Exercise 9 1. Suppose that the left pan of a balance contains x+5 pounds and the right pan 15 pounds. Let the 5 pounds be taken from the left pan. Then what must be done to the right pan to maintain a balance? Then x equals how many pounds? 2. The left pan contains x-12 pounds and the right 7 pounds. Add 12 pounds to the left pan. Then what must be done to maintain a balance? Then x equals how many pounds? 3. If 3x pounds just balance 12 pounds, x pounds will balance how many pounds? What must be done with 3x to get x? What must be done with 12 to get the number that balances x? 4. Answer similar questions if 5x pounds balance 20 pounds, that is, if 5x=20. 5. Answer similar questions if pounds balance 12 pounds, that is, if x=12; if x=8; if 7x=21. 6. If the weights in the two scale pans balance, tell what must be done to the weights in the other pan (a) If a certain number of pounds is added to the weight in one pan. (b) If a certain number of pounds is subtracted from the weight in one pan. (c) If the weight in one pan is divided by a certain number. (d) If the weight in one pan is multiplied by a certain number. 7. If we wish to keep the two members of an equation equal, what must be done to one member (a) If a certain number is added to the other? (b) If a certain number is subtracted from the other? 10. Principles concerning equal numbers. In solving equations much use is made of certain principles concerning equal numbers. Exercise 10 1. In a certain room there are d seats in desks and 5 chairs, in which to seat m men and 10 women. What is the total number of seats? What is the total number of persons? 2. Suppose that the number of seats is the same as the number of persons. State this fact as an equation. 3. If three more persons enter and three seats are brought in, how many persons are there? How many seats? How does the number of persons compare with the number of seats? State this as an equation. How 4. If now seven persons leave and seven seats are taken out, how many seats and how many persons remain? do these numbers compare? State this fact in symbols. 5. If a and b+2 are equal numbers and 3 is added to each of them, what are the results? Which is the greater? State this fact in symbols. 6. If n is added to each of two equal numbers what is true of the sums? 7. If k is subtracted from each of two equal numbers what is true of the remainders? 8. I am thinking of two equal numbers. I add the same number to each of them. What is true of the sums? 9. I am thinking of two equal numbers. I subtract the same number from each of them. What is true of the re mainders? 10. Suppose that a and b are equal numbers. Multiply each by 6. What are the products? How do they compare in value? State this fact in symbols. 11. If each of two equal numbers is multiplied by the same number what is true of the products? 12. If each of two equal numbers is divided by the same number what is true of the quotients? It has been illustrated in the above exercises that if we start with equal numbers, we shall obtain equal results if we add to them the same or equal numbers; subtract from them the same or equal numbers; multiply them by the same or equal numbers; divide them by the same or equal numbers. These truths are usually stated in the following form: Principle I. Equal numbers added to equal numbers give equal sums. Principle II. Equal numbers subtracted from equal numbers give equal remainders. Principle III. Equal numbers multiplied by equal numbers give equal products. Principle IV. Equal numbers divided by equal numbers give equal quotients. We shall find much use for these principles in what follows. Exercise 11 1. Which of the above principles is illustrated in exercise 5, page 20? In exercise 7? In exercise 10? In exercise 12? Give the reason for the conclusion in each of the following: 2. a=4, b=5; then a+b=9. 11. Operations used in solving equations. The principles just given are used in the operations performed in solving equations. The pupils must now learn which of the operations to perform in solving a given equation. The operations to be used are those which, in the given equation, will result in leaving the unknown standing alone as one member of the equation. Exercise 12 1. What number is 6 greater than x? 7 greater than y? 9 greater than 2x? 2. Which is the greater b or b+2? How much? Answer similar questions about a and a+5; 2x and 2x+8; 5m+75 and 5m; 8r+45 and 8r. 3. What number is 4 less than m? 8 less than x? 16 less than 3y? 68 less than 5x? 4. Which is the greater a-2 or a? swer similar questions about r-3 and r; and 3v-21; 6z and 62-80. How much? An y-47 and y ; 3v 5. What must be added to or subtracted from : (a) x-12 to get x? (b) x+12 to get x? (c) 3x+7 to get 3x? (d) 9y-53 to get 9y? 6. What must be done to a+32 to get a? To r-14 to get r? To 8t-27 to get 8t? To 31b+965 to get 31b? 7. Suppose that a+1-6. What must be subtracted from a+1 to get a as a result? If the same number is subtracted from 6 what is then true of the two results? What is a then equal to? Which of the above principles may be used here? 8. Ask and answer questions similar to those in the previous exercise if a+7=12. 9. If n+5=20, what can be done to both members of the equation to find the number that n equals? Answer a similar question if y+14=36. 10. Suppose that x-5=7. What must be added to x-5 to get x as a result? If the same number is added to 7 what is true of the two results? Hence what number is x equal to? Which of the above principles have you used? 11. Ask questions similar to those in the previous exercise if x-9=17. 12. If y-3-8, what can be done to both members of the equation to find the number that y equals? Answer a similar question if y-10-22. What principle applies? 13. Tell what each of the following must be divided by to get x 12x; 9x; 70x; 956x; ax; 8.6x; 4.5x ; .6x; 21x ; 3x; x; fx. 14. In the equation 2n=10, by what must 2n be divided to get n as a quotient? Then n equals what number? Which of the above principles is used here? 15. Ask questions similar to those in the last exercise if 3n=24; if 7n=14. 16. The two numbers 3x and 30 are equal. Each is divided by 3. What are the quotients? How do the values of the quotients compare? Then x equals what number? 17. If you know that 7a=56, how can you find the number that a equals? What principle is used? 18. By what must each of the following be multiplied to get y y; y; ty; Tooy; .ly; .01y? 19. In the equation x=6, what must be multiplied by to get x as a product? Then x equals what number? What principle is used? 20. Answer questions similar to those in the last exercise if x=7; if x=20. 21. The numbers in and 2 are equal. Each is multiplied by 6. How do the values of the products compare? Then n equals what number? What principle is used? 22. If you know that 4y=14, how can you find what number y equals? What principle is used? |