COLLEGE 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: College Algebra. This section of material only is downloadable for a nominal fee at << T.B.A. >>. See "Tell me more about Keely's GM book".
Linear Systems of Equations
GOLDen Mathematics - College Algebra: N/A, but covered in
GM - Elementary Algebra: Section 5.1-5.2
Supplemental Sites:
MathOL Links - Calg
5.1
Review Topics (from elementary algebra)
2x2 linear systems:
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
Solving a 2x2 linear system by graphing manually
Solving a 2x2 linear system graphically on a calculator
using INTERSECTION Calc Guide
- Intersection Points
Solving a 2x2 linear system algebraically by substitution method or
elimination method
Special cases: no solution vs. an infinite number of solutions
Recognizing special cases algebraically and graphically
3x3 linear systems:
Solving a 3x3 linear system algebraically
Solving linear systems on calculator using SIMULT
Calculator Guide
- Linear Systems
Applications of linear systems:
2x2 mixture word problems
3x3 and higher linear system word problems
Business and Economic applications: cost, revenue, profit,
break-even, supply, demand, equilibrium
Topics of Importance (for college algebra)
Solve 2x2 and 3x3 systems of linear equations algebraically by the
substitution or elimination methods
Solve 2x2 systems graphically by the intersection method
Solve 3x3 and higher systems on a calculator using a simultaneous systems solver
Solve applications of linear systems
Comments and Cautions
Solving linear systems algebraically by substitution or elimination methods and the graphically
by the intersection method should all be review from an elementary algebra
course.
Text Notes (These notes refer to College Algebra 5th ed by Blitzer sections 5.1-5.2.)
 | ch 5.1 begins with a review of 2x2 linear systems from an elementary algebra course. You should already know how to do all of the things listed under "Review Topics" the most important of which are reviewed in the text examples 2-4. |
| ch 5.1 examples 5-6 are the two "special cases" (no solution and infinitely many solutions). Be sure you can recognize these cases algebraically and graphically. |
| ch 5.1 pg 490-492 "Break-Even Analysis" is OPTIONAL. |
| ch 5.2 covers 3x3 linear systems and their applications which should all be review from an elementary algebra course. Some of the items listed under "Review Topics" above are elaborated upon here. Be sure you can solve these systems both algebraically and electronically using SIMULT or RREF on a calculator. If you don't have a calculator see the supplemental sites listed above for an online row reducer java applet. |
Nonlinear Systems of Equations
GOLDen Mathematics - College Algebra: Section 5.1
Supplemental Sites:
MathOL Links - Calg 5.1
Topics of Importance
Solving systems of nonlinear equations algebraically
Graphical significance of the Real solutions to nonlinear systems
Comments and Cautions
Our focus is to extend the algebraic and graphic methods of solving
linear systems to nonlinear systems. Recall that nonlinear
systems are systems of equations that include variables to powers higher than
one, negative powers on variables, non-integer powers on variables, variables in
the denominator, or variables in a radicand. As you work through this material
be sure to try solving both algebraically and graphically. When solving
algebraically, concentrate on when the elimination method would work or when the
substitution method is the way to go.
Text Notes (These notes refer to College Algebra 5th ed by Blitzer section 5.3-5.6.)
| ch 5.3 see Lecture Notes - Partial Fractions |
| ch 5.4 Concentrate on solving these non-linear systems algebraically. You should know that the real solutions are the points of intersection between the graphs of the original equations and it would be good (but not necessary) to be able to verify those graphically. |
| ch 5.5-5.6 SKIP |
Gaussian Elimination
GOLDen Mathematics - College Algebra: Section 5.2
Supplemental Sites:
MathOL Links - Calg 5.2
Topics of Importance
Introduction to matrices and matrix terminology
Solve linear systems by Gaussian elimination
Solve linear systems by Gauss-Jordan method
Determine if system is consistent, inconsistent, or dependent (on calc using
RREF)
Dependent system solutions
Comments and Cautions
In todays lesson we express linear systems in augmented matrices and solve
them using matrix methods. We use matrix row operations to transform the system
to REF form (a diagonal of ones with a triangle of zero below the diagonal) and
use back-substitution to solve the system. This is the Gaussian Elimination
method. Or we could go further using matrix row ops to get a second triangle of
zeros above the diagonal of ones (RREF) and reading the answers straight from
the resulting matrix. This is Gauss-Jordan method. REF and RREF can be conducted
using a graphing calculator (be sure to read my online
Calculator Guide:
Echelon Form and work through the examples on your own calculator), but
primarily concentrate on the algebraic "by hand" approach. However, when
determining if a system is consistent, inconsistent, or dependent you may rely
solely on the calculator if you wish according to:
| If the RREF of an augmented matrix has a row containing zeros on the left of the vertical bar and a nonzero number on the right of the vertical bar, then that indicates an inconsistent system (i.e., no solution). |
| If the RREF of an augmented matrix has a row containing all zeros, then that indicates a dependent system (i.e., an infinite number of solutions) and the solution should be written in terms of an appropriate parameter. |
This is a heavy-weight section of material that forms the basis for the chapter so spend some time getting the terminology, algebraic processes, and calculator features down well!
Text Notes (These notes refer to College Algebra 5th ed by Blitzer section 6.1-6.2.)
| ch 6.1 example 3 The text forces the diagonals of ones before getting the lower triangle of zeros. Personally I prefer to get the lower triangle of zeros before forcing the diagonal of ones. This will delay having to work with fractions as long as possible. |
| ch 6.2 is applications of systems including consistent, inconsistent, and dependent linear systems. Concentrate on setting up the system of equations then solve them electronically (some are quite large systems and tedious to solve by hand). |
Matrix Algebra
GOLDen Mathematics - College Algebra: Section 5.3
Supplemental Sites:
MathOL Links - Calg 5.3
Topics of Importance
Matrix terminology
Matrix equality
Determinants
Matrix arithmetic: matrix addition, scalar multiplication, matrix multiplication
Inverse of a matrix
Solve linear systems by the matrix equation method
Comments and Cautions
Lots of fun today! Everything you learned to do with numbers like
multiplying or reciprocating you get to learn to do today with matrices :)
Practice working the problems by hand and use the calculator when the technology
comes in particularly handy like when finding the inverse of a large square
matrix. Be sure to read my online
Calculator Guide:
Matrix Algebra,
Calculator Guide: Inverse Matrices, and
Calculator Guide: Determinants working through the examples with your
calculator.
By the end of this section you should have at your disposal the following methods to solve square consistent linear systems:
| Algebraically using the substitution or elimination methods |
| Gaussian elimination to transform to row echelon form (REF on a calc) and then using back substitution |
| Gauss-Jordan elimination to transform to reduced row echelon form (RREF on a calc) |
| Matrix equation method AX=B -> X=A-1B |
Text Notes (These notes refer to College Algebra 5th ed by Blitzer section 6.3-6.5.)
| ch 6.3 covers matrix algebra and has plenty of terminology to memorize and operations on matrices to learn! Leave the applications for another day. | ||||||
| ch 6.4 covers finding inverses of matrices as well as solving systems using matrix equations. Again, leave the applications for another day. | ||||||
| ch 6.4 The shortcut for finding an inverse of a matrix shown in the blue box on page 596 only works on a 2x2 matrix. 3x3 and larger matrices require the method shown in the blue box on page 598. | ||||||
| ch 6.5 I recommend that to find the determinant of a:
| ||||||
| ch 6.5 pg 610-612 SKIP the "adjunct" method of finding a determinant (including the two blue boxes on page 610). | ||||||
| ch 6.5 pg 608-609, 612-614 SKIP "Cramer's Rule" |
Matrix Applications
GOLDen Mathematics - College Algebra: Section 5.4
Supplemental Sites:
MathOL Links - Calg 5.4
Topics of Importance
Various applications of matrices, matrix algebra, and
determinants
Comments and Cautions
Applications of matrices, matrix algebra, and determinants are sprinkled
throughout the chapter. Be sure to try a nice variety as well as working the
problems in this handout:
Matrix Applications Handout Discuss these examples on the main classroom
board!
Text Notes (These notes refer to College Algebra 5th ed by Blitzer sections 6.2-6.5 ... apps at the end of each section.)
| I recommend that you try one of each of the following types of
applications. Discuss these on the main classroom board!
|
Originally written: 2006-09-04
Last revision:
2009-10-25 09:33 AM