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3. But 8-x being a whole number, it may be

*. rejected, and the remainder,


, 15

made equal to a whole number w.

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must be

4. By the fifth part of the rule, the unknown; quantity, a, in this remainder, having more than 1 for a coefficient, therefore, multiply... by 7, and the product will be



which being added to x,




or which is the same 15x 70-14x 70+x

as follows,



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thing, to

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6. Then 70+x=15w; .. x=15w—70;

7. Let w be taken 5;.

8. Then x=5, and y=3, two numbers that will answer the conditions of the question.

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1. Given 3x+2y=26, to find the values of a and y in whole positive numbers.

Ans. 6, and y=4.

2. Given 4x-5y=10, to find the values of x and y in whole numbers.

Ans. 5, and y=2.

3. Given 12x=138-7y, to find the values of x and

y in whole numbers.

Ans. x8, and y=6.

4. Given 87x+256y-130814102, to find the least value of x, and the greatest of y, in whole numbers. Ans. 30, and y=50.

5. Divide 100 into two such parts, that the one may be divisible by 7, and the other by 11.

Ans. 56 and 44.

6. Two boys having together 100 marbles; one says to the other, When I count my marbles by eights, there is an overplus of 7' the other replies, "If I count my marbles by tens, I find the same overplus of 7. How many marbles had each?

Ans. The first boy had 23, and the second 77.

7. A draper bought two sorts of cloth, consisting of blue and black, for £88. 10s.: for the blue he paid 31s. per yard, and for the black 21s. per yard. How many yards did he buy of each ?

Ans. 9 yards of blue, and 71 of black, 40 do.

or 30


8. In what year of our Lord was the cycle of the sun 23, the cycle of the moon 14, and the cycle of the Roman Indiction 4 ?

Ans. 1666.

9. How much wheat at 5s. 4d. per bushel, and rye at 3s. 7d. per bushel, will compose a mixture worth 4s. 7d. per bushel ?

Ans. 4 bushels of wheat, and 3 of rye,

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N. B. Any two numbers in the proportion of 4 to

3 will answer the question.



When the given equation contains three or more unknown quantities.


Find the limit of that quantity which has the greatest coefficient, and then find the different values by Case I. as in the following example.


Given 3x+4y+5z=62, to find the values of x, y, and z, in whole numbers.

Here, in order to find the limit or number of lues that z may have (z having the greatest coefficient), take the values of x and y, each 55=11, the li

Sequal to 1; then z

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No To mit or number of values that z can have, though they may be considerably less.

Then, by Case I,

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Dy Zz


by the fifth part of the first rule, the remainder, must be a whole number, but having more than 1 for a coefficient, and the 2z being a negative quantity, this remainder must be added to z in order to exterminate the coeffi

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3z 2-y-2z +2 which

cient; thus, + 3
must be a whole number;

dr. let

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w; then 2+2y=3w 3/550 iół bas

v. y=z+2—3w; and by taking w=0, y=z+2. Now, by the limit, z may be taken any number from 1 to 11, that will answer the conditions of the question;

if z = 1, y will = 3, and x = 15, be add if=12, y will = 4, and ♫ — 12, ki sxum ifz = 3, y will

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5, and 19's .bol z = 4, y will 6, and

if z

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1. Given 3x+5y-12z-3, to find the values of a, y, and z, in whole numbers.>

Ans. x=4, y=3, and z=2.

12. Given 5x-2y+32=22, to find the values of x, y, and z, in whole numbers.

Ans. x=3, y=4, and z=5..

3. Given 4x-6y+3z=2, to find the values of x, y, and 2, in whole numbers.....

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*Ans. a=2,{y=4, and 2=6.

OFA silver-smith has three kinds of silver, the first

oz., the second of 5 oz., and the third of 41⁄2 oz. fine, per mark of 8 oz. How many ounces of each must he take to form a mixture to weigh 30 marks, at 6 oz. ?

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Ans. 12,15, and 3; or 14, 10, and 6; or 16, 5, and 9.

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5. Required three numbers such, that if the first be multiplied by 3, the second by 5, and the third by 7, the sum of their products will be 560; but if the first be multiplied by 9, the second by 25, and the third by 49, the sum of their products will be 2920.

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-i90s Ans. 15, 82, and 15; or 50, 40, and 30.

6. A person bought oxen, cows, calves, and sheep, to the amount of 100 in number, for £100 for each ox he gave £10, for each cow £5, for each calf £2, and for each sheep 10s. How many did he buy of each?

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0 = nor or fork ofwor or ser yðr ór Oxen, 1, 1,

Ans. Cows, 1, 2, 3, 4, 5, 6,


7, 8, 1, 2.

Calves, 24, 21, 18, 15, 12, 9, 6, 3, 5, 2.
Sheep, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92.


7. How many different ways thay wine at 6d. 8d. 10d. and 16d. per quart, be mixed so as to make 100 quarts worth 12d, per quart?

Ans. 312.

THE APPLICATION OF ALGEBRA eniado Þ slyuR.11 1,303 bbger

Thế ĐẦM 1 TO THE SOLUTION OF A qua sự ban enrol>


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In order to solve Geometrical Problems Algebraically, draw a figure to represent the conditions of the problem, and consider it the true one.hapati gd

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Then, let the unknown line or lines, which appear the easiest to find, be represented by the common algebraical symbols; connect these, by the different rules and theorems in Euclid's Elements, with the known quantities or lines, and form as many independent equations as there are unknown quantities, which may be solved by the rules already given. Talsoibroqraq aut me fal amit vode bdt mi

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In some cases, after the figure is drawn, it requires to be prepared for solution, by producing and drawing such lines as may be wanted, in order to obtain a solution; but as no general rule can be given for preparing the figures, and making the proper substitu ons, so as to be bring m out the simple conclusions, it may not be improper in the learner to attempt the solution of the same problem different ways, and apply that method which succeeds the best to similar) daynol adi aa 24dafont cola ad #I dibond 94 von 16- 47 godi



Before the Student attempts the solution of Geometrical Problems by Algebras he must be tolerably versed with / the

of Euclid.

The Author gives preference to Bonnycastle's Euclid ̧í as

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