First degree equations

First degree equations linear equations) have the form... 

A set of n first-degree equations in n unknowns is solved in a similar fashion by multiplication and addition to eliminate n - 1 unknowns and then back substitution. Second-degree equations in 2 unknowns may be solved in the same way when two of the following are given the product of the unknowns, their sum or difference, the sum of their squares. For further solutions, see Numerical Methods. ... 

To solve a linear inequality, isolate the letter and solve the same as you would in a first-degree equation. Remember to reverse the direction of the inequality sign if you divide or multiply both sides of the equation by a negative number. 

Changes in the vapour pressure of tetranitromethane depending on temperature may be expressed by a first degree equation ... 

The conditions of the minimum are that partial derivatives on the right-hand side with respect to j, 2 and 3 must be equal to zero. This gives three first-degree equations with respect to. 4i, . 2 and ds, from which they can be defined. 

Case i. The equation can he split up into factors. If the differential equation can be resolved into n factors of the first degree, equate each factor to zero and solve each of the n equations separately. The n solutions may be left either distinct, or combined into one. 

The equation of state of gas B is a first-degree equation in Vm and therefore can never model critical behavior, the process of liquefaction, or the existence of a two-phase region. 

Equations of the First Degree. Equations diat define functions showing a linear dependence between variables are known as equations of the first degree or first-degree equations. These functions describe a dependence commonly called the... 

Mathematically equation (A2.1.25) is the direct result of the statement that U is homogeneous and of first degree in the extensive properties S, V and n.. It follows, from a theorem of Euler, that... 

Linear Equations A hnear equation is one of the first degree (i.e., only the first powers of the variables are involved), and the process of obtaining definite values for the unknown is called solving the equation. Every linear equation in one variable is written Ax + B = 0 or X = —B/A. Linear equations in n variables have the form... 

Two quadratic equations in two variables can in general be solved only by numerical methods (see Numerical Analysis and Approximate Methods ). If one equation is of the first degree, the other of the second degree, a solution may be obtained by solving the first for one unknown. This result is substituted in the second equation and the resulting quadratic equation solved. 

Equations with Separable Variables Every differential equation of the first order and of the first degree can be written in the form M x, y) dx + N x, y) dy = 0. If the equation can be transformed so that M does not involve y and N does not involve x, then the variables are said to be separated. The solution can then be obtained by quadrature, which means that y = f f x) dx + c, which may or may not be expressible in simpler form. 

Linear Equations A differential equation is said to be linear when it is of first degree in the dependent variable and its derivatives. The general linear first-order differential equation has the form dy/dx + P x)y = Q x). Its general solution is... 

If the dependent variable y(jt) and all of its derivatives occur in the first degree and do not appear as products, the equation is said to be linear. In effect, the solution of a differential equation of order n necessitates n integrations, each of which involves an arbitrary constant. However, in some cases one or more of these constants may be assigned specific values. The results, which are also solutions of the differential equation, are referred to as particular solutions. The general solution, however, includes all of the n constants of integration, whose evaluation requires additional information associated with the application. 

A simple example will show how higher-degree linear equations reduce to a system of first-order equations. 

The volume function then is homogeneous of the first degree, because the parameter X, which factors out, occurs to the first power. Although an ideal solution has been used in this illustration. Equation (2.31) is true of all solutions. However, for nonideal solutions, the partial molar volume must be used instead of molar volumes of the pure components (see Chapter 9). 

As told earlier for scalar equations, we must be well aware that there are unsolvable IVPs for which the solution may, for example, blow up , i.e., have a pole before reaching the right endpoint of the target interval [a, 6], Likewise, an IVP may have multiple solutions, especially if the number of initial conditions is less than the order of the DE or the dimension of the first-degree system. But under mild mathematical assumptions of continuity and Lipschitz14 boundedness F(x,y)—F(x,y) < L y—y of the function F in (1.12), every ODE system y = F(x, y) with y = y(x) R — Rn and with n specified initial conditions y(xo), y (xo),. .., /" 1 -1 (xq) 6 R" at xq R will have a unique solution we refer the reader to the literature quoted in the Resources appendix at the end of this book. 

As usual we first transform the coupled two second-order DEs in (5.68) and (5.69) into one first-degree system with 4 first-order differential equations. 

The polymerization kinetics of m-dodecalactam in chemical processing has been investigated in a more limited number of publications in comparison with e-caprolactam. It is not really possible to compare different data therefore only the results of Ref.37 will be discussed below. It was shown that up to a degree of conversionp 0.7, the kinetics of anionic activated polymerization of co-dode-calactam is described by the first-order equation ... 

This equation is consistent with the concept that the Gibbs energy is a homogenous function of the first degree in the mole numbers. Comparison of this equation with Equation (7.49) together with the relation that Pt = Py( shows that... 

Other equations can now be obtained. The energy of a region is still a homogenous function of first degree in the entropy, volume, and mole numbers. Then... 

The internal energy is homogeneous of degree 1 in terms of extensive thermodynamic properties, and so equation 2.2-8 leads to equation 2.2-14. All extensive variables are homogeneous functions of the first degree of other extensive properties. All intensive properties are homogeneous functions of the zeroth degree of the extensive properties. 

Any equation of first degree (i.e. involving only x (andhaving no x2, x3 etc. terms)) can be represented by a straight line graph. The general equation takes the form ... 

In the reactive case, r is not equal to zero. Then, Eq. (3) represents a nonhmoge-neous system of first-order quasilinear partial differential equations and the theory is becoming more involved. However, the chemical reactions are often rather fast, so that chemical equilibrium in addition to phase equilibrium can be assumed. The chemical equilibrium conditions represent Nr algebraic constraints which reduce the dynamic degrees of freedom of the system in Eq. (3) to N - Nr. In the limit of reaction equilibrium the kinetic rate expressions for the reaction rates become indeterminate and must be eliminated from the balance equations (Eq. (3)). Since the model Eqs. (3) are linear in the reaction rates, this is always possible. Following the ideas in Ref. [41], this is achieved by choosing the first Nr equations of Eq. (3) as reference. The reference equations are solved for the unknown reaction rates and afterwards substituted into the remaining N - Nr equations. 

Now the Gibbs-Duhem equation must be satisfied (recall that the Gibbs-Duhem equation is a simple consequence of the fact that the function, or functional, that delivers the free energy is homogeneous to the first degree see Section V). For arbitrary 5rt/ resulting in some 6fi/, one must have... 

If there is a finite point in the xy plane which is symmetrically, situated with respect to a conic section, the latter is called a central conic (ellipse, circle, hyperbola). If this point is taken as the origin, the equation of the conic contains no terms of the first degree in x and y. The parabola is a non-central conic, since the centre is at an infinite distance. 

The equation of the eighth degree is thus reduced to two of the first degree and two cubic equations. It follows that one solution is... 

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