Page 37 - Standard Handbook Of Petroleum & Natural Gas Engineering

26 Mathematics

First-degree equations in 2 unknowns are of the form
alxl + b,x, = c1
a2xl + b,x, = c2

The solution is found by multiplication of Equations a and b by some factors
that will produce one term in each that will, upon addition of Equations a and
b, become zero. The resulting equation may then be rearranged to solve for
the remaining unknown. For example, by multiplying Equation a by a2 and
Equation b by -al, adding Equation a and Equation b and rearranging their sum

and by substitution in Equation a:

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 substitu-
tion. Second-degree eyuations 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 dif-
ference, the sum of their squares. For further solutions, see “Numerical Methods.”

Determinants
Determinants of the second order are of the following form and are evalu-
ated as

and of the third order as

and of higher orders, by the general rule as follows. To evaluate a determinant
of the nth order, take the eiements of the first column with alternate plus and
minus signs and form the sum of the products obtained by multiplying each of
these elements by its corresponding minor. The minor corresponding to any
element en is the determinant (of the next lowest order) obtained by striking
out from the given determinant the row and column containing en.
Some of the general properties of determinants are:
1. Columns may be changed to rows and rows to columns.
2. Interchanging two adjacent columns changes the sign of the result.