We obtain by multiplying the first equation by b', the second by b, and subtracting one from the other, In order to eliminate z, we multiply the first equation by c', the second by c, and subtract the second from the first. It becomes - (a c' — c a') x + (b c — c b') y = d c' — c d'. . . (4). Combining, in like manner, the second equation with the third, we find (a' c'' — c′ a'') x + (b' c'' — c' b'') y = d' c" — c' d'' . . (5). c' In order to eliminate y, we must multiply equation (4) by b' c" — c′ b', and equation (5) by bc-cb', and then subtract, which gives [(a c' — ca') (b'c" — c′ b'') — (a' c'' — c' a") (bc' — cb')] x = (dc' — cd') (b'c" — c' b') — (c' d" — d' c'') (b c' — c b'); by performing the operations indicated, reducing and dividing the two members by c', we have (a b'c" — a cb" + c a' b" — b a' c" + bc' a" — c b' a") x = db'c"-dcb" + cd' b" —b d' c" be d" Then finally, + c - cb' d". By performing analogous operations for eliminating x and z, and and y, we shall find for y and z, afterwards As beginners may not be much used to making all possible abbreviations in the calculation, we will here give a method of passing from the value of x to that of y and z, without being obliged to go over all the preceding operations from the beginning. We observe that the set of equations (1), (2), and (3), will remain the same if we substitute for x, a, a', a", the quantities y, b, b', b', and the reverse; then, if in the expression which gives the value of x, we change x into y, then a, a', a", which are the coefficients of x, into b, b', b", which are the coefficients of y, and the reverse, we shall obtain a result which will be no other than the value of y. or, changing the signs of the numerator and denominator, and writing in each, the last three terms first, and the first three terms last, y= a d' c'' —a c' d' + c a' d' —d a' c''+d c' a". -c d' a" ba' c' + b c a'' In like manner, we should obtain the value of z by changing x, a, a', a" into z, c, c', c', and the reverse. We can easily see the method which must be pursued, if we had four equations with four unknown quantities, &c. N. B. By reflecting a little on the manner in which these formulas have been obtained, we shall easily perceive that for any number whatever of equations, containing an equal number of unknown quantities, x, y, z..., there can exist, in general, but one set of values of x, y, z ..., which will verify the equations. un First, the proposition is evident for an equation with one known quantity axb. There is only the value which can satisfy it. b a Let us consider two equations with two unknown quantities." After we have multiplied the first equation by the coefficient of y in the second, and the reverse, the result which we obtain, by subtracting one from the other, may be substituted in one of the two proposed equations. Now this result, containing only one unknown quantity, admits for that unknown quantity but one value, which, carried back into one of the proposed equations, will likewise give but one value for y. The same reasoning will apply to three equations with three unknown quantities. 67. The use of accents, in the notation of coefficients, has led to the observation of a law, according to which we can easily find the preceding formulas, without being obliged to perform the elimination. Let us consider, first, the example of two equations with two unknown quantities. We have found, for the values, (1.) To obtain the common denominator to these two values, form with the letters a and b, which designate the coefficients of x and of y, in the first equation, the two arrangements a b and ba, then interpose the sign, which gives ab-ba; finally, accent in each term, the last letter; (2.) In order to obtain the numerator relative to each unknown quantity, replace, in the denominator, the letter which designates the coefficient of that unknown quantity, by the letter which designates the known quantity, still leaving the accents in the same place. Thus, a b' — b a′ is changed into c b' — b c', for the value of x, and into a cc a', for the value of y. Let us consider the case of three equations with three unknown quantities, a, b, c, designating the coefficients of x, y, z, and d the known quantity. nator a b (1.) In order to have a common denominator, take the denomiba, which corresponds to the case of two unknown quantities, (the accents being omitted); introduce the letter c into each of the terms a b and b a, at the right, in the middle, and at the left; then place between them alternately the signs plus and minus; there results abc-acb+cab-bac+bea-cba. Afterwards place, in each term, the accent' over the second letter, and the accent over the third letter; and we have for the denomi nator, a b' c' -a c'b'c a' b ba'cbc' a" c b'a". (2.) In order to form the numerator of each unknown quantity, replace in the denominator, the letter which designates the coefficient of that unknown quantity, by the letter which designates the known quantity, leaving the accents in the same place. Thus for x change a into d; for y, b into d; and for z, e into d. This law, which may be regarded as the result of observation for two or three equations, is capable of being extended to any number of equations; but the demonstration is very complicated, and does not belong to the elements of algebra. The learner may consult, for this purpose, the second part of the Algebra of Garnier, which refers to one of M. Laplace. This demonstration is taken from the Memoirs of the Academy of Sciences, 1772. 68. Let us attend to the use which may be made of these formulas, in particular applications. = a = 5, b = 7, c 34, a' = 3, b'—13, c'6. in the place of a, b, c, a', b', c', these values; and we have 5 X-6-34 X 3 (2.) y = 5×—13 — (—7) × 3 and x = 11, y = posed equations. 3, are the values which will satisfy the two pro We can immediately assure ourselves of this, by substituting them in the equations. But, in order that the demonstration may be independent of every particular example, we remark, that in order to pass from the formulas relative to the equations ax + by = c and ax + b' y = c, to those which belong to the equations it is sufficient (65) to change b into — b, b' into b', and c' into cx — b' — (— b) × — c' X= ax-b'-(-b) xa, y = ax - c— cx a' ax-b'-(-b) x a'; and, in order to deduce from these new general formulas, the values which belong to the particular equations, we must make a = 5, b = 7, c = 34, a' = 3, b' = 13, c = 6. Then, finally, to obtain the values relative to the proposed equations, it is sufficient to make, in the general formulas before obtained, a = 5, b=—7, c = 34, a' = 3, b′ — — 13, c' — —6, then to perform the calculations, according to the rules laid down for simple quantities. = The rule consists, in general, in substituting, in the place of the coefficients, a, b, a', b' . . ., .., their particular values considered with the signs by which they are affected in the particular equations, and in performing all the operations indicated according to the established rules. These applications justify again the necessity of extending to simple quantities the rules of the signs established for polynomial quantities, since it is the means of rendering general formulas of the first degree, applicable to every particular example. We proceed to the discussion of these formulas. 69. It results from inspection, that in particular applications, we can obtain four kinds of values for answers to problems of the first degree, that is, positive values, negative values, values of 0 0 the form the couriers has given rise to these four results, which we now propose to explain in a general manner. in fine, values of the form. The problem of First, the positive values are usually answers to questions in the sense of their enunciation. However, we observe that for certain problems all the positive values do not satisfy the enunciation. If, for example, the nature of the problem requires that the numbers sought should be whole numbers, and that we should find fractional numbers, the problem cannot be resolved. Sometimes, also, the nature of the problem does not permit that the unknown numbers should exceed the numbers known and given a priori, or that they should be less than other numbers. If the values obtained, although positive, do not satisfy that condition which the enunciation requires, but which cannot be expressed by an equation, the problem cannot be resolved. Thus the positive values of unknown |