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Linear Algebric Equations

OUTLINE
6.1 Elementary Solution Methods
6.2 Matrix Methods for Linear Equations
6.3 Cramer’sMethod
6.4 Underdetermined Systems
6.5 Overdetermined Systems
6.6 Summary
Problems
, ..:
Linear algebraic equations such ‘~s
5x – 2y = 13
7x + 3y = 24
occur in many engineering applications. For example, electrical engineers use them to predict the power requirements for circuits; civil, mechanical, and aerospace engineers use them to design structures and machines; chemical engineers use them to compute material balances in chemical processes; and industrial
engineers apply them to design schedules and operations. The examples and homework problems in this chapter explore some of these applications. Linear algebraic equations can be solved “by hand” using pencil and paper, by calculator, or with software such as MATLAB. The choice depends on the circumstances. For equations with only two unknown variables, hand solution is easy and adequate. Some calculators can solve equation sets that have many variables. However, the greatest power and flexibility is obtained by using software.

For example, MATLAB can obtain and plot equation solutions as we vary one or more parameters. Without giving a formal definition of the term linear algebraic equations, let us simply say that their unknown variables never appear raised to a power other than unity and never appear as products, ratios, or in transcendental functions
such as In(x), e”, and cos x. The simplest linear equation is ax = b, which has the solution x = bf a if a i’ O.
In contrast, the following equations are nonlinear: which has the solutions

x = ±.J3, and
sinx = 0.5
r
which has the solutions x = 300
, 1500
, 3900
, 5100
, •••• In contrast to most nonlinear
equations, these particular nonlinear equations are easy to solve. For example,
we cannot solve the equation x +2e-x – 3 = 0 in “closed form”; that is,
we cannot express the solution as a function. We must obtain this solution numerically,
as explained in Section 3.2. The equation has two solutions: x = -0.5831
and x = 2.8887 to four decimal places.
Sets of equations are linear if all the equations are linear. They are nonlinear
if at least one of the equations is nonlinear. For example, the set
8x – 3y = 1
6x +4y = 32
is linear because both equations are linear, whereas the set
6xy – 2x = 44
. 5x – 3y =-2
is nonlinear because of the product term xy. Systematic solution methods have been developed for sets of linear equations. However, no systematic methods are available for nonlinear equations because
the nonlinear category covers such a wide range of equations. In this chapter we first review methods for solving linear equations by hand, and we use these methods to develop an understanding of the potential pitfalls that can occur when solving linear equations. Then we introduce some matrix notation that is required for use with MATLAB and that is also useful for expressing solution methods in a compact way. The conditions for the existence and uniqueness of solutions are then introduced. Methods using MATLAB are then treated in four sections: Section 6.2 covers equation sets that have unique solutions; Section 6.3 covers ‘Cramer’s method; Sections 6.4 and 6.5 explain how to determine whether a set has a unique solution, multiple solutions, or no solution at all.