ELEMENTARY & INTERMEDIATE ALGEBRA
These brief notes are intended to guide you through the textbook and/or other course readings/materials. As you read the textbook pay particular attention to the "topics of importance" and be sure you know how to accomplish each. The "supplemental sites" may provide additional resources on the internet that supplement the topics. Note: This material is extensively elaborated upon in my optional e-book GOLDen Mathematics: Elementary Algebra. This section of material only is downloadable for a nominal fee at www.lulu.com/content/431047. See "Tell me more about Keely's GM book".
2x2 Systems of Linear Equations
GOLDen Mathematics - Elementary Algebra: Section 5.1
Supplemental Sites: MathOL Links - Alg 5.1
Topics of Importance
Definition of linear system; what it means to be a solution to
a system
Check if given values are actually a solution to a given system
Solve a 2x2 linear system by graphing (manually and on calculator using INTERSECTION)
Solve a 2x2 linear system algebraically by Substitution or Elimination method
Special cases: no solution vs an infinite number of solutions.
Recognize
special cases algebraically and graphically.
Comments and Cautions
When we studied lines, the
equation was a single equation in two variables like y=-2x+5. Today's lesson
involves solving a "system" of linear equations meaning that there are two or
more equations you are solving simultaneously (at the same time). A 2x2 (read
"two by two") system is two equations in two variables, a 3x3 system is three
equations in three variables, etc. The solution to a 2x2 system is point
in (x,y) form.
 | We will solve 2x2 systems graphically (by finding the intersection point of the graphs of the linear equations, see my Calculator Guide: Intersection Points) and by two different algebraic methods (the substitution method and the elimination method). These two algebraic methods are the most important processes to learn in this chapter. (The elimination method is sometimes also called the addition method or the addition-elimination method.) |
| We will solve 3x3 systems algebraically (usually by the elimination method, but sometimes the substitution method will work). |
| We will solve any size system but using a calculator's system solver feature but this will be particularly useful for 4x4 and larger systems. |
Not all systems actually have solutions. A 2x2 system that has a single solution point is called a consistent system. A 2x2 system that has no solution (because the graph of its linear equations are parallel lines and thus never intersect) is called an inconsistent solution. A 2x2 dependent system is one whose equations are actually equivalent, has a graph where one line lies atop the other, the lines intersect each other at every point along that line, and thus the system has an infinite number of solutions. For 3x3 systems (and larger) we will only look at the consistent kind.
We will look at applications of systems of linear equations later, but let's get the algebraic processes down first. This material is very useful in the real world where there are often several constraints (conditions on the variables) that need to be met simultaneously. In fact when designing a plane at Boeing the engineers typically work with 140.000 equations in 140.000 variables! (Thank goodness for technology, aye?)
Text Notes (These notes refer to Introductory & Intermediate Algebra for CS 3rd ed by Blitzer sections 4.1-4.3.)
| ch 4.1 covers checking if a point is a solution to a system, solving linear systems by graphing, graphically, determining the number of solutions to a system (see chart pg 264), the no solutions and infinite number of solutions cases, and applications of linear functions. Lots to discuss on the boards! |
| ch 4.2 covers solving systems algebraically by the substitution method and ch 4.3 covers solving systems algebraically by the elimination method. Be sure that you can also recognize the no solution/infinite number of solutions cases using both methods. |
| ch 4.2 You can SKIP "supply and demand models" such as example 5. |
| ch 4.2-4.3 As you work through the examples and exercises, try checking your answers by graphing. That will provide you practice with the calculator. |
| You can SKIP "curve fitting" and "linear regression" throughout chapter 4 (and the entire course). |
3x3 Systems of Linear Equations
GOLDen Mathematics - Elementary Algebra: Section 5.2
Supplemental Sites: MathOL Links - Alg 5.2
Topics of Importance
Solve a 3x3 linear system algebraically
Solve linear systems on calculator using SIMULT/RREF
Comments and Cautions
Now that we have the basic
process of solving systems down we are ready to extend our knowledge to include
larger linear systems (3x3, 4x4, etc.). The larger the systems get the more
useful using technology to solve them becomes. Although you should know how to
solve a 3x3 system algebraically, I strongly recommend that you solve 4x4
and larger systems electronically. This is pretty cool ... today we will learn to
use a calculator to solve linear systems ... a whole new feature of your
calculator! If you have a TI-86 then use the SIMULT feature to solve systems
(see my Calculator Guide: Linear Systems). If you have a TI-84/89/92 use the RREF command
instead (see my Calculator Guide: Echelon Form).
If you don't have a graphing calculator then use an online row reducer (one is
listed at my
College Algebra Links: Systems and Matrices).
Text Notes (These notes refer to Introductory & Intermediate Algebra for CS 3rd ed by Blitzer section 4.5.)
| ch 4.5 I will only test you on the systems that has a single point as an answer (the “consistent” type). |
| ch 4.5 pg 309-310 SKIP the “dependent” and “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 again in math 105, math 111, or calculus. |
| ch 4.5 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." |
Applications of Linear Systems
GOLDen Mathematics - Elementary Algebra: Section 5.3
Supplemental Sites: MathOL Links - Alg 5.3
Topics of Importance
2x2 mixture word problems
3x3 and higher linear system word problems
Comments and Cautions
Ready for more word problems and applications? Ready or not here they come!
Today we will study translating
word problems into linear systems and solving them using the systems solver on
your calculator.
Text Notes (These notes refer to Introductory & Intermediate Algebra for CS 3rd ed by Blitzer section 4.4.)
| ch 4.4 The "mixture problems" are particularly important. You can SKIP the "motion problems" (these are covered in math 095). |
| ch 4.4 Concentrate on translating the word problem into system of equations. Then just use a calculator to solve the resulting system :) |
Systems of Linear Inequalities
GOLDen Mathematics - Elementary Algebra: Section 5.4
Supplemental Sites: MathOL Links - Alg 5.4
Topics of Importance
Definition of system of linear inequalities
What it means to be a solution to a system of linear inequalities
Graphing the solution to a system of linear inequalities including:
finding
intersection pts, finding the final shaded solution set
Finding the system of linear inequalities given the solution set
Comments and Cautions
Continuing in chapter 8.9 we
synthesize skills from
several previous lessons: Solving
Compound Linear Inequalities (particularly the "and" case), Equations
of Lines (particularly graphing by the slope-intercept method), Graphing
Linear Inequalities, and Solving 2x2
Systems of Linear Equations (graphically and algebraically). Be sure you are
familiar with all this information (especially Graphing
Linear Inequalities) before working through this lesson!
Text Notes (These notes refer to Introductory & Intermediate Algebra for CS 3rd ed by Blitzer section N/A.)
| THIS TOPIC IS OMITTED FROM THIS CLASS. IT IS COVERED IN A FINITE MATH CLASS. |
Originally written: 2006-006-15
Last revision:
2009-01-16 11:29 AM