Solution of the Simultaneous Equations

The solution of the simultaneous equations is accomplished by assuming a series of values of x, then calculating corresponding values of y from equations (1) and (2), then interpolating on a graph to find the correct values. The correct values are... 

In all of these applications, a series of simultaneous equations is solved for two unknowns (K and AH or K and ee or K and d, etc.). A graphical procedure for illustrating the solution of the simultaneous equations has been developed (26) and is illustrated in Fig. 3 a. Each line represents a different set of acid and/or base concentrations. It is essential to make a plot of vs. e or AH° or 6 to visually examine the intersections and the variations in the slopes of the lines which are... 

Gas pressure drop across the spout, APv, and gas flow rate through the spout, Ggv, are the more important variables for the design and operation of the V-valve. From Eqs. (6) and (7), we can see that in order to calculate APv and Ggv, the gas-solids relative velocity (uro - up0) must be known. The value of (Wf0 - uPo) can be calculated by solving the simultaneous differential Eqs. (2) to (5) for the trapezoidal spout, with the boundary conditions of simultaneous equations consists of a trial-and-error process for the numerical method. 

Using this ccmpcsiticn the intensity of several other peaks not used in the solution of the simultaneous equations and assumed to originate from these degradation products only were calculated. 

These are the equilibrium equations of the y-tp model that are to be solved for the unknown variables among T, p, x, y. Hi, or from the known variables simultaneously with the mass balances and/or other applicable constraint equations. A common method of solution of the simultaneous equations is to use K values. From Equation (4.494) the K values are formed. 

If the four molar absorption coefficients are known and absorption is measured at the two wavelengths, the concentrations Ci and C2 then can be determined by solution of the simultaneous equations (2.18) and (2.19). If three components are present, it is necessary to make absorption measurements at three wavelengths, and three simultaneous equations are solved, The number of wavelengths used must be at least as many as the number of components. 

Thus a transformation that minimizes A, the free energy of the uncoupled transformed system, Ho = Hj + H , hopefully minimizes the effects of the system-bath coupling term. If the transformation depends on some set of parameters c, this minimum is defined by the solution of the simultaneous equations... 

The solution of the simultaneous equations (32) requires some care because of their size. The number of independent elements of the matrix (/ which describes the perturbed orbitals is, for closed-shell SCF wavefunctions, the product of the number of occupied and the number of virtual molecular orbitals. This can be a large number—several thousand for a calculation on a big molecule. This makes it difficult to solve (32) by the conventional techniques used for small sets of simultaneous equations. However, an iterative approach introduced by Pople et al. provides the answer. This method constructs the solution 1/ as a linear combination of trial vectors... 

We now turn to methods which do not attempt to obtain the characteristic values without the characteristic vectors. Of course, if the characteristic values have been obtained by the methods of Secs. 9-3 or 9-4, one may insert any given Xj, and solve the simultaneous equations. However, if one requires characteristic vectors as well as the X s, for problems where r > 3, it is much better to use one of the methods to be described in this and the following sections. Furthermore, the subsequent methods can all be used in conjunction with the principle described in Sec. 9-6, namely, that an approximate solution of the simultaneous equations, i.e., characteristic vector, V, substituted in Eq. (18), Sec. 9-6, yields a relatively accurate estimate of the corresponding X. ... 

The solution of the simultaneous equations (4.6) and (4.7) provides the best-fit values of -In K /n and 1/n. Variances and covariances can also be calculated which set the shape and inclination of the confidence ellipses drawn in Figure 4.2. Ellipse size is taken to include two standard deviations. 

For a set of simultaneous equations that are all equal to zero (like equations 12.30), there are mathematical ways of finding solutions. Linear algebra allows for two possibilities. The first is that all of the constants, in this case and c, 2> ar exacdy zero. Although this possibility would satisfy the equations 12.30, it provides a useless, or trivial, solution ( = 0 exactly a wavefimction that has been rejected previously for its uselessness). The other possibility can be defined in terms of the coefficients on the cs in equations 12.30, the expressions involving the M s and the S s. Linear algebra allows for a nontrivial solution of the simultaneous equations 12.30 if the determinant formed from the coefficient expressions of equations 12.30 is equal to zero ... 

According to equation 12.31, the nontrivial solution of the simultaneous equations found by minimizing the energy will be given by... 

As an example, let us consider an equation for the hydrogen-ion concentration in a solution of a weak acid in which the hydrogen ions from the ionization of water cannot be ignored. We must solve simultaneous equations for the ionization of the weak acid and ionization of water. If activity coefficients are assumed equal to unity, solution of the simultaneous equations gives the result ... 

Computer programs are available for finding the numerical solution of the simultaneous equations. In one of these programs, /i2 and T are calculated from input data of A, and R (or of A, dij/, and SR/R) by the Newton-Raphson method of successive iteration. The calculation starts by assigning arbitrary values to the parameters and t and calculating the theoretical values of... 

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