Addition of Similar Monomials. 1. Find the sum of 3 a, 2 a, a, 5a, 7 a. The sum of the coefficients is 3+2+1+5+ 7 = 18. 2. Find the sum of 5c, c, 3c, 4c, 2 c. - The sum of the coefficients is 5-1-3-4-2 = Hence, the sum of the monomials is 15 c. - 15. 3. Find the sum of 8x, - 9x, — x, 3x, 4x, — 12x, x. -- The sum of the positive coefficients is 8 + 3 + 4 + 1 = 16. The sum of the negative coefficients is 9-1-12 = — 22. The difference between 16 and 22 is 6, and the sign of the greater is negative. Hence, the sum is 6x. Therefore, 72. To Find the Sum of Similar Monomials, Find the algebraic sum of the coefficients, and annex to this sum the letters common to the terms. 7.3xz, 7 xz, -2xz, xz. 15. c2, 2 c2, -15 c2, - 18 c2. - 9 ac, 3 ac. 16. 4m2, 11m2, -7 m2, 5m2. Express in one term each of the following: 17. 9x27 x2+4x2-3x2 + 3x2- 5 x2. 18. 3a2 18 a2 + a2 - 5 a2+6a2 - 10 a2. 19. 5a3x+7 a3x-9 a3x-29 a3x + 4a3x. 20.5 a22 + 7 a22 + 11 a22 - 4 a2b2 9 a2b2. - 21 ax3 +20 ax3 -- - 11 abcx + 3 abcx 3y-27 y*z* — 2 y*z* + 41 y*z* + y*z*. 24. -4x3y+18 x3y3 + 27 x3y3 — 43 x3y3 — x3y3. 25. - 31 abz3 + 17 abz3· 26. 47 abz3 +61 abz3 + abz3. aa75a + 8 a − ‡ a + a. 27. § a2-5a2 + § a2 + 7 a2 — § a2 + † a2. 28. xy - xy + ‡ xy − xy + } xy − }} xy. 29. -- 2 x + x + 30.ayz ayz + -- } } ≈ - § z. ayz + 31. 14 a7a-3 a 15 a + 12 a. ayz - 7 ayz. 32. 4 a'c10 a2c + 6 a2c - 9 a2c + a2c. 33. 3x2y 4x2y + 2x2y - x2y + 5 x2y. Subtraction of Algebraic Numbers, 73. In order to subtract one algebraic number from another, we begin at the place in the series which the minuend occupies, and count, in the direction opposite to that indicated by the sign of the subtrahend, as many units as there are in the absolute value of the subtrahend. Thus, the result of subtracting +1 from + 3 is found by counting from +3 one unit in the negative direction; that is, in the direction opposite to that indicated by the sign +before 1, and is, therefore, +2. The result of subtracting -1 from +3 is found by counting from +3 one unit in the positive direction, and is, therefore, + 4. The result of subtracting +1 from 3 is found by counting from 3 one unit in the negative direction, and is, therefore, 4. The result of subtracting counting from 3 one unit in the positive direction, and is, therefore, -2. If a and b represent any two numbers, we have 74. From these four cases we see that subtracting a positive number is equivalent to adding an equal negative number; and that subtracting a negative number is equivalent to adding an equal positive number. Therefore, 75. To Subtract One Algebraic Number from Another, Change the sign of the subtrahend, and add the result to the minuend. 76. To Subtract a Monomial from a Similar Monomial, Change the sign of the coefficient of the subtrahend; then add the coefficients, and annex the common letters to the result. Multiplication of Algebraic Numbers. 77. Multiplication is generally defined in Arithmetic as the process of finding the result when one number (the multiplicand) is taken as many times as there are units in another number (the multiplier). This definition fails when the multiplier is a fraction. In multiplying by a fraction, we divide the multiplicand into as many equal parts as there are units in the denominator, and take as many of these parts as there are units in the numerator. If, for example, we multiply 6 by , we divide 6 into three equal parts and take two of these parts, obtaining 4 for the product. The multiplier, 3, is of 1, and the product, 4, is of 6; that is, the product is obtained from the multiplicand precisely as the multiplier is obtained from 1. 78. Multiplication may be defined, therefore, As the process of obtaining the product from the multiplicand as the multiplier is obtained from unity. 79. Every extension of the meaning of a term must be consistent with the sense previously attached to the term, and with the general laws of numbers. This extension of the meaning of multiplication is consistent with the sense attached to multiplication when the multiplier is a positive whole number. |