The process of elimination, when the number of unknown quantities and equations is greater than three, may be conducted by any of the known methods, but most conveniently by that of reduction. The employment, however, of accented letters to represent the coefficients of the unknown quantities and the algebraic sums of the given quantities has led to the observation of a law, by means of which the formula which satisfy any number of equations may be obtained without the necessity of elimination. n 89. Resuming the equation ar=n, or x=; the numerator of the value of x is the algebraic sum of the terms which are entirely composed of known quantities, and the denominator is the coefficient of the unknown quantity. Next, the two equations, ax+by=n, d'x+b'y=n', give the results, nb'-bn' an' -na' x= “ab' —ba" y=- ab-ba In these values of x, y the denominator, which is the same for both unknown quantities, is composed of a, a, b, b', the coefficients of the unknown quantities. It can be formed by writing the letter a before b, thus, ab; then after b, thus, ba; connecting these expressions by the sign, and placing an accent over the second letter of each term. Thus the denominator of the values of x and y is ab'-ba'. To form the numerator of the value of x, a, a' are changed, in the expression ab-ba', into n, n'; and to form the numerator of the value of y, ̃ ̄b, b are changed in the same expression into n, n'. Thus, the numerator of x is nb'-bn', and the numerator of y is an'—na'. To obtain the denominator which is common to the values of three unknown quantities, assume the denominator ab-ba of the preceding case (omitting the accents). Into each of the terms ab, ba, introduce the letter c at the end, in the middle, and at the beginning; make the signs alternately and, and place in each term one accent on the second letter and two accents on the third letter. The denominator thus formed is ab'c'—ac'b'+ca′b”—ba ́c′′ +bc'a′′"—cb'a′′. To form the numerator of the value of each unknown quantity, replace in the denominator the letter which represents the coefficient of that unknown quantity by the letter which represents the quantity wholly known, leaving the accents in the same place. Thus, for a change a into n, for y change b into n, and for z change c into n. If the number of unknown quantities and equations is four, the coefficients being a, b, c, d, a', b', c', d', &c. &c., and the known quantities n, n', n", n'""', it is necessary to introduce the coefficient d into each of the six products abc-acb+cab-bac+bca―cba, and to make it occupy in succession all the places. The product abc, for example, gives abcd-abde+adbc-dabc. Proceeding in the same manner with the five remaining terms, the complete denominator will contain 24 terms, in each of which the second letter will have one accent, the third two, and the fourth three. The denominator which is common to the values of all the unknown quantities being formed, the numerator of the value of each unknown quantity is obtained from it in the same manner as in the case of three unknown quantities, namely, for each unknown quantity by replacing in the denominator the letter which represents the coefficient of that unknown quantity by the letter which represents the quantity wholly known, leaving the accents as in the denominator. This law can be extended to any number of equations. The demonstration of it is complicated, and by no means elementary. The manner of applying these formula to the resolution of numerical equations of the first degree has been already explained (Art. 55). DISCUSSION OF THE GENERAL FORMULE FOR THE RESOLUTION OF EQUATIONS OF THE FIRST DEGREE. 90. The general equations of the first degree involving two unknown quantities being ax+by=n dx+by=n' nb'-bn' na'-an' the value of x= ab'—ba', and the value of y=ab-ba In these values of x, y, let the denominator ab′—bd=0; Therefore x=" 0 ,y= 0 Whence in this case the values of x and y are infinite. From the equality ab′-ba=0 is obtained, by transposition and division, or ab'x+bb'y bn' or b'(ax+by)=bn'; the first member of this result is the first member of equation 1 multiplied by b; therefore n, the second member of equation 1, if multiplied by b', ought to be equal to bn'; that is, b'n ought to be equal to bn', or no equal to bn'. But if nb'=bn', nb′-bn'=0, or the numerator of the value of x is equal to zero, which is contrary to the hypothesis. In this manner the impossibility of finding values of x and y which at once satisfy the equations, is made evident. nb'-bn' na'-an' But the impossibility is more distinctly marked by the infinite values which not only show its existence, but also that it arises from the circumstance that the values of the unknown quantities are greater than any which can be assigned. If the value of the expression ab'-ba' is very small, the values of x, y are very great, but such as satisfy the equations. This being true, if, when ab'-ba'=0, the verification cannot be made, the reason why it cannot is, that then x, y are greater than any assignable magnitudes. It ought to be observed, that values = ∞, deduced from equations, may give the true solution of the question the conditions of which are expressed by these equations. Of this the application of algebra to geometry affords many instances. To give one: when an angle is unknown, and an infinite value is found as the expression of its tangent, it is certain that the angle is a right angle. Second, let the denominator ab′-ba′=0, and one of the numerators nb′-bn'=0; then shall the other numerator an'-na be also equal to zero. ab' For since ab-ba′=0, a'=' and since nb′-bn'=0, n'= ̄• Substituting these values of a', n' in the expression an'-na', it becomes For the hypothesis ab'-ba'=0, an'-na'=0, it can be in like manner proved that nb'—bn'=0. Whence the common denominator, and one of the numerators of the values of x, y being zero, the other numerator is also zero; and the values of x, y are 0 0 x= These symbols indicate indeterminate quantities, and by returning to the equations it can be proved that this case is indeterminate. al' nb' The values of a', n', found above, are b If these values are substituted for a', n' in equation 2, nb' ab' b -x+b'y= b This equation can be obtained by multiplying both members of the equation ax+byn by Therefore all the values of x, y which satisfy one equation, must also satisfy the other. Now, if to x arbitrary values are given, and these values are substituted in the first equation, corresponding values of y can be obtained from it; thus are obtained solutions which satisfy the first equation. But these values must also satisfy the second equation; therefore it follows that the proposed equations admit an indefinite number of solutions, or, in other words, are indeterminate. The indetermination does not, however, permit an arbitrary value of x to be taken, and also an arbitrary value of y; for the preceding explanation shows that the value of one of the unknown quantities ought always to be calculated by means of the arbitrary value of the other. This case comprehends that in which n=0, n'=0, ab'—ba'=0, for by these conditions 0 x=0 0 and y=0 By the hypothesis n=0, ax+by=n is reduced to ax+by=0. Whence the two values of y are equal for every value of x, and by consequence there is a case of indetermination. It is proper to observe that the ratio of y to x in these expressions is determinate; except for x=0. In this case y must also have been equal to zero, and the that is, indeterminate. In employing the expressions a= coefficients a, b are different from zero. If at the same time b=0, b'=0, the two expressions ab'-ba', nb′—bn' must be equal to zero, and this without giving any determinate value of an'-na'. 0 an' -na' In this case the values of x, y are x= O' y= In like manner, if a, a' are at once equal to zero, the results for x, y are nb'-bn' 0 This particular case is, however, hardly admissible, for by it the given equations are reduced to two equations to one unknown quantity, namely, ax=n, a'x=n', for b and b'=0, while the subject under consideration is that of two equations to two unknown quantities. 91. The general formulæ relative to more than two equations are more complicated, and the discussion of the conditions of impossibility and indetermination more laborious and difficult. Denoting the numerator of the value of x by N, and the denominator by D, if by the application of the general formulæ to a system of equations, a result x= is obtained, this result is a character of impossibility. But the N 0 N 0 result =D=0' x= is sometimes a character of indetermination, sometimes of impossibility; and it sometimes indicates the existence of a factor common to N and D. To ascertain the precise signification of these results, there is no simpler method than to determine the values of the unknown quantities from the system of equations by the method of Article 55. The proposed equations being, x+9y+6z=16 2x+3y+2z=7 1 2 But if the second equation is multiplied by 3, and the first equation is subtracted from the product, that is, if from 6x+9y+6z=21 x+9y+6z=16 is subtracted, the remainder 5x = 5, .*.x=1. Substituting this value of x in the proposed equations, the results are 1st. 9y+6z=15, or 3y+2z=5. The value is x, consequently determinate and equal to 1; but the values of y and z are indeterminate, since there is only one equation, 3y+2z=5, to the two unknown quantities, y, z. SECTION V. OF THE INDETERMINATE ANALYSIS OF THE FIRST DEGREE. 92. In resolving an equation of the first degree involving two unknown quantities, if any value whatever is given to one of the unknown quantities, the equation gives a corresponding value of the other unknown quantity. Considered in this manner, the equation admits of an infinite number of solutions. The number of solutions is more limited if it is required that the values of x, y shall be whole numbers; and still more if it is required that the values shall be both integer and positive. Conditions of this kind cannot be expressed by equations. Any equation of the first degree to two unknown quantities can be reduced to the form ax+by=c, in which a, b, c are any whole numbers, positive or negative. The factors common to a, b, c may be suppressed; and when it is required that the values of x, y shall be integer, it is necessary, after this suppression of the common factors, that a, b be prime to each other. For, supposing a and b both prime to c, but not to each other, and that any integer values of x, y are substituted in the equation ax+by=c, the first member of the resulting equality can be divided by the common factor of a, b; but c, which is prime with a, b, cannot be divided by it. Consequently, under this hypothesis, the equality is impossible. 93. Having given the equation ax+by=c 1 in which a, b, c represent any whole numbers, positive or negative (with the limitation that a, b are prime to each other), the problem is, to find all the systems of integer values of x, y capable of satisfying equation 1. If the coefficient of one of the unknown quantities, y for example, is 1, the equation becomes ax+y=c. ..y=c-ax, a result in which, if any whole number is substituted for x, the corresponding value of y is also a whole number. When, therefore, the coefficient of either of the unknown quantities is 1, the problem presents no difficulty. Next, let a>b and b>1, the value of y given by equation 1 is y= c-ax Let a be divided by b, let q= the quotient, and r= the remainder. If any whole number is substituted for x, the expression -qr must be a whole number. But in order that y also may have an integer value, it is necessary that C-rx b shall be a whole number. Let t denote a new indeterminate quantity, and make The question is thus reduced to the resolution of equation 2 for integer values of x and t. If r=1, the investigation is ended. Suppose r>1. Resolving equation 2, with respect to x (since r, the coefficient of x, is less than b, the coefficient of t), Divide b by r; let q' represent the quotient, and the remainder. |