Here ax + by, or 4 × 3 + 6 × 7 = 12 + 42 = 54. Here, ab, or 9 X 4 =36, which, divided by c or 3, gives 12, the value of the first term. Then ax, or 9 x 872, from which subtracting d or 8, there remains 64; which, divided by b, or 4, gives 16, the value of the second term. Therefore the sum of the first and second terms is 28. Then ab2, or 9 × 4 x4 = 144, the square root of which is 12, the value of the third term; and this subtracted from the sum of the former terms, because connected by the sign, gives 16, the value of the whole expression. = ADDITION. 38. Addition in Algebra is the method of finding the sum of several algebraic quantities, and connecting them together by their proper signs. This rule is generally divided into three cases. CASE I. To add like quantities with like signs. RULE. Add the coefficients of the several quantities together, and to their sum prefix the common signs and annex the common letter or letters. 294 28a 206 xy 59xy 4a3 2a3 93 —73/x—20xy 21xy-15x+20ab CASE II. To add like quantities with unlike or different signs. RULE. Add all the positive or plus quantities into one sum, and all the negative or minus into another sum; subtract the less of these sums from the greater; to their difference prefix the sign of the greater sum, whether or and annex the common letter or letters. Ex. 1. Ex. 2. Ex. 3. Ex. 4. Ex. 5. ILLLE 2a sum. 2x sum. la sum. +3y sum. 3yx sum. NOTE. In the 6th example, the sum of the positive or plus quantities exceeds the sum of the negative by 11a; consequently the sign is, according to the rule. In the 7th example, the sum of the positive or +(plus) quantities is less by 7a than the sum of the negative or (minus) quantities; consequently, the sign is, according to the rule. In the 9th example, the sum of the positive terms is 23a2, and the sum of the negative ones is 13a; their difference, therefore, is + 10a2, which is the sum required. The other examples are wrought in a similar manner. If the positive and negative quantities be equal, the sum is nothing, and they are said to destroy each other. See example 7, right hand column. 3ab 6x 3аx 5a2 86 ab 2x+7ax +4a2. 26 10ax 2a + 7x Ex. 19. 6ax2+5x+ 6x+ -2ax2 +3ax2 10x+ ·2ab+7x-4ax 3a2+9b + ax2 + 11x+ Ex. 20. 6x+4x2 +4ab 4x aNx 8a2 26 8 Ex. 21. 7√y-4(a+a) Ex. 22. a(a+b) -- 7 Ny-3(a+b) 5 +8x + x2 5x-3x2 +7x-5x2 14x-3x2 CASE III. 6y+2(a+b) -4a(a+b) + ? (a-x) 2y+ (a+b) -2a(a+b) — 8 (a—x) 16/y-4(a+b) To add quantities when some are like and others unlike; or when all the quantities are unlike. RULE. Add the like quantities together, according to cases 1 and 2; connect the unlike quantities in any order, with their proper signs and coefficients prefixed. NOTE. When quantities with literal coefficients are to be added placing the coefficients, with together, it may be done by proper signs, under a their ax+b ax2+bx cx+d (a+c)x+b+d (a+c) x2+(b—d)x vinculum, or between brackets, and then subjoining the common quantity to the sum or difference thus arising, as in the margin. Ex. 1. Ex. 2. x2+adz x2-nz bx2+cez bx-cy 2. Add 5x-3a+b+7, and -4a-3x+26-9, together. |