Key to Robinson's Algebra: Containing Also a Short Treatise on the Indeterminate and Diophantine Analysis and Some Miscellaneous ExamplesJ. Ernst, 1852 |
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1+y¹ 1st Divisor 2d Divisor a+2r+s acres algebraic answer the conditions approximate cube root arithmetical progression c+by Clearing of fractions coefficient Complete the square Diophantine analysis Double equation gives equations become example expression a square extract square root factors find one value Find three numbers Find two numbers geese Given greater number integer values integral last equation last expression least number less Let x represent Let x+y Let x=A's Multiply number added number of cows number of solutions original expression perceive primitive equation problem proper fraction Put x=P Put x=vy QUADRATIC EQUATIONS r(r+A Reduced gives remainder represent the numbers resolved second equation shillings square number stage wagon Subtract Take p=1 transpose trial we find UNKNOWN QUANTITIES value of x verify whole number x and y x=a+y x²-y x²+y² xy² xy³ represent
Popular passages
Page 112 - A person bought two cubical stacks of hay for 4 1 £ ; each of them cost as many shillings per solid yard as there were yards in a side of the other, and the greater stood on more ground than the less by 9 square yards. What was the price of each ? 69.
Page 105 - H, ±1, are the numbers ; but we can multiply them all by the same square number 64, and their arithmetical relation will not be changed, and they will still be squares ; hence 1, 25, and 49 may be the numbers, or 4, 100, and 196, 5. Find two whole numbers, such that the sum and difference of their squares, when diminished by unity, shall be a square. Let a:-{-l=one number, and y= the other.
Page 114 - A square public green is surrounded by a street of uniform breadth. The side of the square is 3 rods less than 9 times the breadth of the street ; and the number of square rods in the street, exceeds the number of rods in the perimeter of the square by 228. What is the area of the square 1 Ans.
Page 105 - To find three whole numbers such, that if to the square of each the product of the other two be added, the three sums shall be all squares. Ans. 9, 73, and 328.
Page 102 - The preceding are some of the most comprehensive and general methods yet known ; but there are cases in practice where no general rules will be so effectual, as the operator's own judgment and penetration. Much, very much will depend on skill and foresight displayed at the commencement of a problem, by assuming convenient expressions to satisfy one or two conditions at once, and the remaining conditions can be satisfied by some one of the preceding rules. EXAMPLES. 1. It is required to find three...
Page 88 - Or, y=am — cy'. (2) In these theoretical equations, (1) and (2), m has different values, it being an arbitrary number taken at pleasure, so that ex' may be greater than bm, and am greater than cy? to render x and y positive. But if no such value of m can be found, it is proof that values of x and y do not exist in positive integers, and on the contrary as many suitable values of m as can be found, so many solutions will the equation admit of, and no more. Now as . - - mb <[ ex', and am ^> cy
Page 115 - He finds that if he buys 120 acres of cleared land, and lays out the rest of his money for that which is not cleared, he will not get the quantity of land he wants by 25 acres, but, if he buys 220 acres of uncleared land, and then buys a sufficient number of acres of cleared land to make up the number of acres he wants, he will have 4 dollars left. How many acres of each must he buy to have the quantity he wishes, and lay out all his money? (Harney page 203. Ans. 20 acres cleared, 218 uncleared....
Page 107 - Find two numbers, such that if the square of each be added to their product, the sums shall be both squares. Ans. 9 and 16. 13. Find three integral square numbers in harmonical proportion. Ans. 25, 49, and 1235. 14. Find two numbers in the proportion of 8 to 15, and such that the sum of their squares shall be a square number. Ans. 136 and 255. Bonnycastle's answer, 476 and 1080. 15. Find two numbers such that if each of them be added to their product, the sums shall be both square.
Page 107 - To find three numbers in geometrical progression, such, that if the mean be added to each of the extremes, the sums, in both cases, shall be squares. Ans. 5, 20, and 80. 24. To find two numbers such, that not only each of them, but also their sum and their difference, when increased by unity, shall be all square numbers.
Page 85 - Let t=0, then r=8,z={42, and 29z-f-10=N=4148, the number. 10. Required the least number that can be divided by each of the nine digits without remainders. Let x= the number. „., xxxxxxxx Then 2' 3' 4' 5' 6' 7' 8' 9