x y

(1′) and (2′) are found by putting x = in equation (1) and (2) of 2736. Hence, according to what precedes, the eliminant of two equations of the first degree, (1) and (2), is the determinant of the coefficients of the unknown quantity and the constant terms, or of the coefficients of the unknown quantities in case the given equations are written in the form of homogeneous equations of the first degree. The resultant is found by placing the eliminant equal to 0.

REMARK.-The results can be described as follows: If the equations (1) and (2) are compatible, or what amounts to the same thing, in case these equations are satisfied by the same values of x, then the identity (3) exists among the coefficients of these equations.

738. The Solution of two Equations of the First Degree in the Case when the Determinant of the Unknown Numbers is not 0. Let the equations be

Multiply equation (1) by b, and equation (2) by -b, and add; then

we get (3)

=

(abab)x c1b2 — c2b1

Multiply equation (1) by -a, and equation (2) by a, and add, and get (a,b, ab)y a ̧¢ ̧—à ̧¢ ̧1;

(4)

RULE. -The denominator of x and y is the determinant of the coefficients of the unknown quantities of equations (1) and (2); the numerator of x is found by substituting in the denominator the terms c1 and C2 in the second members of (1) and (2) for the coefficients a and a, of x; similarly the numerator of y is found by substituting c, and c, for the coefficients b1 and b, of y.

2

The correctness of the solutions in (5) can be verified thus: Substitute in (1) and (2) the values of x and y in (5), and they will become

or arranging the numerators with respect to the c's,

That is, the values in (5) satisfy equations (1) and (2).

739. Homogeneous Equations.

1. With two Unknown Quantities. If c,

=
2
=

0, then equations (1) and (2), 738, are homogeneous and give x = 0, y = 0 as solution of the equations when a,b,- ab2 0.

2

2. With Three Unknown Quantities.-Substitute in equations (1) and (2), 1738,

(6)

(7).

Y
y =
Z

Y

x=

and they become

Z

Then, instead of equations (5), we obtain the equations

Hence X,

- Y, Z are proportional to the determinants of the table

found by striking out the first, then the second, and finally the third column.

740. Determinants of Nine Elements.-The expression.

and is called the determinant of the elements (a, b, c,), (a, b, c), (a, b, c). The term a,b,c, is called the principal term of A. The definition of columns and rows given in 8734 is adopted in case of a determinant of nine elements.

Formation of the Determinant A. Place the first and second columns to the right and next the given determinant or the first and second rows immediately below the third row of the given determinant, thus:

Now form the six products of three elements which lie on lines drawn through the two diagonals and the lines parallel to them, neglecting lines which pass through but one or two elements. Place the sign+before the principal term and the remaining products of lines running from the left above toward the right below, and before the other products place the sign -.

REMARK.-Each of the six terms of a determinant of nine elements contains an element from each column and each row.

Write the following polynomials in the form of determinants:

11. ab-ba+ab ̧―ba+ab,—abs⋅

2

3

13. abeam2 + bn2 + cp2.

15. Find the value of x in the equations

12. a3 + b3 + c3 — 3abc.

14. 2pq-p-pq2.

One can therefore write a determinant of the second order in the form of a determinant of the third order and in some cases conversely. Suppose that one uses stars (*) instead of the arbitrary elements; then one has