Solving Systems of Equations

Unit 9 contains some of the ways one can solve a group of several linear equations in more than one unknown so that the solutions satisfy all the given equations simultaneously. Note the stress on the word ‘linear’ because, in such cases, the highest power of each of its unknowns is 1.

In any case, if the number of linear equations $= m \geq n =$ the number of unknowns, then any such system will have either no, one, or infinitely many solutions. There are no other possibilities because of the underlying Euclidean geometry.

However, some systems of equations may not be classified as ‘linear systems’ because one of more of the equations may have other kinds of functions in them, besides polynomials, or may contain higher order polynomial equations. There is a group of such non-linear systems which may be converted to a linear system using appropriate substitutions of variables. Ways to solve those kinds of systems will be discussed in the online tutorials.

Learning Objectives

After you have completed Unit 9, you should be able to:

define a system of linear equations in two or more variables;
distinguish between a linear system and a nonlinear system of equations;
solve a system of linear equations in two or more variables using either the substitution or the elimination methods;
outline the procedures for finding an equivalent system of linear equations to a given one;
distinguish between a consistent and an inconsistent system of linear equations and to know if a system is a dependent system;
interpret the meaning graphically/geometrically of the solution set of a system of linear equations in two or more variables; and
model and solve real-world situations using systems of linear equations.

Readings and Practice

Chapter 11 (pp. 401–420) of the textbook

Online graphing tool: Graph

Systems of Linear Equations in Two Variables

Textbook Readings

pp. 402–409

Practice

(pp. 410–412): 1, 2, 3, 4, 5, 9, 13, 17, 23, 31, 35, 39, 55, 59, 63, 69

Answers

pp. 539–540

Terms to Understand

linear equation in two variables/unknowns; system of linear equations in two variables; solution set of a system of linear equations in two variables; definitions of a consistent, an inconsistent and a dependent system of linear equations in two variables

Figures/Tables

Substitution Method (p. 402)
Elimination Method (p. 403)
Graphical Method (p. 404)
Number of Solutions of a Linear System in Two Variables (p. 405)
Guidelines for Modeling with Systems of Equations (p. 407)

Online Tutorials—the Overview

Pertinent to “Systems of Linear Equations in Two Variables”

None

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Systems of Linear Equations in Several Variables

Textbook Readings

pp. 412–418

Practice

(pp. 418–420): 1, 2, 3, 5, 9, 15, 17, 21, 27, 31, 37, 43

Answers

p. 540

Terms to Understand

linear equation in several variables/unknowns; system of linear equations in several variables; solution set of a system of linear equations in several variables; definition of equivalent systems of linear equations in several variables

Figures/Tables

Operations That Yield an Equivalent System (p. 414)
Number of Solutions of a Linear System (p. 415)

Online Tutorials—the Overview

Pertinent to “Systems of Linear Equations in Several Variables”

None

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the ‘Basic Algebra’ topic page.

Unit 9 Review

None

Unit 9 Test

None

Figures/Tables

Formulas to Remember (pp. iv and v at the beginning of the text)