Consistency and Independence in Simultaneous Equations

We can understand consistency and independence in simultaneous equations by looking at the graphs of the equations. Each of the equations represents y as a function of X. With linear equations, these functions are represented by straight lines. Figure 3.3 shows the two lines representing the two equations of Example 3.15. The two equations are consistent and independent, and the lines cross at the point... 

We can understand consistency and independence in simultaneous equations by looking at the graphs of the equations. Each of the graphs represents y as a linear function of x. 

The pure component adsorption equilibrium of ethane and propane are measured on Norit AC at three temperatures (30, 60 and 90 °C). All experimental data of two species at three temperatures are employed simultaneously to fit the isotherm equation to extract the isothermal parameters. Since an extended Langmuir equation is used to describe the local multicomponent isotherm, the maximum adsorbed capacity is forced to be the same for ethane and propane in order to satisfy the thermodynamic consistency. The saturation capacity was assumed to be temperature dependent while the other parameters, bo and u], are temperature independent but species dependent. The derived isotherm parameters for ethane and propane are tabulated in Table 1. The experimental data (symbols) and the model fittings (solid lines)... 

Under the first assumption, each electron moves as an independent particle and is described by a one-electron orbital similar to those of the hydrogen atom. The wave function for the atom then becomes a product of these one-electron orbitals, which we denote P (r,). For example, the wave function for lithium (Li) has the form i/ atom = Pa ri) Pp r2) Py r3). This product form is called the orbital approximation for atoms. The second and third assumptions in effect convert the exact Schrodinger equation for the atom into a set of simultaneous equations for the unknown effective field and the unknown one-electron orbitals. These equations must be solved by iteration until a self-consistent solution is obtained. (In spirit, this approach is identical to the solution of complicated algebraic equations by the method of iteration described in Appendix C.) Like any other method for solving the Schrodinger equation, Hartree s method produces two principal results energy levels and orbitals. 

If an equation is written in the form f(x) = 0, where / is some function and jc is a variable, solving the equation means to find those constant values of x such that the equation is satisfied. These values are called solutions or roots of the equation. We discuss both algebraic and numerical methods for finding roots to algebraic equations. If there are two variables in the equation, such as F(jc, y) — 0, then the equation can be solved for y as a function of 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) = 0, is required, and the two equations must be solved simultaneously. In general, if there are n variables, n independent and consistent equations are required. 

When a system of linear equations has more equations than unknowns, it is not generally possible to find values of the unknowns which simultaneously satisfy all the equations when these are independent. But if a set of values does simultaneously satisfy a system of (m+n) equations in n unknowns, then m of the equations are not independent and the system is said to be consistent. 

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