find the equation of this line in slop intercept form.
  • y = mx + b
    parallel lines have the same slope
    m = − [1/4]b = ?
  • 7 = − [1/4]( − 5) + b
  • 7 = [5/4] + b
  • 7 = 1[1/4] + b
  • 5[3/4] = b
y = − [1/4]x + 5[3/4]
A line is parallel to the line whose equation is
y = − [3/7]x + 6
This line also passes through the point ( - 4,2)
Find the equation of this line in slop intercept form.
  • y = mx + b
    parallel lines have the same slope
    m = − [3/7]b = ?
  • 2 = − [3/7]( − 4) + b
  • 2 = [12/7] + b
  • 2 = 1[5/7] + b
  • [2/7] = b
y = - [3/7]x + [2/7]
Are the lines determined by the equations
3x − 6y = 18
4x − 2y = 12
parallel, perpendicular, or neither?
  • 3x − 6y = 18
  • − 6y = 18 − 3x
  • y = [(18 − 3x)/( − 6)]
  • y = [18/( − 6)] − [3x/( − 6)]
  • y = − 3 + [x/2]m = [1/2]
  • 4x − 2y = 12
  • − 2y = − 4x + 12
  • y = [( − 4x + 12)/( − 2)]
  • y = 2x − 6m = 2
  • ([1/2]) ≠ − 1
  • ([1/2]) ≠ (2)
neither
Are the lines determined by the equations
5x − 2y = 10
2x + 12y = 24
parallel, perpendicular, or neither?
  • 5x − 2y = 10
  • − 2y = − 5x + 10
  • y = [( − 5x)/( − 2)] + [10/( − 2)]
  • y = [5/2]x − 5m = [5/2]
  • 2x + 12y = 24
  • 12y = − 2x + 24
  • y = [( − 2x)/12] + [24/12]
  • y = − [x/6] + 2m = − [1/6]
  • (−[1/6])([5/2]) ≠ − 1
  • (−[1/6]) ≠ ([5/2])
neither
A line passes through the point ( - 4, - 3) and is perpendicular to the line whose equation is y = 1/5 x − 6. Find the equation of this line in slope intercept form.
  • y = mx + b
    m = [1/5]
    negative reciprocal = - 5
  • − 3 = − 5( − 4) + b
  • − 3 = 20 + b
  • − 23 = b
y = − 5x − 23
A line passes through the point (5,4) and is perpendicular to the line whose equation is y = − 7/10 x + 3. Find the equation of this line in slope intercept form.
  • y = mx + b
    m = − [7/10]
    negative reciprocal = [10/7]
  • 4 = [10/7](5) + b
  • 4 = [50/7] + b
  • 4 = 7[1/7] + b
  • - 3[1/7] = b
y = [10/7]x − 3[1/7]
A line is perpendicular to the line whose equation is 2x − 4y = 8. This line also passes through the y - intercept of the graph of x − 5y = 20. Find the equation of this line in slope intercept form.
  • 2x − 4y = 8
  • − 4y = − 2x + 8
  • y = [( − 2x + 8)/( − 4)]
  • y = [1/2]x − 2
  • m = [1/2]
    slope of perpendicular line = - 2
  • x − 5y = 20
  • − 5y = − x + 20
  • y = [( − x + 20)/( − 5)]
  • y = [1/5]x − 4b = − 4
y = − 2x + 4
A line is perpendicular to the line whose equation is 3x − 7y = 42. This line also passes through the y - intercept of the graph of 6x + 3y = 30. Find the equation of this line in slope intercept form.
  • 3x − 7y = 42
  • − 7y = − 3x + 42
  • y = [( − 3x + 42)/( − 7)]
  • y = [3/7]x − 6
    negative reciprocal = − [7/3]
  • 6x + 3y = 30
  • 3y = − 6x + 30
  • y = [( − 6x + 30)/3]
  • y = − 2x + 10
    b = 10
y = − [7/3]x + 10
A line is parallel to the line whose equation is
y = − [2/5]x − 6
This line also passes through the point ( - 10,12)
Find the equation of this line in slop intercept form.
  • y = mx + b
    parallel lines have the same slope
    m = − [2/5]b = ?
  • 12 = − [2/5]( − 10) + b
  • 12 = [20/5] + b
  • 12 = 4 + b
  • 16 = b
y = − [2/5]x + 16
A line passes through the point ( - 6, - 4) and is perpendicular to the line whose equation is y = − 2/3 x + 8. Find the equation of this line in slope intercept form.
  • m = − [2/3]
    negative reciprocal = [3/2]
  • − 4 = [3/2]( − 6) + b
  • − 4 = [( − 18)/2] + b
  • − 4 = − 9 + b
  • 5 = b
y = [3/2]x + 5

*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

Linear Equations in Two Variables

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

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

Transcription: Linear Equations in Two Variables

Welcome back to www.educator.com.0000

In this lesson we are going to continue on with our linear equations and look at how we can build these linear equations for ourselves.0002

Specifically, some of things that we will look at is we will look at the point slope form on a line.0014

It will help us when building those equations.0020

For some of our examples, we will also have to know a little bit more about parallel and perpendicular lines.0023

Watch out for my explanation on what these are and how you can tell if two lines are parallel or perpendicular.0029

Earlier, we looked at two forms of a line.0038

We looked at standard form and we looked at slope intercept form.0041

Remember both of these forms are important because they were good for graphing.0045

When we want to build our own line, sometimes we can do this as long as we have enough information.0057

To form that we use for building our line is known as point slope form.0064

The reason why it is called that is because those are the two bits of information that we need.0068

We need to know at least one point on a line and we need to know what the slope of that line is.0072

Once we have both those bits of information and we can drop it into point slope form.0077

This form here requires a little bit of explanation.0086

You will see that there is an x and y that has some subscripts on it and that is where the point goes in terms of its x and y.0090

The slope is the n here so we can just put that in and then we also have a couple of other x’s and y’s which do not have subscripts.0098

With the ones that do not have any subscripts whatsoever, you will not be putting any values in for those.0108

Those simply stay in our equation.0114

One last thing to note is even though we will use point slope to actually build the equation on a line.0116

Usually you take point slope and end up converting it into slope intercept form because you are looking to do some other things without a line,0122

other than just than build it or maybe you want to eventually graph it.0129

It is good to go ahead and move it in to one of those other forms.0132

To build lines using point slope form, we want to substitute in our point and we will go ahead and substitute in our slope.0143

I already have point slope form, I will put it over here and I have a point and I have a slope.0151

Let us go ahead and just substitute these in and see how it works.0157

I will have y - -3 then the m represent the slope, I will put in ¾, x – xy.0161

Notice how the x and y which did not have any subscripts are still in there.0176

Once we have everything substitute in here, we want to go ahead and start rewriting this0181

either into standard form or slope intercept form so we can do some other stuff with it.0186

I’m going to go ahead and put this into slope intercept form by just getting y all by itself and maybe cleaning up a few other terms.0190

I'm subtracting a negative on the left side and that will be the same as adding 3.0198

I will go ahead and distribute my ¾ here.0203

I have ¾ x and -3, looking much better.0206

I will go ahead and subtract 3 from both sides.0216

Y = 3/4x – 3 – 3 is 6.0219

Our point slope gets our foot in the door so we can actually build the equation and then put it into a different form, maybe a slope intercept down here.0227

We can go ahead and graph it and do some other things with it.0241

Sometimes we will be given information about another line in order to build the one we want.0247

And information about that other line, we may know that it is parallel or perpendicular to the one we want.0253

What exactly does that mean?0258

We can say that two lines are parallel if they had exactly the same slope.0261

In my little picture here, you can see what that does.0266

You will end up with two lines and they usually have a little bit of a gap or space in between them, but they have exactly the same slope.0269

They are going in the same direction.0275

You might also have lines that are perpendicular.0279

With this one, they end up meeting at a right angle.0282

We do not talk much about angles in this course, another way that you can say two lines or perpendicular is0286

if their slopes are negative reciprocals of one another.0291

Let me show you what that means.0295

I suppose this blue line had a slope of 4/5, if the red one was perpendicular then I wanted to be the negative reciprocal of the blue line.0296

I made it negative and I flipped it over.0312

With perpendicular lines, they will always be different in sign, one will be positive and one will be negative and they will be reciprocal when flipped over.0314

Another way that you can test if two lines are perpendicular or not is actually you take both of their slopes and just multiply them together.0325

Two things could happen if you multiply them together and their slope is negative or their combined value is -10333

then you will know for sure that they are perpendicular.0342

If you multiply them together and you get anything else other than -1 then you will know that they are not perpendicular.0346

It is a nice and easy task you can use to know how the two lines are related.0352

We will definitely know this as we get into more of those examples.0358

Let us start off with example 1, in this one we want to write the equation on a line using the given point and the slope.0363

Since the two bits of information that I need for point slope, I will just go ahead and nearly drop it into the formula.0372

y -y1 equals slope x - x1.0378

Let us put in the y value first, y - 3 equals my slope is -2/3x - -6.0386

You will know the formula has a minus sign in there since the x value is a negative, go ahead and put that negative sign in there as well.0399

Now all we got to do is clean it up a little bit and maybe turned it into a different form.0408

I will see what we can do.0412

y – 3 = -2/3, when I subtract the negative that is the same as addition.0413

x + 6 and I think I will go ahead and distribute this -2/3 in there.0422

-2/3x – 12 ÷ 3 and a -12 ÷ 3 = -4, it looks pretty good.0430

Let us add 3 to both sides and now we have that same line written into slope intercept form.0449

Again we use point slope form to go ahead and create the equation of the line and then probably put it into some other form.0462

This one is a little different, in this one we simply want to determine whether the two lines are parallel0472

or maybe the perpendicular or maybe they do not fall into either of those two cases.0477

With these ones, notice how many of them are written not in one of the forms that we have covered.0482

In other words they are not in slope intercept form but the second one is in standard form but we have to be able to figure out what their slope is.0489

A way to determine what the slope of the line is to go ahead and rewrite it into our slope intercept form and we can just read it out of the equation.0501

Let us do that first before we actually compare what these slopes are.0509

Starting with this first one here, I can move the 2 to the other side by subtracting 2 so y =5x – 2.0512

Let us see what the second one, let us go ahead and move the x to the other side, -x – 15 and then we will divide it by 5.0521

Here is our first line, here is our second one.0539

Now they are both in slope intercept form we can say that the first one has a slope of 50545

since it is right next to x and the other one has a slope of -1/5.0550

Looking at the two slopes, they are definitely not the same, they are not parallel.0555

It looks like one is positive and one is negative and they are reciprocals.0562

I think these two are perpendicular.0567

If you want to test out feel free to just take the two slopes and multiply them together and see that when you do this you get a -1.0570

You will know that these two lines are perpendicular.0579

Let us try this process with the next pair of lines.0590

I will begin by just putting this into slope intercept form.0595

This one is almost in slope intercept form.0599

I just have to reverse the y and putting it first.0602

This one I will go ahead and move the 3x to the other side, so now I have my two lines right here.0606

y = 2x + 1 and y = -3x + 4, in this form, I can read off what both of their slopes are.0616

Looking at their slopes I can see that they are not the same, they are definitely not parallel.0627

One is positive and one is negative, but they are not reciprocal so they are not perpendicular either.0633

They are not parallel or not perpendicular, I will put this in the neither categories.0638

They are just two lines hanging around on the graph.0643

Let us try that same process with another pair of lines.0650

Let us see if we can find a couple of that which actually are parallel.0652

We will begin by putting these into a better form so we can read off that slope.0656

I'm moving the 4x to the other side, this one the y is still not completely all by itself.0662

Let us go ahead and divide everything through by 2, now we have one of our lines.0668

The second one let us go ahead and rearrange things.0678

The y is on the left and let us get it completely isolated by multiplying everything through by -1.0683

Now we have both of our lines.0693

Now that they are in this form, the slope of the first one is -2 and the slope of the second one is -2.0702

I can see that they are exactly the same and we will call these pair of lines parallel.0709

Onto one more pair of lines, let us see what they are, parallel, perpendicular, or neither, we will find out.0719

I need to get my y isolated I will start off by moving the 4x to the other side.0726

I will divide both sides by 3, y =4/3x +2.0734

The second one, let us go ahead and move the 2 over.0745

I have 3x + 2 and I will divide everything by the 4, ¾ x + ½.0749

Here is equation 1 and here is equation 2.0760

If you look at them in this form, we can pick out what each of their slopes are, I have 4/3 and 3/4.0767

These ones are pretty close, they are definitely reciprocals of one another since you would take 4/3 and flip it over and get ¾.0775

When those have been both positive so we can not say that these are perpendicular.0783

Even though they are reciprocals they are not negative reciprocals of one another.0788

They are definitely so they are not parallel.0792

Even though they are close, they do end up in the neither category.0795

They are not parallel or not perpendicular, they are simply neither.0799

They are just two lines.0802

With this one we want to write the equation of a line that happens to be parallel to the given line and also goes through the given point.0809

This one is a little different, notice how we need to know what the slope of the line is0818

and we need to know what the point is but they have not quite given us what the slope is.0822

We are going to have to extract that out of this other line that they have given us by knowing that it is parallel to the one we want to build.0827

Let us hunt down what the slope of this other line is.0834

We will do that by dividing everything through by 4.0837

It looks good so y = ¼ x + 5.0845

Let us see, I know that the slope is ¼.0850

Now that I have a point, I have a slope, I can use point slope form and build our line.0855

Y - -3 = ¼ x – xy and now let us clean it up and see what line we have.0861

y + 3 = ¼ x, ¼ of 2 would be -1/2.0872

Now we will subtract 3 from both sides.0890

y = ¼ x – ½ - 3, get a common denominator over there.0893

Our line is y = ¼ x – 7/2.0908

The way we built this line, I know for sure that it goes through the given point 2, -3.0917

I already explored what the slope of the other line was so I know it has a slope of 1/4.0923

Let us try this one, write the equation of a line which is perpendicular to the given line and it goes through the given point.0932

Similar to the other one, but it have not given us the slope directly.0939

It just told us it is perpendicular to this other line over here.0943

Let us figure out what slope is, y equals, I will move the 8 to the other side, -8x + 3.0948

The slope of this line is -8 which is good but that is not exactly the slope we want to use.0958

Our line is perpendicular to this line, so we need to use the slope -1, 1/8.0966

That way it is the negative reciprocal of the other.0979

Now we will drop it into our formula.0984

Let us go ahead and put in our y value.0994

We have our slope and our x value.0997

We will clean it up and see what line we have.1002

y - 8 = 1/8 - 4/8, my x in there.1005

y – 8, 1/8x – ½ and we will go ahead and 8 to both sides, find a common denominator and combine the last of our like terms.1017

1/8x + 15/2.1044

We can see that its slope is the negative reciprocal of the other line.1053

I know it is perpendicular and the way we build that it, for sure it goes through the point 4/8.1057

This last one is a little bit more of the word problem but you can see how we can still use these techniques in order to build the equation of the line.1066

We want to build a line that represents the following problem.1074

It cost a $20 flat fee to rent a drill + $2 every day starting with the first day.1077

Let x represent the number of days that we rent this drill and y represent the charge to the users.1085

How much will we end up willing to get from them?1091

When we are all done, let us see if we can write this line in a slope intercept form.1095

Using the techniques that we know about so far, we need to figure out what our slope is and what point this will go through.1100

Let us see, one thing that we can think of is the slope being what changes or your variable costs.1108

The $20 that you have to pay up front is not a variable.1120

It is going to be there no matter what, what does change is a $2 every single day.1123

The variable cost, we can think of that as our slope.1134

Let us see what we can do with that.1144

What happens with that flat cost?1146

We will be charged of that no matter what, that is like our y intercept.1151

If this one, we can just drop in our variable cost and we can drop in our flat fee.1158

What we are left here is the equation of the line but it represents how much we end up charging the person1177

and we can see this by substituting some values.1183

Suppose they rent this drill for one day, it cost $2 for that day + $20 flat fee, it cost them $22.1186

If they rent it for two days, we can put that in for x, $4 for the variable + $20 flat fee, $24.1196

You can go on and on figuring out how much it cost for each of the different days.1206

But in the end, this formula right here would represent how much we need to charge them.1211

Just simply plug in the number of days for x and y would give you the cost.1215

You can see that building a line is not so bad and using point slope form is handy in that entire process.1221

If you do use point slope form, you need to know the slope of the line and you need to know a point on that line.1228

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