10 - The solution of simultaneous algebraic equations

Abstract

If there are two variables in an equation, such as F(x, y) — O, then the equation can be solved for y as a function of x or x as a function of y, but in order to solve for constant values of both variables, a second equation, such as G(x, y) — O, is required, and the two equations must be solved simultaneously. If there are n variables, n independent and consistent equations are required. In this chapter, various methods for finding the roots to sets of simultaneous equations are discussed. The chapter discusses simultaneous equations with more than two unknowns. Solution by matrix inversion is then described. The use of Mathematica to solve simultaneous equations is explained. The chapter presents several methods for solving simultaneous equations. The method of substitution is first considered, but is found not to be practical for more than two or three equations. Several methods which can apply to sets of linear inhomogeneous equations are discussed in the chapter. Cramer's method is a method which uses determinants to obtain the roots. The methods of Gauss elimination and Gauss–Jordan elimination are also presented. Finally, linear homogeneous equations are examined.