Parentheses. 39. A parenthesis preceded by the sign +. If a man has 10 dollars and afterwards collects 3 dollars and then 2 dollars, it makes no difference whether he puts the 3 dollars and the 2 dollars together and adds their sum to his 10 dollars, or adds the 3 dollars to his 10 dollars, and then the 2 dollars. The first process is represented by 10+ (3 + 2). Hence, 10 + (3 + 2) = 10 + 3 + 2. (1) If a man has 10 dollars and afterwards collects 3 dollars and then pays a bill of 2 dollars, it makes no difference whether he pays the 2 dollars from the 3 dollars collected and adds the remainder to his 10 dollars, or adds the 3 dollars collected to his 10 dollars and pays from this sum his bill of 2 dollars. The first process is represented by 10+ (3-2). Hence, 10 + (3-2) = 10 + 3 – 2. If we use general symbols in (1) and (2), we have, and a + (b + c) = a + b + c, a + (b - c) = a + b − c. (2) Hence, We have the general rule for a parenthesis preceded by +: If an expression within a parenthesis is preceded by the sign +, the parenthesis may be removed without making any change in the signs of the terms of the expression. Instead of a parenthesis, any other sign of aggregation may be used and the same rule will apply. 40. A parenthesis preceded by the sign -. If a man with 10 dollars has to pay two bills, one of 3 dollars and one of 2 dollars, it makes no difference whether he takes 3 dollars and 2 dollars at one time, or takes 3 dollars and 2 dollars in succession, from his 10 dollars. The first process is represented by 10 (3+2). Hence, 10 - (3 + 2) = 10 — 3 — 2. (1) If a man has 10 dollars consisting of two 5-dollar bills, and has a debt of 3 dollars to pay, he can pay his debt by giving a 5-dollar bill and receiving 2 dollars. This process is represented by 10 -5 +2. Since the debt paid is 3 dollars, that is, (5-2) dollars, the number of dollars he has left can be expressed by Hence, 10 (5-2). 10- (5-2) = 10 – 5 + 2. If we use general symbols in (1) and (2), we have, a - (b + c) = a - b — с, and a - (b - c) = a - b + c. (2) Hence, We have the general rule for a parenthesis preceded by -: If an expression within a parenthesis is preceded by the sign, the parenthesis may be removed, provided the sign before each term within the parenthesis is changed, the sign + to and the sign to +. NOTE. If the vinculum is used, the sign prefixed to the first term under the vinculum must be understood as the sign before the vinculum. and Thus, a + b chas the same meaning as a + (b - c), b-chas the same meaning as a a (b - c). EXERCISE 3. Remove the parentheses and combine: 1.9+ (3+2). 5.9 - (8-6). 2.9+(3-2). 6.8 - (7-5). 3.7+(5+1). 7.9 - (6+1). 4.7+(5-1). 8. 8 - (3 + 2). 9. (3-2) - (2 - 1). 10. (7-3) — (3 – 2). 11. (8-2) - (5 – 3). 12. 15 - (10 -3- — 3 — 2). If a = 10, b = 5, c = 4, d = 2, find the value of: Product of a Compound by a Simple Factor. 41. In finding the product of 4 (5 + 3), it makes no difference in the result whether we multiply the sum of 5 and 3 by 4, or multiply 5 by 4 and 3 by 4 and add the products. By the first process, 4 (5 + 3) = 4 × 8 = 32. By the second process, 4(5+3) = (4 × 5 + 4 x 3) = 32. In like manner, 4 (5-3) = 4 × 2 = 8, 4 (5-3) = (4 × 5-4 × 3) = 8. and In general symbols, a (b + c) = ab + ac, and a (b - c) = ab — ас. This is called the distributive law for multiplication. 42. The order of the factors is immaterial. 4 (5+3) = 4×5+4×3= 32, Thus, and (5 + 3) 4 = 5 × 4+3 × 4 = 32. In general symbols, ab = ba. This is called the commutative law for multiplication. Perform the indicated operations : 1. x + 3 (a - b). 2. x - 3 (a - b). 1. x + 3 (a - b) = x + (3 a − 3 b) = x + 3a-3b. EXERCISE 4. Perform the indicated operations, and find the numerical value of each expression, if a = 5, b = 4, c = 3 : Quotient of a Compound by a Simple Expression. 43. In finding the quotient of (8+4)÷2 it makes no difference in the result whether we divide the sum of 8 and 4 by 2, or divide 8 by 2 and 4 by 2, and add the quotients. By the first process, (8 + 4) + 2 = 12 ÷ 2 = 6. By the second process, (8+4)÷2=(8+2+4+2) = 6. This is called the distributive law for division. Perform the indicated operations: 1. x + (3a +36) ÷ 3. 2. x - (3a +36) ÷ 3. 1. x + (3a + 3b) ÷ 3 = x + (a + b) = x + a + b. 2. x - (3a + 3b) ÷ 3 = x - (a + b) = x - a - b. EXERCISE 5. Perform the indicated operations, and find the numerical value of each expression, if a = 8, b = 4, c = 2: CHAPTER II. SIMPLE EQUATIONS. 44. Equations. An equation is a statement in symbols that two expressions stand for the same number. Thus, the equation 3x + 2 = 8 states that 3x + 2 and 8 stand for the same number. 45. That part of the equation which precedes the sign of equality is called the first member, or left side, and that part of the equation which follows the sign of equality is called the second member, or right side. 46. An equation containing letters, if true for all values of the letters involved, is called an identical equation; but if it is true only for certain particular values of the letters involved, it is called an equation of condition. Thus, a + b = b + a, which is true for all values of a and b, is an identical equation; and 3x + 2 = 8, which is true only when a stands for 2, is an equation of condition. For brevity, an identical equation is called an identity, and an equation of condition is called simply an equation. 47. We often employ an equation to discover an unknown number from its relation to known numbers. We usually represent the unknown number by one of the last letters of the alphabet, as x, y, z; and the known numbers by the first letters, a, b, c, and by the Arabic numerals. 48. Simple Equations. Equations which, when reduced to their simplest form, contain only the first power |