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Algebra I / Elem. Algebra

The Basics

1.1 Intro to Algebra
1.2 Real Number System and Calculators

Linear Equations and Inequalities

Functions and Graphs I

3.1^s Introduction to Graphing
3.2 Solving Equations Graphically
3.3 Functions I - Introduction
3.4 Functions II - Domain, Range, & Operations

Lines and thier Graphs

4.1 Intercept Points
4.2^s Slopes
4.3 Equations of Lines
4.4 Graphing Linear Inequalities

Algebra II / E&I Algebra

Linear Systems

5.1 2x2 Systems of Equations I - Graphically
5.2 2x2 Systems of Equations II - Substitution Method
5.3 2x2 Systems of Equations III - Elimination Method
5.4^s 3x3 Systems of Equations
5.5 Applications
5.6 Systems of Linear Inequalities

Exponents & Polynomials

Intermediate Algebra starts here!

Factoring

Rational Expressions

Rational Equations and Applications

Algebra III / Inter. Algebra

Radical Expressions

10.1 Roots & Radicals
10.2 Fractional Exponents
10.3 Operations I - Simplify & Multiply
10.4 Operations II - Add, Subtract, & Divide
10.5 Operations III - Distribute, Foil, & Factor
10.6^s Complex Numbers

Nonlinear Equations and Applications

11.1 Radical Equations
11.2 Quadratic Equations I - Root Method
11.3^s Quadratic Equations II - Complete the Square
11.4^s Quadratic Equations III - Quadratic Formula
11.5^s Applications - Geometry & Pythagorean Theorem

Functions and Graphs II

12.1 Parabolas I - Algebraic Approach
12.2 Pareabolas II - Graphical Approach
12.3^s Functions III - Composite & Inverse

Exponential and Logarithmic Functions

13.1^s Exponential Functions & Graphs
13.2 Logarithmic Functions & Graphs
13.3 Exponential & Logarithmic Equations

Liberal Arts Maths

Under Construction

College Algebra

Equations and Inequalitites

Functions and Graphs

Polynomial and Rational Functions

3.1 Polynomial Functions I - Introduction, Zeros
3.2 Polynomial Functions II - Division, Remainder Thm, Factor Thm
3.3 Polynomial Functions III - Complex Zeros, Fundamental Thm of Algebra, Descartes Rule
3.4 Rational Functions & Graphs
3.5 Partial Fractions

Exponential and Logarithmic Functions

Systems and Matrices

Conic Sections

Sequences and Series

7.1^s Introduction to Sequences and Series
7.2 Arithmetic Sequences and Series
7.3 Geometric Sequences and Series

Trigonometry

The Six Trigonometric Functions

Right Triangle Trigonometry

Circular Functions

Graphs of Trigonometric Functions

4.1^s Basic Graphs
4.2^s Amplitude & Period
4.3^s Vertical Translations & Phase Shift
4.4 General Equation & Graph to Equation
4.5^s Combination Graphs & Harmonic Motion
4.6^s Inverse Trigonometric Functions

Trigonometric Identities

5.1 Proving Trig Identities
5.2 Sum/Difference Identities
5.3 Double-Angle Identities
5.4^s Half-Angle Identities
5.5 IDs Involving Inverse Trig Functions

Trigonometric Equations

Oblique Triangles and the Laws

Vectors

Complex, Parametric, and Polar Forms

9.1 Trigonometric Form of Complex Numbers
9.2 Products and Quotients in Trig Form
9.3 Powers and Roots in Trig Form
9.4 Parametric Equations and Graphs
9.5^s Polar Equations and Graphs

Calculus I

Limits and Continuity

Derivatives

2.1 Definition of Derivative
2.2 Derivatives of Basic Functions
2.3 Motion and the Derivative
2.4 Product Rule, Quotient Rules, & Deriv. of Trig Functions
2.5 Chain Rule
2.6 Implicit Differentiation
2.7 Derivatives of Exponential and Logarithmic Functions
2.8 Logarithmic Differentiation
2.9^s Derivatives of Inverse Functions
2.10^s Derivatives of Inverse Trig Functions
2.11 Related Rate Applications

Analysis of Curves

3.1 Extreme Values
3.2 Mean Value Theorem for Derivatives
3.3 First Derivative Test & Increasing-Decreasing Functions
3.4 Second Derivative Test & Concavity
3.5^s Curve Sketching
3.6 Fun with Reform Calculus
3.7 Optimization Applications

Antiderivatives

Calculus II

Transcendental Functions

5.0^s Review of Transcendental Functions and thier Derivatives
5.1 Integrals Resulting in Logarithmic Functions
5.2 Integrals Resulting in Inverse Trig Functions
5.3 Hyperbolic Trigonometric Functions
5.4 Numeric and Electronic Integration

Geometry

Physics

Integration Techniques

Calculus of Infinity

Parametric, Polar, and Conic Curves

Calculus III

Under Construction

S = contains supplemental resources

Course: Algebra II / Elementary Algebra
Topic: Linear Systems
Subtopic: 3x3 Systems of Equations

Overview

Now that we have the basic process of algebraically solving 2x2 systems down, we are ready to extend our knowledge to include larger linear systems (3x3, 4x4, etc.). Solving a 3x3 system algebraically still uses elimination/substitution methods, but the idea is to eliminate one variable/equation to transform the 3x3 system down to a 2x2 system, solve it, then back substitute to get all there variable answers.

The larger the systems get the more useful using technology to solve them becomes. I strongly recommend that you solve 4x4 and larger systems electronically. Because solving a 3x3 system algebraically is time-consuming, you may want to solve those electronically too (especially on a test), but be sure you know HOW to solve a 3x3 system algebraically.

Solving a linear systems electronically (on a handheld grapher or using a system solver program) requires using a command called RREF. How you do this will depend what technology you have available to you. See the supplemental resources below for suggestions.

Objectives

By the end of this topic you should know and be prepared to be tested on:

5.4.1 Solve 3x3 linear systems algebraically (usually by the elimination method, but sometimes the substitution method will work).
5.4.2 Solve 4x4 and larger linear systems electronically using a system solver such as RREF.
5.4.3 When solving a system electronically using RREF, recognize consistent, inconsistent, and dependent systems.

Terminology

Define: back substitution, matrix, row of a matrix, column of a matrix, size of a matrix (a.k.a. dimension, order), 3x3 system (and 4x4 system, etc.), augmented matrix, row reduced echelon form (RREF)

Text Notes
Text: Intro & Inter Algebra for CS 3ed by Blitzer, sect. 4.5

For 3x3 and larger systems, you will only be asked to actually solve those that have a single point answer (the consistent type). For the inconsistent and dependent types, the only thing you need to know is how to recognize them electronically based on the last line of the result from RREF.
pg 309-310 SKIP the inconsistent and dependent types of systems. It’s neat to look at the geometric aspect of 3x3 systems (i.e. intersecting planes), but if you don’t completely understand them, no problem, it’s covered in later courses.
pg 310-311 SKIP the modeling applications such as example 4. This includes any problems of the form, "Find the quadratic function y=ax²+bx+c whose graph passes through the given points."

Supplementary Resources

Directions for solving a larger linear system electronically:

If you have a TI-84/86/89/92 calculator (or other handheld grapher) use the RREF command. See Calculator Guide: Echelon Form.
If you have a TI-86 calculator, you may use the SIMULT command to solve consistent systems still need the RREF command to identify inconsistent or dependent systems. See Calculator Guide: Linear Systems.
If you don't have a handheld graphing calculator then use an online row reducer.

Prof. Keely's Lesson Notes for Mathematics Online