Equations in one unknown

This is eontinued n — 1 times until the entire eoeffieient maPix has been eonverted to an upper triangular matrix, that is, a maP ix with only zeros below the principal diagonal. The b veetor is operated on with exactly the same sequenee of operations as the eoeffieient matrix. The last equation at the very bottom of the triangle, aititXit = bn, is one equation in one unknown. It ean be solved for whieh is baek-substituted into the equation above it to obtain x i and so on, until the entire solution set has been generated. 

Logarithms 21. Binomial Theorem 22. Progressions 23. Summation of Series by Difference Formulas 23. Sums of the first n Natural Numbers 24. Solution of Equations in One Unknown 24. Solutions of Systems of Simultaneous Equations 25. Determinants 26. 

Matrix algebra was invented by Cayley22 as a systematic way of dealing with systems of linear equations. The single equation in one unknown... 

We know Qcol, U, Acoil, TR and TCm. Combining the two equations above gives one equation in one unknown, the temperature of the coolant leaving the coil Tc,out. However, the log term precludes an analytic solution, so an iterative interval halving solution method is used. 

In the example, a total mass balance involves only one unknown, m, while benzene and toluene balances each involve both m and x. By writing first a total balance and then a benzene balance, we were able to solve first one equation in one unknown, then a second equation, also in one unknown. If we had instead written benzene and toluene balances, we would have had to solve two simultaneous equations in two unknowns the same answers would have been obtained, but with greater effort. 

In this and the next several sections, we will discuss methods for solving one nonlinear equation in one unknown. Extensions to multivariable problems will be presented in Section A.2i. 

The alert reader will notice that although the left-hand-side of this equation depends only on x, the right-hand-side depends only on t. So both sides must be equal to the same constant. Now you have two easy ordinary differential equations in one unknown each. Also you have an unidentified flying parameter, namely the constant that both sides of the equation must equal. In the grand tradition of calculus textbooks, let us call this constant C. So now we have two separate equations to deal with, each in only one variable. The first one is ... 

This represents one equation in one unknown (Tj), assuming that the IPs and J s are known. 

When given the temperature and pressme of a gaseous mixture, and the parameters a and b, then to find the specific volume you would have to solve the cubic equation of state for specific volume, v. This represents one algebraic equation in one unknown, the specific volume. 

Now the mass balance can be considered a single equation in one unknown (c) Q... 

Two equations (Equations 5.57 and 5.58) have been found for [A Jacmai and [HA]actuai respectively and these only involve known stoichiometric concentrations, plus [H30+]actuai and [OH ]actuai- These latter concentrations are related through the ionic product for water. The equations for [A lactuai and [HAJactuai thus contain one unknown only, and these can be substituted into the equilibrium expression to generate an equation in one unknown. 

These approximations convert the cubic into a quadratic equation in one unknown. These quadratic equations must be used in situations where the pH is less than 4 (Equation 5.61). or greater than 10 (Equation 5.62). In Equation (5.61) the [H30 ]actuai is likely to be comparable to both [HAJtotai and [A ]totai and the term in [HsO Jactuai in the right hand side of Equation (5.61) must not be dropped. Similarly, for pH values greater than 10 the term in [OH lactuai will become comparable to both [HAJtotai and [A Jtotai and must not be dropped,... 

This is a quadratic equation in one unknown, and can be solved to give s. ... 

Equation (1-13) consists of one equation in one unknown, the temperature. The form of the implicit function X,(T) generally requires that the solution of Eq. (1-13) for the bubble-point temperature be effected by a trial-and-error procedure. Of the many numerical methods for solving such a problem, only Newton s method2,5 is presented. In the application of this method, it is convenient to restate Eq. (1-13) in functional forip as follows... 

This system of nonlinear equations is readily reduced to one equation in one unknown (say Vt) in the following manner. First observe that the total material balance expression (a dependent equation) may be obtained by summing each member of the third expression of Eq. (1-26) over all components to give... 

The relationships given by Eq. (1-26) may be reduced to one equation in one unknown in a variety of ways, and a variety of forms of the flash function may be obtained. One form of the flash function is developed below and a different form is developed in Chap. 4 in the formulation of multiple-stage problems. Elimination of the yf, s from the last expression given by Eq. (1-26) by use of the first expression, followed by rearrangement, yields... 

The first step of this method is to manipulate one equation to give one variable as a function of the other. This function then is substituted into the other equation to give an equation in one unknown which can be solved. The result is then substituted into either of the original equations, which is then solved for the second variable. 

It is important to stress that the system (7.7) is linear (and thus a simpler problem) and that it is reduced to a very simple equation in one unknown X4. 

This analyzes bubble point calculation, a special case in point. To calculate the bubble point, the problem is reduced to a single equation in one unknown temperature. Now, for the same mixture of toluene and 1-butanol, the problem is to calculate flash separator conditions when vapor flow is V = 25. 

The critical point can be found also as one equation in one unknown, for details we refer to [28]. The same applies to the critical endpoint (CEP), which corresponds to the q value where CP and TP coincide it is the lowest q where stable liquid is possible. See the extended discussions on liquid windows as related to the CEP in [28, 29]. 

There are many problems which require the finding of the root of a single nonlinear equation in one unknown. For example, the heat capacity of carbon dioxide is given as a function of temperature as... 

---