(34.) Like or similar terms are terms composed of the same letters affected with the same exponents. Thus, in the polynomial 4ab+9ab+5a’c—12aʼc, the terms 4ab and Sab are similar, and so also are the terms 5a'c and -12a2c. . But in the binomial 6ab2 +5a2b, the terms are not similar; for, although they contain the same letters, the same letters are not affected with the same exponents. (35.) The reciprocal of a quantity is the quotient arising from dividing a unit by that quantity. 1 Thus, the reciprocal of 3 is ; the reciprocal of a The following examples are designed to exercise the pupil upon the preceding definitions and remarks. (36.) Examples in which words are to be translated into algebraic symbols. Ex. 1. What is the algebraic expression for the following statement? Five times the square of a mul. tiplied by the cube of b?. Ans. 5a’bo. Ex. 2. Six times the square of a multiplied by the cube of b, diminished by the square of c, multiplied by the fourth power of d. Ans. 6a’be -cod. Ex. 3. The second power of a increased by twice Quest.--What are similar terms ? What is the reciprocal of a quan tity ? the product of a and b, diminished by c, is equal to nine times d. Ans. a® +2ab-c=9d. Ex. 4. Three quarters of x increased by five, is equal to two fifths of b diminished by eleven. Ans. Ex. 5. The quotient of three divided by the sum of X and four, is equal to twice b diminished by eight. Ans. Ex. 6. One third of the difference between five times x and four, is equal to the quotient of six divided by the sum of a and b. Ans. Ex. 7. The quotient arising from dividing the sum of a and b by the product of c and d, is equal to four times the sum of 2 and y. Ans. (37.) Examples in which the algebraic signs are to be translated into common language. atb. NX Ex. 1. € +mx=ā Ans. The quotient arising from dividing the sum of a and b by c, increased by the product of m and x, is equal to the quotient arising from dividing n times x by d. Ex. 2. 7a+b(c,d)=x+y. Ans. atb. 2 c. Ex. 3. 3+x'a d+2° Ans. 6+2c Ex. 4. 4 Vab-17= SECTION II. ADDITION. (39.) Addition is the connecting of quantities together by means of their proper signs, and incorpora. ting such as can be united into one sum. If it is required to add a number represented by a to four times itself, we write it 3+4x, which may be reduced to 5x. If it is required to add a number x to m times itself, we write it 2+mx, which two terms can not be united in one. If it is required to add a number represented by a to three times itself, and also four times itself, we write it x+3x+4x, which may be reduced to 8x. If it is required to add a number x to m times itself, and also to n times itself, we write it x+mx+nx, which three terms can not be united in one, and this is called algebraic addition. (40.) It is convenient to consider this subject under three cases. CASE I. When the quantities are similar, and have the same signs. QUEST.What is Addition? How many cases are there in Addition? What is case first ? . C RULE. Add the coefficients of the several quantities togethe er, and to their sum annex the common letter or letters, prefixing the common sign. Thus the sum of 3a and 5a is obviously 8a. . . . Ans. 19a. Ex. 2. What is the sum of 4xy, 8xy, xy, and 3xy ? Ans. 16xy. Ex. 3. What is the sum of 3ab, 7ab, ab, and 12ab? Ans. 23ab. Ex. 4. What is the sum of 2mx, 9mx, 4mx, and 5mx? Ans. 20m2 Add together the following terms: (5.) (6.) (7.) (8.) 25+33 2a+ y 5a + xy 3axt m 56+73 5a+2y® 9a+3xy 12ax+7m 5+23 Sa+3y 3a+8xy 11ax +5m 46+32 4a+by 7a+4xy 4ax+9m Ans. 126+15x. (41.) We proceed in the same manner when all the signs are minus. Thus the sum of — 3a and -5a is – 8a; for the minus sign before each of the terms shows that they are to be subtracted, not from each other, but from some quantity which is not here expressed ; and if 3a and 5a are to be successively subtracted from the same quantity, it is the same as subtracting at once 8a. Quest.--Give the rule. How do we proceed when all the signs are negative ? |