The rules of Algebra, therefore, make not only known certain contradictions, which may be found in the enunciations of problems of the first degree; but they still indicate their rectification, in rendering subtractive certain quantities which we had regarded as additive, or additive certain quantities which we had regarded as subtractive, or in giving for the unknown quantities affected with the sign.. Hence, it follows, that we may regard as forming, properly speaking, but one question, those whose enunciations are connected to one another in such a manner, that the solution which satisfies one of the enunciations, can, by a simple change of the sign, satisfy the other. We must nevertheless observe that we can make upon the signs and values of the terms of an equation, hypotheses which do not agree with the enunciation of a concrete question, whereas the change which we will make in this enunciation might be always represented by the equation. These principles, which will be illustrated by examples, are applicable to equations of all degrees, and to determinate equations containing many unknown quantities. The question which conducts to the equation ax+b=cx+d, is not well enunciated for a>c, and b>d, since the first member is greater than the second. gives for a negative value; but by rendering the unknown negative, the equation is changed into the following, b-ax=d-cx, which is possible under the above relations between a and c, b and d, and which gives then for x an absolute value. If we have b>d and c>a, the two subtractions become impossible in the formulæ d-b a-c but in order to resolve the equation, let us subtract ca+b from both members, which would be impossible, because that cx+b is greater than each of the two members: we must therefore, on the contrary, take away ax+d from both sides, and it becomes b―d—cx—ax ;* from whence we deduce b-d Ca This formulæ, compared to the preceding, differs from it in this, that the signs of both terms of the fraction, are changed. We may therefore conclude, that we can operate on negative isolated quantities, as we would do if they had been positive. 199. These principles will be clearly elucidated, when we come to treat of the solutions of Problems producing simple Equations: we shall now proceed to illustrate the Rules in the preceding Section, by a variety of practical examples. Multiplying both sides of the equation by 16, the least common multiple of 16, 8, and 2, we shall 336+3x-11=10x-10+776-56x; have .. by transpostion, 3x-10x+56x-11-10+776-336, value of x. Multiplying both sides of the equation by 6, the product of 2 and 3, which is the least common multiple, we have 6x+9x-15=72-4x+8; by transposition, 6x+9x+4x=72+8+15, 3 12x- -24 -(4x In this example, when the fraction multiplied by 5, the result is 8)=4x+8, or which is the same thing, when the sign stands before a fraction, it may be transformed, so that the sign + may stand before it, by changing the sign of every term in the numerator; therefore, we make the above step -4x+8, and not 4x-8. 40-5x+510x+4x-4+240, by transposition, 40x-5x-10x-4x-240-4—5, or, 40x-19x=231; or -7x=6, by transposition, 4x-x-10x=-4-2, Ex. 5. Given 3ax-2bx=3b-a, to find the value of x. Here, 3ax-2bx=(3a-2b)x, by collecting the coefficients of x. Therefore, Ex. 6. Given bx+x=2x+3a, to find the value by transposition, 3bdx+adx-4abdx-2abx=abcd, or (3bd+ad-4abd-2ab) x=abcd, .. by division, x= abcd 3bd+ad-4abd-2ab* of x. product becomes Multiplying by 30, the product of 5 and 6, the 6x-5x+5a=30b+30c; by transposition, 6x-5x=30b+30c-5a, Multiplying by 9, the least common multiple, 96-8x=45x-42—3x, by transposition, -45x-8x+3r=−96—42, by changing the signs, 45x+8x-3x=96+42, or 50x=138, Multiplying by 12, the least common multiple of the denominators, and the equation will become, 3ax-3b+4a=-6bx-4bx+4a, (1). by taking away 4a from each member, we shall have 3ax-3b=6bx-4bx=2bx, |