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From 1 Suppose the cask contains x gallons; after the first drawing there gallons of wine and 9 gallons of water. At the second drawing of. Suppose that in parts of bronze there are x parts of copper a; of zinc ; also suppose that in the fused mass there are a parts of brass, and parts of bronze. Also parts of bronze contain parts of copper, 45 parts of zinc, and parts of tin. Then the number of races on the a;"' day varies as the product x a- x Subtract 2.

Also there are 20 then the no. We proceed as in the last Example, and express in the scale of three. Thus the number This is. But r - 1 is even ; hence is even, and therefore and S are either both even or both odd.

Hence the required result follows. Giving to y any real value, we find two real values for x: or giving to x any real value we find two real values for y. In this and the following examples, the solution obtained by taking the negative value of y satisfies a modified form of the given equation. Examples 31 to 34 may be solved as in Art. Examples 35 37 are reciprocal equations and may be solved by method of Art. Example 38 may be solved by Art.

Combine with the second equation. From 1 , 3a;-2! Divide 2. Substituting for. Or, multiply the second equation by 2, the third equation by By multiplication,! Hence one of the quantities x, y, or z must be zero. Hence from 3 the remaining quantity must be equal to a, Combining 1. Let X denote the number of shillings, y the number of sixpences; have aU values from to , and therefore.

The general. When the particular man is included we have to select 3 men out of the remaining 23 this can be done ;. Suppose the letters a, u fastened together ; then they count as one and we have six things to arrange. This can be done in ways; but since a, u admit of two arrangements among themselves we must multiply this result by 2. The no.

Here we have 3 places in which two letters are to be placed; this Then the four consonants can be arranged in gives rise to 3 x 2 or 6 ways. Suppose the vols, of the same work inseparable, then we have 4 works to be arranged taken as a whole ; since vols, of each work can be arranged in any order, we get 14 x 13 x IS x 12 x 12, or Suppose the best and worst papers fastened together, then the no.

We must subtract this no. If we write down all the positive signs there will he p-1 places between them in which a negative sign may be placed. Also the row may begin and end with a negative sign. The first place can be occupied in n ways, and then the second place 7. The first thing may be given in two ways ; so may the second ; so the third, and so on.

Hence we have 2 x 2 x 2 x 2 x 2, or 32 ways ; but this includes two cases in which either person has all the five things. If we reject these the number of ways will be The first ring can be placed in fifteen different positions; so may the second ; so may the third. Hence there are 15 x 15 x 15 different positions possible, only one of which is the right one ; therefore the number of unsuccessful attempts possible is Each number is to consist of not more than 4 figures; and we may suppose each number to be written with four figures, because if we have less than 4 we can insert ciphers to begin the number with.

Thus 24 may be written Therefore every possible arrangement of 4 figures out of the given 8 will furnish one of the required numbers, except Thus by Art. Then taking each arm in succession and combining the different positions each is capable of, we ultimately get iK From this result we must subtract 1 for the case in which each arm is in the position of rest.

The first. By Art. There are 6 letters of four different sorts, namely , s; e,e; In finding arrangements of three, these may be classified as follows 1 2. But this includes the case where all the books are rejected. Thus the sum arising from the digit 7 alone is Of the p like things we may take 0, 1, 2, The r unlike things may each be disposed of in 2 ways and Hence, combining therefore the r things may be disposed of in 2' ways.

Of the m letters a we can take 0, 1, 2, 3, Then the other n unlike letters may each be dealt with in two ways, either taken or left.

The terms of the two series are numerically the same, but in the the terms are all positive, and in the second they are alternately positive and negative. Therefore the i"" and the T,. See Example. Thus the coefficient required. Hence the result. Examples 1 to 14 are too easy to require full solution; the following six solutions will suffice. Solving these simultaneous equations,. Let A be the number of pounds paid annually, then present value of an annuity to commence at once and to run 10 years.

The second of these factors is always positive, but the first is only positive so long as a;. It is easily seen that r n - r. Thus the expression a. As in the preceding case, ; that is when x' 1. The expression then. Hence the expression. Hence p-r of the quantities a, 6, c, d Let Equating coefficients,.

Let 1. Equate coefficients of 7. Therefore the equation is an identity. Whence, by equating the coefficients of , ij, f on the two sides, we obtain the required relations. With the same notation we have 1 - 8a! Now proceed as in the last Example, and we get 9. By trial, one solution is 9. By substitution. Thus t can only have the values and 1. Thus y of the number in the denary scale, is equal to Since and have no common factor, no two divisions will be If a is the length of the two rods, then the distance from the first is.

Let a. Hence 29? Since there are to be 6 solutions, t is restricted to the values 0, 1, 2, 3, 4, 5. Since zero solutions are inadmissible, a must lie between 1 and 13, and 5 must be greater than 5 x 19 and less than 6 x This Example includes Examples The factors of are 1, ; 3, 35; 5, 21; 7, 15; the solution may easily be completed as in Art.

Further, Hendriek bought 23 more than Catriin; thus Hendriek bought 32 hogs, and Catriin 9 hogs; also Claas bought 11 more than G-eertruij thus Claas bought 12 and Geertruij 1 hog. Hence Cornelius bought 8, and Anna 31 hogs therefore we have the following arrangement ;.

And as in Art. Whence S is easily found. This last expression consists of two parts, the. As in the first part of the question the expression is divisible by 5; thus it ia divisible by 2 4 3 2 5 or Every number x is one of the forms 3g, 3gl.

Let r, s represent any two of the numbers 1, 2, 3,


Complete Solutions to Hall & Knight's Higher Algebra With Questions

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Solutions for Hall & Knight

From 1 Suppose the cask contains x gallons; after the first drawing there gallons of wine and 9 gallons of water. At the second drawing of. Suppose that in parts of bronze there are x parts of copper a; of zinc ; also suppose that in the fused mass there are a parts of brass, and parts of bronze.


Solutions of the Examples in Higher Algebra

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