operations as it would be necessary to employ in order to verify the value of the unknown quantity if that value were determined. Two different algebraic expressions having the same assumed value, and involving the symbol of the unknown quantity, are formed by these means; and the equation of the question is obtained by putting these expressions equal to each other. If the question involves more unknown quantities than one, and these are not given in terms of each other, for each unknown quantity the algebraic translation of the question must contain (as will be hereafter shown, Art. 50,) two different expressions of the same assumed value. 47. The second part of the solution of a question, in which the values of the unknown quantities are deduced from the equations obtained by the first part of the solution, is made by fixed and invariable rules. To resolve an equation it is necessary to disengage the unknown quantity from the known quantities with which it is combined, for the purpose of obtaining an equation of which the unknown quantity alone shall form one member (generally the first) and given quantities the other member. The unknown quantity may be combined with the known quantities by 1st. Addition, as x+a=b, or by several of these forms of combination, or by all of them. a. It has been already stated as a general principle that if two quantities are equal, and the same arithmetical operation is performed upon them both, the results are equal. Therefore, if from both members of the first equation a is subtracted, the remainders are equal; that is, b. And in like manner, if to both members of the second equation c is added, the sums are equal; or, Therefore, to free the unknown quantity from known quantities connected with it by the sign of addition, subtract these quantities from both members of the equation; And to free the unknown quantity from known quantities connected with it by the sign of subtraction, add these quantities to both members of the equation. Hence, a term which is connected additively or subtractively with one member of an equation may be cancelled in that member, provided it is introduced into the other member with a contrary sign. By this means the known quantities which are connected with the unknown quantity by the signs + or are transferred to the other side of the equation. An unknown quantity found in the second member is transferred to the first by the same process, which is called Transposition of terms. A term found in both members with the same sign disappears by the effect of transposition. c. If both members of the third equation are divided by m, the quotients are equal; or d. And if both members of the fourth equation are multiplied by d, the products are equal; or xd=qd, or x=qd. Whence, to free the unknown quantity from known quantities connected with it by the sign of multiplication, divide both members of the equation by the multiplier of the unknown quantity; And to free the unknown quantity from known quantities connected with it by the sign of division, multiply both members of the equation by the divisor of the unknown quantity. Consequently, the known quantities which are connected with the unknown quantity by multiplication or division are disengaged by dividing or multiplying all the terms of the equation by these quantities. By these principles a given quantity which is a factor of all the terms of an equation, or of both members of an equation, may be cancelled. The signs of all the terms of an equation may be changed by multiplying every term by -1. The denominators (being given quantities) of any terms, whether in x or not, may be taken away. 48. Equations in which the unknown quantity is connected with the given quantities by several or all of these forms of combination are resolved by the successive application of the appropriate rules. Thus in the equation 3x+5=20-2x, the terms 5 and 2x are transposed by Articles 47 a, b, and the coefficient of x is made to disappear by Article 47. c. The operation is represented thus, Similarly, in the equation ax+b-cx=d-ax+g, the unknown quantity ax is transferred to the first member of the equation, and the given quantity b to the second, by the rules of Article 47. After these transpositions the equation is, ax-cx+ax=g—b. To disengage the known quantities a and c from x, it is to be observed that ax-cx+ax=2ax-cx=(2a-c)x. Now, if both members of the equation are divided by 2a-c, the coefficient of the unknown quantity, x alone is left in the first member, and the second becomes g-b whence x= g-b 2a- -c 2a-c Consequently, when the terms in r have been transferred to one side of the sign, and the given quantities connected with x by the sign + or to the other side, the given quantities connected with x by multiplication may be formed into a coefficient of x. Dividing both members of the equation by this coefficient, an equation is obtained of which the first member is x, and the second is composed of known quantities. To obtain the value of r in the equation x 9 21 x 324 6' it is necessary, by Rule 47, to multiply the terms of the equation by 3, to multiply the terms of the resulting equation by 2, &c., in the manner following, 9 = 21 x 2 4 6' 27 63 3x x + 2 2x+27= Multiplying by 3, 2, 4, 6, Transposing, 126 6x 4 6 24x 6 756-24x. 8x+108 126 - 48x+24x=756-648, or 72x=108. 108 3 x= 72-2-13. By dividing the denominators, when possible, instead of multiplying the numerators, the result is simplified. obtained are, Thus the equations successively = From multiplication of 1st equation by 3, x+2-42 27 63 x The most simple method, however, of removing the denominators, when several terms of an equation have a fractional form, is to reduce all the terms to the same denominator (which does not disturb the equation), and to cancel the denominators, by Article 47 d. In reducing the terms to the same denominator, the abridged processes of Article 30 a. may be employed. Resuming the last example; the least common multiple of the denominators is 12; reducing all the terms to the common denominator 12, 4x 54 12+ 12 = 63 2x 12 12' 4x+54=63-2x. 6x=9 and x=&=1}. 49. These principles (Arts. 47 and 48) apply to equations of all degrees, and are themselves sufficient for the resolution of equations of the first degree with one unknown quantity. General rule for the resolution of equations of the first degree with one unknown quantity. 1st. Remove the denominators of all the terms which have a fractional form, and execute in both members of the equation all the algebraic operations which are indicated. Each member of the equation is thus rendered an integer polynomial. 2d. Transpose to one member all the terms which contain the unknown quantity, and to the other member the known terms. 3d. Reduce to a single term all the terms in a, if the equation is numerical; and if literal, reduce all these terms to a single product composed of two factors, of which one is x, and the other the algebraic sum of all the quantities by which a is multiplied. 4th. Divide both terms of the equation by the numerical or literal coefficient of x, and perform the division of the second term, if it is possible. Find the value of x in each of the following equations: 4a2-3ab 16th. 2ax-bx+2ab=4a2 — ab—3ax ?..................... Ans. x= 5a-b 7ab-3a2 17th. (3a-x)(a−b)+2ax=4b(a+x)? .........Ans. x= a 9x Ꮖ 5x a-36. 39ab-14a2 +3- = + ?Ans. x=27ab-9b+12. +3ab=0? ac(1-3ab) ..Ans. x= c-ad. 50. When two equations of the first degree involving two unknown quantities are given, the method employed to determine the values of these quantities is to deduce from the two proposed equations a new equation containing only one unknown quantity, and from which the value of this unknown quantity may be obtained. The process by which one of the unknown quantities is banished is named Elimination. The elimination of one of the unknown quantities from two equations which contain two unknown quantities can be effected in different ways. To explain these, let be two equations which contain the unknown quantities, x, y, and the known quantities, a, a, b, b', c, c', which may be either additive or subtractive. Assuming that there are two numbers which, being substituted for x, y, satisfy equations 1, 2, and considering r, y as the representatives of these numbers, the equations 1, 2 may be regarded as equalities. c-by Now these equations give x= a c'-b'y x= |