Being Systematic with Systems of Equations

M My riting an equation to use for solving a story problem is more than half the battle. Once you have a decent equation involving a variable that represents some number or amount, then the actual algebra needed to solve the equation is typically pretty easy. 

Many word problems lend themselves to more than one equation with more than one variable. It s easier to write two separate equations, but it takes more work to solve them for the unknowns. And, in order for there to be a solution at all, you have to have at least as many equations as variables. 

Most of the problems in this chapter deal with the more typical two-equation solutions, but I include a section on dealing with three or more equations, too. 

Word problems often deal with how many of two or more coins, how many ducks and elephants, how much to invest in this or that, how many red and green jelly beans, and so on. You let variables represent the numbers of coins or ducks or dollars or jelly beans. When working with two different equations written about the same situation, then you have two different variables and need to do some algebra to knock that down to one equation. That s where substitution comes in. 

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Chapter 17 Being Systematic with Systems of Equations 231... 

The flowsheet shown in the introduction and that used in connection with a simulation (Section 1.4) provide insights into the pervasiveness of errors at the source, random errors are experienced as an inherent feature of every measurement process. The standard deviation is commonly substituted for a more detailed description of the error distribution (see also Section 1.2), as this suffices in most cases. Systematic errors due to interference or faulty interpretation cannot be detected by statistical methods alone control experiments are necessary. One or more such primary results must usually be inserted into a more or less complex system of equations to obtain the final result (for examples, see Refs. 23, 91-94, 104, 105, 142. The question that imposes itself at this point is how reliable is the final result Two different mechanisms of action must be discussed ... 

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

Combine these reac tion equations so as to ehminate from the set all elements not present as elements in the system. A systematic procedure is to select one equation and combine it with each of the other equations of the set so as to ehminate a particular element. This usually reduces the set by one equation for each element ehminated, though two or more elements may be simultaneously eliminated. 

Note that application of a systematic approach enables us to resolve a material-balance system into a number of independent equations equal to the number of unknowns that it needs to solve for. The following steps should be followed with any material-balance system, regardless of complexity ... 

To properly describe electronic rearrangement and its dependence on both nuclear positions and velocities, it is necessary to develop a time-dependent theory of the electronic dynamics in molecular systems. A very useful approximation in this regard is the time-dependent Hartree-Fock approximation (34). Its combination with the eikonal treatment has been called the Eik/TDHF approximation, and has been implemented for ion-atom collisions.(21, 35-37) Approximations can be systematically developed from time-dependent variational principles.(38-41) These can be stated for wavefunctions and lead to differential equations for time-dependent parameters present in trial wavefunctions. 

In this work. My was found to be more sensitive than Xa to systematic errors in Dt. This can be expected in other polymer-solvent systems as well. Thus, a bias in Dt values are manifested in the a terms of equations 7 and 8, yielding a relative error in the denominator of equation 7 that is of the same order of magnitude as the relative error induced in the numerator of equation 8. However, the cubic form of equation 7 results in more severe consequences for My compared with Xa. ... 

Equation (51) may be directly applied in the interpretation of ECL efficiency data for DPA. With the ECL efficiencies of this system [122] one can obtain Ao values in the range 0.13-0.22 eV. Similar values are obtained from the calculation according to Eq. (6). Assuming the effective radii of both DPA radicals n = T2 r = 0.48 nm (from the molar volume of DPA) and reaction distance ri2 = 0.55 nm, the following values can be obtained Ao = 0.15eV in 1,2-dimethoxyethane and 0.20 eV in acetonitrile solutions (Ao = 0.38 eV in N,N-dimethylformamide solutions has been found in [130] for electron exchange between anthracene and its radical anion). These calculations must be treated only as a semiquantitative approach until the systematic temperature study of eci efficiency has been done, especially as the DPA ECL system has been studied only in a limited number of solvents. 

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