An Equation in which the unknown quantity is found in every term, with different exponents in different terms, may often be reduced to a simple Equation by dividing it by some power of the unknown quantity. (113...5). Thus if 2x3=1Qx2, by dividing by x2 we have 2x=10; hence x=5. Literal Equations. x- -b 27. Given ax-c= to find the value of x. a+c Clearing the equation of fractions, by multiplying it by the denominator a+c, we have Remark. In an identical Equation the unknown quantity has no determinate value, since any quantity whatever may be substituted for it, and the equation will be satisfied. Thus in the equation 3x-5-3x-5, the two members will be equal whatever be the value of x. (113...3) PROBLEMS In Simple Equations of one unknown Quantity. (119.) A Problem is a question proposed for solution; and the solution of a problem by Algebra consists in forming an Equation which shall express the conditions of the problem, and then solving the equation. The general method of forming the Equation of a problem, is, to represent a required quantity by x, or y, &c., and then to perform or indicate the same operations that would be necessary to verify the value of x or y, supposing that value to have been found. 1. What number is that to the double of which if 13 be added, the sum will be 75? Let x represent the required number; then 2x will represent twice the number; and, by the conditions of the problem, the equation will be 2x+13=75. The value of x in this equation is the number required. х Ans. 31. 2. Find a number such that if it be multiplied by 5, and 24 be subtracted from the product, the remainder will be 36. Ans. 12. 3. What number is that to of which if 25 be added, the sum obtained will be equal to the number itself minus 39? Ans. 96. 4. Find a number such that if of it be subtracted from three times the number, the remainder will be 77. Ans. 28. 5. Find what number added to the sum of one half, one third, and one fourth of itself will equal 4 added to twice the number. Ans. 48. 6. Divide the number 165 into two such parts that the less may be equal to of the greater. Let x represent the less part; then 165-x will represent the greater; and the equation will be 165-x x= 10 Ans. 15 and 150. Ans. 25 and 75. 7. Divide the number 100 into two such parts that six times the less may be equal to twice the greater. 8. It is required to divide 75 into two such parts that 3 times the greater may exceed 7 times the less by 15. Ans. 21 and 54. 9. What sum of money is that to which if $100 be added, of the amount will be $400 ? Ans. $500. 10. A prize of $100 is to be divided between two persons, the share of the first being of that of the other. 11. A post is of its 15 feet above the water. What are the shares ? length in the mud, of it in the water, and What is the length of the post? Ans. 36 feet. 12. Find a number such that if it be divided by 12, the divisor dividend and quotient together shall make 64. Ans. 48. 13. In a mixture of wine and cider, of the whole plus 25 gallons was wine, and part minus 5 gallons was cider. What was the whole number of gallons in the mixture? Ans. 120. 14. After a person had expended $10 more than he had $15 more than of it remaining. of his money, What sum had he at first? Ans. $150. 15. Divide the number 91 into two such parts that if the greater be divided by their difference, the quotient may be 7. Ans. 49 and 42. 16. A and B had equal sums of money; the first paid away $25, and the second $60, when it appeared that A had twice as much left as B. What sum had each? Ans. $95. 17. After paying away of my money, and then of what was left, I had $180. What sum had I at first? Ans. $300. 18. A line 37 feet in length is to be divided into 3 parts, so that the first may be 3 feet less than the second, and the second 5 more than the third; what are the parts? Ans. 12, 15, and 10 feet. 19. A can perform a piece of work in 12 days, and B can perform the same in 15 days. In what time could both together do the work? Let x represent the number of days. Then since A could do of the work, and B of it, in 1 day, х will represent the part of the work A could do.in x days, 12 ac will represent the part of the work B could do in x days 15 20. If A could mow a certain meadow in 6 days, B in 8 days, and 5 days, in what time could the three together do it? Ans. 2 days. 21. Out of a cask of wine, which had leaked away a third part, 20 gallons were afterwards drawn, and the cask was then found to be but half full; how much did it hold? Ans. 120 gallons. 22. It is required to divide $300 between A, B, and C, so that A may have twice as much as B, and C as much as the other two together. Ans. A $100, B $50, C $150. What 23. A gentleman spends of his yearly income in board and lodging, of the remainder in clothes, and then has $20 left. is the amount of his income? Ans. $180. 24. A person at the time he was married, was 3 times as old as his wife, but 15 years afterwards he was only twice as old. What were their ages on their wedding day? Ans. 45 and 15 years. 25. Two persons, A and B, lay out equal sums of money in trade; the first gains $126, and the second loses $87, and A's money double of B's; what did each lay out? is now Ans. $300. 26. A courier, who travels 60 miles a day, had been dispatched 5 days, when a second is sent to overtake him, who goes 75 miles a day, in what time will he overtake him? Ans. 20 days. 27. An island is 60 miles in circumference, and two persons start together to travel the same way around it: A goes 15 miles a day; and B 20; in what time would the two come together again? Ans. 12 days. 28. A man and his wife usually drank out a cask of beer in 12 days, but when the man was from home it lasted the woman 30 days; how many days would the man alone be in drinking it? Let x be the number of days; then 1 is the part that he would drink in 1 day; and since the woman would drink of it in 1 day, the equation will 29. If A and B together can do a piece of work in 9 days, and A alone could do it in 15 days, in what time ought B alone to accomplish the work? Ans. 22 days. 30. The hour and the minute hand of a clock or watch are exactly together at 12 o'clock; when are they next together? Ans. 5 minutes past one |