Page 212 - Excel for Scientists and Engineers: Numerical Methods

Chapter 9

Systems of

Simultaneous Equations

Sometimes a scientific or engineering problem can be represented by a set of
n linear equations in n unknowns, for example
x+2y= 15
3x+ 8y= 57
or, in the general case

allxl + a12x2 + a13x3 + “’ + al&,x,, = c1
+
~~21x1 ~22x2 + ~23x3 + + ~2,&,, = ~2

a17lX1 + am + a,43 + * * + a,,,&,, = c,
where xl, x2, x3, ..., x, are the experimental unknowns, c is the experimentally
measured quantity, and the aii are known coefficients. The equations must be
linearly independent; in other words, no equation is simply a multiple of another
equation, or the sum of other equations.
A familiar example is the spectrophotometric determination of the
concentrations of a mixture of n components by absorbance measurements at n
different wavelengths. The coefficients ay are the E, the molar absorptivities of
the components at different wavelengths (for simplicity, the cell path length,
usually 1.00 cm, has been omitted from these equations). For example, for a
mixture of three species P, Q and R, where absorbance measurements are made
at hl,h2 and h3, the equations are

E 1, [PI + E?, [QI + E:, [RI =An,

E:~ [PI + E:, [QI + &f2 [RI =A,,
&I, [PI + &?? [QI + ~n”, [RI =A,,
This chapter describes direct methods (involving the use of matrices) and
indirect (iterative) methods for the solution of such systems. The chapter begins

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