Now, in order to pass from these formulas, ax-by=c, 1st. To those which agree to the equations dx+fy=g, it is only necessary to change b into —b, which gives x= cf+bg ag-cd ax-by=c, £d. To the formulas relative to the equations {dx-fy=g, it will be sufficient to change +b to-b, and +f to -f, which gives the formulas, x= -cf+bg_bg—cf The demonstration being precisely the same as in the preceding example, we will not repeat it. IV. General discussion of Problems and Equations of the First Degree. 66. In order to generalize the discussion of problems of the first degree, involving one or more unknown quantities, we will proceed to find formulas which can represent the values of the unknown quantities, for any system of equations whatever, containing the same number of them. In the first place, every equation of the first degree, involving but one unknown quantity, can be reduced to the form ax=-b; a denoting the algebraic sum of the quantities by which the unknown quantity is multiplied, and b the algebraic sum of all the known terms. From this equation we deduce, x= b a. Secondly. Every equation of the first degree, involving two unknown quantities, may be put under the form ax + by=c. For if the proposed equation contained denominators, we would make them disappear. (44). Afterwards, collect all the terms involving x, and those involving y, in the first member, then transpose all of the known terms into the second; the algebraic sum of the first can be designated by ax, that of the second by by, and the third by c. Hence, a, b, c are entire. Multiplying the first by b', and the second by b, and subtracting one from the other, we have (3). ax + by + cz = d a'x+by+cz d' = a"x+by+c"z=d" To eliminate z, multiply the first equation by c', and the second by c, and subtract the second from the first; there results (ac-ca')x+(bc-cb')y=dc-cd'. . . . (4). Combining the second equation with the third in the same manner, we find (a'c"—c'a')x+(b'c'"—c'b')y=d'c'—c'd". (5). To eliminate y, multiply equation (4) by b'c"-cb", and equation (5) by bc-cb', then subtract one from the other, and we have [(ac'—ca') (b'c'—c'′b′′)—(a'c'—c'a′′) (bc'—cb')] x= (dc'—cd') (b'c"—c'b')—(c'd"—d'c'') (bc'—cb') ; or, by performing the operations indicated, reducing and dividing by c', (ab'c'—acb" +ca'b"-ba'c" +bc'a"—cb'a") x= Hence, x= db'c"-dcb" + cd'b'-bd'c"+bc'd"-cb'd" By performing analogous operations, to eliminate x and z, and y and z, we will find, ad'c"-ac'd" +ca'd"-da'c'" + dc'a"-cd'a" y=ab'c"—ac'b" +ca'b"—ba'c" + bc'a"—cb'a" ab'd"-ad'b"+da'b"-ba'd" +bd'a"-db'a" ab'c"-acb" +ca'b"-ba' c" +bc'a"-c b'a" z= As the beginner will not have had much practice in abbreviating the operations as much as possible, we will indicate a method of deducing the values of y and z, from that of x, without being obliged to go through all of the preceding operations. It will be observed, that the system of equations (1), (2) and (3), remain the same when we substitute y, b, b' and b", for the quantities x, a, a', a", and reciprocally; therefore if in the expression for the value of x, we change x into y; also a, a', a", which are the coefficients of x, into b, b', b", which are the coefficients of y, and reciprocally. The result will be, the value of y, viz. : y= da'c"-dc'a" +cd'a"-ad'c"+ac'd"—ca'd" ba'c"-be'a"+cb'a"-a b'c"+ac'b"-ca'b" or, changing the signs of the numerator and denominator, and writing in each the three last terms first, and the three first last, ad'c"-ac'd" +ca'd"—da'c" +dc'a"—cd'a" ab'c"-acb" + cb'b"-ba'c"+bc'a"-cb'a" In a similar manner we might obtain the value of z, by changing x, a, a', a" into z, c, c' c". and reciprocally. This is sufficient to indicate the course to be pursued, in the case of four equations involving four unknown quantities, &c. N. B. By a little reflection upon the manner in which these formulas have been obtained, we are sensible, that for any number of equations whatever, containing a like number of unknown quantities x, y, z . . . . there can be, in general, but one system of values, which will verify the equations. b This proposition is evident for an equation involving but one unknown quantity, ax=b. There is but one value, which will satisfy it. In the case of two equations involving two unknown quantities, after having multiplied the first equation by the coefficient of y in the second, and reciprocally, the result obtained by subtracting one from the other, may be substituted in one of the two proposed equations. Now as this equation will then contain only one unknown quantity, it will admit of but one value for this unknown, which, being substituted in one of the equations, will in like manner give but one value for y. The same reasoning will apply to three equations involving three unknown quantities. 67. From the use of accents in the notation of the coefficients, a law has been observed, from which the preceding formulas can be deduced, without the aid of the rules for elimination. In the case of two equations involving two unknown quantities we have found 1st. To obtain the common denominator of these two values, take the letters, a, and b, which are the coefficients of x, and y, in the first equation, and form the two arrangements ab and ba, and connect them with the sign —, which gives ab-ba; then accent the last letter of each term; and we have ab'-ba'. 2nd. To obtain the numerator relative to each unknown quantity, take this denominator, and replace the letter which denotes the coefficient of this unknown quantity, by the letter which represents the known term of the equation; taking care to leave the accents as they were. In this way, ab—ba' is changed to cb'-bc', for the value of x, and to ac'-ca', for the value of y. Let us now consider the case of three equations involving three unknown quantities, a, b, c, denoting the coef1st. To find the comficients of x, y, z, and d the known term. mon denominator, take the denominator ab-ba, which agrees to the case of two unknown quantities (excepting the accents); introduce the letter c into each of the terms ab and ba, in the following places; viz. on the right, in the middle, and on the left, then interpose alternately the positive and negative signs; the result will be abc-acb+cab-bac+bca-cba. Now over the second letter of each term place the accent ('), and over the third letter the accent ("). The denominator will be ab'c"-ac'b", +ca'b"-ba'c" + bc'a" + cb'a". 2d. To form the numerator for the value of each unknown quantity, take the denominator, and replace the letter denoting the coefficient of this unknown, by the letter which denotes the Thus for x, change known term, leaving the accents the same. a into d; for y, b into d; and for z, cinto d. This law which may be regarded as resulting from observation, in the case of two or three equations, can be extended to any number of equations, but the demonstration of it is very complicated. (See the second part of Garnier's Algebra.) 68. To show the use of these formulas in the applications to particular cases, we will take the equations 5x-7y=34, 3x-13y=-6. Comparing them with the two general equations ax+by=c, a'x+b'y=c', we have a=5, b——7, c=34, a'=3, b'=—13, c'=-6. Substitute these values in the place of a, b, c, a', b', c', in the formulas 1st. x= 34x-13-(-7) × −6 -34 x 13-7x6 5x-6-34x3 2d. y=5x-13-(−7) × 3 And the values x=11, and y=3, will satisfy the proposed equations. We might assure ourselves of this at once, by substituting them in the equations. But, in order that, the demonstration may be independent of any particular example, we will remark that we may pass from the formulas relative to the equations ax+by=c, a'x+b'y =c' to those which agree to the equations ax-by=c, and a'x-b'y=-c', by (65) changing b to −b, b' to-b' and c' to-c' which gives x= cx —b'-(—b) × −c' y= ax-c'-cxa' ax-b'-(-b)× a' ' ax-b'-(-b) xa' and to deduce from these new general formulas, the values which agree to the particular equations, it will be necessary to make a=5, b=7, c=34, a'=3, b'=13, c'=6. Hence, the values relative to the proposed equations may be obtained from the first general formulas, by making a=5,b=—7, c=34,a'=3, b'=-13, c'— — 6, then performing the operations indicated, by the rules established for monomials. In general, the rule is to substitute in place of the coefficients a, b, a', b' - --, their values considered, with the same signs with which they are affected in the particular equations, and then perform the operations indicated. Hence we again perceive the necessity for admitting the rule for the signs relative to monomials, if we wish to render the general formulas of the first degree, applicable to each particular example. |