ELEMENTARY ALGEBRA CHAPTER I INTRODUCTION THE LANGUAGE OF ALGEBRA. THE EQUATION 1. Algebra, like arithmetic, deals with number. Algebra, however, goes farther than arithmetic, and enables one to solve with readiness problems which cannot be solved by arithmetic. Algebra makes use of the symbols of arithmetic. Thus, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, are used in algebra to represent numbers in exactly the same way that they are used in arithmetic. In algebra the sign of addition is +; the sign of subtraction is; the sign of multiplication is x, or a period; the sign of division is, /, or the fractional notation. In arithmetic it is customary to use abbreviations for concrete quantities. Thus, instead of the word cent or cents, the letter c. is used. For example, 54 cents is written 54. Instead of the word acre or acres, the letter A. is used. Thus, 5 acres is written 5 A. word ton or tons, the letter T. is used. Instead of the Thus, 10 tons is written 10 T. In the metric system of measures the linear unit is the meter. The abbreviation for meter is m. Thus, 12 meters is written 12 m. A meter is, approxi mately, 39.37 inches. 12 x 39.37 inches. Hence, 12 m. is equivalent to 2. In algebra, symbols are carried farther than in arithmetic. In algebra, a letter of the alphabet may stand for any number whatever. When a number is written before a letter, it indicates multiplication. Thus, 4 a means 4 times a. If a stands for 5, then 4 a stands for 4 times 5. If a stands for 90, then 4 a stands for 4 times 90. Letters written in succession indicate multiplication. Thus, ab means a times b. abc means the continued product of the numbers a, b, and c, that is, a times b, and this result times c. If a = 4, b=7, and c= 8, then abc-4.7.8, or 4 x 7 x 8. 3. Each of the letters in an indicated product is called a factor of the product. Any one factor is called a coefficient of the number expressed by the remaining factors. Thus, in the indicated product, 9x, 9 is the coefficient of X. In the indicated product, 4 ab, 4 is the coefficient of ab, and 4 a is the coefficient of b. When no coefficient is written, 1 is understood. In the expression x + 3, 1 is the coefficient of x. The sum of 4. Algebra uses a language of its own. two numbers, 7 and 4, is written 7+4. The sum of two numbers, x and y, is written x+y. The difference of two numbers, 18 and 7, is written 18-7. The difference of two numbers, a and b, is written a b. The quotient of 56 divided by 11 is written. The quotient of a divided by b is written a2 is a short method of writing a times a, or aa. a3 is a short method of writing a times a times a, or aaa. at is a short method of writing a times a times a times a, or aaaa. The number indicating how many times a number or letter is taken as a factor is called an exponent. Thus, in a1, 4 is the exponent. It is important that the student distinguish carefully between coefficient and exponent. 4 a means 4 times a. If a = 10, 4a= 40, and a1 = 104 = 10,000. EXERCISE I 1. If x=7, find the value of 2x, 7x, 11x, 19 x, 25 x. 2. If x=11, find the value of 8x, 12 x, 15 x, 24 x, 32 x. 3. If x 8, find the value of 5 x, = 9x, 7x, 14 x. 16 a, 20 a, 30 a. a2, 3 a, a3, 5 a. 6. If b = 12, find the value of b2, 3 b2, b3, 4 b2, 7b2. 8. If a = 4, b = 3, find the value of a+b, a−b, 2 a+b, 3 a-b. 9. If a 4, b= 7, find the value of ab, 5 ab, = 7 a a2 a3 b' b' b = 10. If a 12, 63, find the value of ab2, b' 46' 8b 11. If a = 6, b=1, find the value of a2+ b2, a2 — b2, a2 - ab. 12. If a = 7, a2 - ac, ac2. c = 3, find the value of 2 ac, a2 + ac, 13. If x=8, y=4, find the value of 2, 22 y y 5. In arithmetic a statement in symbols that one quantity equals another quantity is called an equation. (The algebraic meaning of equation will be given later.) For example, 5x + 5 = 83 is an equation. The part of an equation to the left of the sign of equality is called the first member of the equation, the part of the equation to the right of the sign of equality is called the second member of the equation. EXERCISE 2 Find the value of x in each of the following equations: 1. 6x=24. 5. 18x=90. 9. 17x=187. 13. 4x-2. 2. 7x=56. 6. 5x=120. 10. 14x=210. 14. 6x=3. 3. 9x=63. 7. 4x=192. 11. 13x=78. 4. 15x=45. 8. 11x=143. 12. 15x=5. 15. 8x=52. 16. 12x=78. EXERCISE 3 1. How many are 5 apples and 8 apples? How many are 5 a and 8 a? 2. How many are 7 horses and 8 horses? How many are 7 h and 8h? 3. How many are 11 cents and 7 cents? How many are 11 c and 7 c? 4. How many are 15 dollars and 8 dollars? How many are 15d and 8 d? 5. How many are 13 acres and 9 acres? How many are 13 a and 9 a? 6. How many are 17 yards and 8 yards? How many are 17 y and 8y? 7. How many are 19 boys and 11 boys? How many are 196 and 116? 15. What is the sum of 4 ab and 2 ab? 16. What is the sum of 12 bc and 9 bc? 17. What is the sum of 5 c2 and 3 c2? 18. A man had x dollars and earned y dollars. How many dollars did he then 19. Take 4 x from 10 x. have? 29. By how much does 12 exceed 7? |