EXERCISE 3. Remove the parentheses and combine: 1. 9+ (3 + 2). 5. 9-(8-6). 9. (3-2) (2 − 1). 2. 9+(3-2). 6. 8-(7-5). 10. (73) — (3 − 2). 3. 7+ (5+1). 7. 9-(6+1). 11. (8-2) (5-3). 4. 7+ (5-1). 8. 8-(3+2). 12. 15-(10-3-2). If a 10, b = 5, c = 4, d= 2, find the value of: = 13. (a+b)+(c + d). 14. (a+b)-(c — d). 15. (ab)-(c — d). 16. (a - b)+(c–d). 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 x 8 = 32. By the second process, 4 (5+3)=(4 × 5+ 4 × 3) = 32. In like manner, 4(5-3)= 4 x 2 = 8, and 4 (5 − 3) = (4 × 5 − 4 × 3) = 8. In general symbols, a (b+c) = ab + ac, and a (b−c) = ab ac. This is called the distributive law for multiplication. 42. The order of the factors is immaterial. Thus, and 4 (5+3) = 4 × 5 + 4 × 3 = 32, (5+3) 45 x 4 + 3 x 4 = 32. In general symbols, ab = ba. This is called the commutative law for multiplication. Perform the indicated operations: 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, (84) ÷ 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+3b) ÷ 3. 1. x + (3a + 3b) ÷ 3 = x x - 2. (3a+3b) +3. + (a + b) = x + a + b. - (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 x 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 of the unknown numbers are called simple equations, or equations of the first degree. Thus, 7x+5=4x+14, and ax + b = c are simple equations in x. 49. Two or more like terms may be combined to form a single like term by uniting their numerical coefficients. Thus, 3 ax + ax = 4 ax; and 5 ax 3 ax = 2 ax. 50. To Solve an Equation with One Unknown Number is to find the unknown number; that is, to find the number which, when substituted for its symbol in the given equation, renders the equation an identity. This number is said to satisfy the equation, and is called the root of the equation. 51. Axioms. In solving an equation, we make use of the following self-evident truths, called axioms: Ax. 1. If equal numbers are added to equal numbers, the sums are equal. Ax. 2. If equal numbers are subtracted from equal numbers, the remainders are equal. Ax. 3. If equal numbers are multiplied by equal numbers, the products are equal. Ax. 4. If equal numbers are divided by equal numbers, the quotients are equal. Ax. 5. If two numbers are equal to the same number, they are equal to each other. 52. Transposition of Terms. It becomes necessary in solving simple equations to bring all the terms that contain the symbols for the unknown numbers to one side of the equation, and all the other terms to the other side. This process is called transposing the terms. The result is the same as if we had transposed the left side to the right side and changed its sign. 2. Find the number for which x stands when x + b = a. Subtract from each side, x + b-bab. Cancel +bb, - b from (Ax. 2) In this case, we have transposed b from the left side to the right side and changed its sign. We can proceed in like manner in any other case. Hence, the general rule: 53. Any term may be transposed from one side of an equation to the other, provided its sign is changed. It follows from axioms 1 and 2 that: 54. Any term that occurs with the same sign on both sides of an equation may be cancelled. Equation (2) is the same as (1) with the sign before each term changed. Hence: 55. The sign of every term of an equation may be changed without destroying the equality. |