23. What is meant by solving an equation? 24. To solve the equation x+7=10, what is done to each member? Answer the same question for the equation n-9 =6; for the equation 5x=45; for the equation y=4. 25. Give an equation that may be solved by adding the same number to both members; one that can be solved by subtracting the same number from both members. 26. Give an equation that can be solved by multiplying both members by the same number; one that can be solved by dividing both members by the same number. Exercise 13 State what must be done to find x, and find it, in the following. State which principle is used. 1. x+4=19. SOLUTION. Subtract 4 from each member of the equation. Principle 2. Then x=15. Check. 15+4=19. 2. x-9=11. . SOLUTION. Add 9 to each member of the equation. Principle I. Then x=20. Check. 20-9=11. 3. x=8. SOLUTION. Multiply each member by 7. Principle III. Then x=56. Check. of 56=8. 4. 23x=57. SOLUTION. Divide each member of the equation by 2}. Principle IV. Then x=24. Check. 23×24=57. 12. Solving equations. In the following solutions you will understand how to go from one equation to the next if you can answer the following questions: (a) What is done? (b) Why is it done? (c) Why are the results equal? The answers to the above questions when going from (1) to (2) are: (a) 9 is subtracted from each member of the equation. (b) So that the terms not involving n shall not be found in the left member. (c) Equal numbers subtracted from equal numbers give equal remainders. The answers when going from (2) to (3) are : (a) Each member of the equation is divided by 4. (b) So that n shall stand alone in one member. (c) Equal numbers divided by equal numbers give equal quotients. In the following solution the answers to questions (a) and (c) only are given. The pupils should be able to answer question (b). EXAMPLE 2. 7x-5=27+3x. SOLUTION. 7x-5=27+3x. 7x=32+3x. Adding 5 to each member. Principle I. 4x=32. Subtracting 3x from each member. Principle II. 56-5=27+24. 51=51. Exercise 14 Solve each of the following equations for the letter involved. Be prepared to explain the solution as in example 1. Check 11. 106=6. 28. 12y+14=3y+77. 29. 5a+4=4+a. 30. 7r+6=44+3r. 31. 8a=16. 32. .8a=16. 38. 1000n=.001. 39. .001n=1000. 40. b=7. = 20. 9x-3x=81. 33. .08a=16. 34. 2.5n=50. 35. 200r 2.4. 36. 1.46 42. 37. .02y+.06y=36. 41. x=97. 42. k=6. 53. x-x=18. 54. 1.06x+.08x=2508. 55. 23y-y-20=0. 56. 34-44 0. 45. x=5. 52. x+x=20. 57. 5.65x+3.05x=21.75. 58. 9n-22-16-3n. 59. 48+24n=18n+288. 60. .01a+.001.001a+.01. 13. Translating sentences into equations. One of the principal reasons for studying algebra is to learn how to solve problems by the use of equations. One of the chief difficulties in solving problems is expressing the conditions of the problem in equations. This amounts to translating the English sentence into an equation. The pupils should have much practice in such translating. Exercise 15 Express the following statements as equations. 1. If 4 is added to a number the result is 7. SOLUTION. If n is the number, then n+4=7. 2. If 9 is subtracted from ≈ the result is 20. X 3. Seventeen times x is 35. 4. If 12 is subtracted from a the result is 46. 5. If x is divided by 6 the result is 17. 6. If 19 is subtracted from three times b the result is 40. 7. The sum of x and y is z. 8. If a is subtracted from b the remainder is r. 9. If m is subtracted from 4 times k the remainder is c. 10. There are s students in a school. If n leave there will be 34 remaining. 11. An investor has d dollars. He gains g dollars. He then has a dollars. 12. The perimeter of a square of side s is 18. 13. The perimeter of an equilateral triangle of side s is 24. 14. The perimeter of a rectangle of length l and width w is 200. 15. If 95 is subtracted from 42 times n the result is 658. 16. If x is added to its double the sum is 930. 17. One-fourth of n is 35. 18. Twenty hundredths of n is 8. 19. Twenty per cent of n is 8. SUGGESTION. In such a case it is convenient to express the per cent as a decimal. The answer is then written, .20n =8. 20. Twelve per cent of x is 60. 21. A invests m dollars. He gains 20% of this investment. He then has $480. 22. If 35% of n is added to n the sum is. 540. 23. If 60% of x is subtracted from x the remainder is 428. 24. A man has d dollars and spends 40% of it. He then has $90. 25. Sixty per cent of m is 4 more than b. 26. If 10% of x is added to 3 times x the sum is 372. 27. Twice n plus 15% of n plus 1% of n is 55. 28. George is x years old. In 6 years he will be 18 years old. 29. George's father is n years old. Ten years ago he was 28 years old. 30. William is b years old. In y years he will be 27 years old. 31. A man walks at the rate of 3 miles an hour, and travels m miles in 5 hours. 32. A man walks at the rate of 3 miles an hour and goes m miles in h hours. 33. A man walks at the rate of r miles an hour and goes m miles in h hours. 34. A train runs a distance of d miles in h hours when running at the rate of r miles an hour. 35. An automobile goes a distance of 139 miles in h hours while running at the rate of m miles an hour. 36. Write an equation that gives the distance, d, that an object moves in t seconds at the rate of r feet a second. 37. Write an equation that gives the number of seconds, t, in which an object will move a distance of d feet at the rate of r feet a second. 38. Write an equation that gives the number of feet, r, that a body travels in one second, if it travels d feet in t seconds. 39. When a is divided by b the quotient is q. 40. Twice n plus of n minus of n equals 146. 41. One-third of x plus 20% of x equals 98. 42. The cost of an article, c, plus 35% of the cost equals $12.45. |