without destroying the equality. This principle may be stated as follows: If the same number, or equal numbers, be subtracted from equal numbers, the remainders are equal. This law is the subtraction axiom. The solution of the equation w+5=15 may now be arranged in the following brief form: Subtracting 5 from both members we have Check: Left side: 10+5 15 EXERCISES w+5=15 5=5 Solve the following equations, arranging your work as shown in the problem above. Check each solution. Solve the following problems by means of equations: 11. What number added to 82 will give 157? 12. Twice a certain number increased by 24 is 54. What is the number? Solution: Let x be the required number. 13. The sum of two angles is 48°. If one angle is 32° larger than the other, how large is each? 14. A sum of $132 is to be divided so that the larger part is 18 greater than the smaller. Find the two parts. 15. A stick 24 inches long is to be cut into two parts so that one is 6 inches longer than the other. Find the length of each part. Solve the following equations: 22. I am thinking of a certain number. If I treble it and add 6, the resulting number is 30. What is the number? 23. A man has 110 yards of fence with which to inclose a rectangular garden. What are to be the lengths of the sides of the garden if it is to be 24 yards longer than wide? 24. Find two consecutive numbers whose sum is 113. Suggestion: Consecutive numbers are whole numbers which differ by 1, as 3 and 4, 5 and 6. Let x be one of the numbers. Then x+1 is the other. 25. Find two consecutive numbers whose sum is 67. 26. Find three consecutive numbers whose sum is 351. 27. Find two consecutive odd numbers whose sum is 68. Suggestion: Let x be one of the odd numbers. Then x+2 is the other. 28. Find two consecutive odd numbers whose sum is 488. 29. Find two consecutive even numbers whose sum is 230. 30. One angle of a triangle is 32°. The second is 18° larger than the third. How large is each of the three angles? Suggestion: Let x be the number of degrees in the third angle (Fig. 269). In problems 31 to 35 make a sketch before solving: 31. Find the angles of a triangle if the first is 20° greater than the second and the third is 16° less than the second. 32. Find the angles of a triangle if the first is 25° larger than the second and the third is 3 times the second. 33. One of two supplementary angles is 48° larger than the other. Find the angles. 34. One of two complementary angles is 32° larger than the other. Find the angles. 35. The length of a room is 2 feet more than twice the width. The length is 18 feet. Find the width. 36. A boy begins to work in an office with the agreement that he will receive $35 the first month with an increase of $3 each successive month, until a maximum of $65 is reached. In how many months will he be getting $65? 37. Mary and her brother John raised vegetables to sell to the neighbors. Their sales amounted to $36. Mary, having spent only half as much time as John, was to get half as much money as her brother. How is the money to be divided? 158. Use of the addition axiom in solving equations. If one of two complementary angles is 40° less than the other, the two angles may be found as follows: Solution: Let x be the number of degrees in the larger angle. Then x-40 is the number of degrees in the smaller angle. Since the angles are complementary, we have x+x-40=90 If we add 40 to both members of this equation we have 2x-40+40=90+40 or, 2x+40-40=130 ..2x=130 and x = 65 x-40=25. Briefly, this solution may be arranged in the follow When adding 40 to both members of the equation, we are using the following law: If the same number, or equal numbers, be added to equal numbers, the sums are equal. This is known as the addition axiom. EXERCISES In each of the following equations find the exact value of the unknown, and check the results. Arrange the work as shown in the problem above. Solve the following problems by means of equations: 11. A boy is 4 years younger than his brother. The sum of their ages is 16. Find the age of each. 12. One of two supplementary angles is 20° less than the other. Make a sketch of the two angles and find them. 13. If 5 times a number is diminished by 10 the result is 35. Find the number. 14. The perimeter of a triangle is 118 feet. One side is 5 feet shorter than the second, and the third is 12 feet longer than the second. Find the three sides. 15. One of two complementary angles is 22° smaller than the other. Find the two angles. 16. If 4 times a number is decreased by 5, the result is 55. Find the number. 17. A sum of $20 was divided between two persons. One received $4 less than the other. How much did each receive? 18. The width of a basketball court is 20 feet less than the length. The perimeter is 240 feet. Find the dimensions. |