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10. Given 5x-3y=x+y+20, and 2+y=11x-5y, to find x and y.

4y+20
4

From the first equation x=(- =) y+5; whence x2= (y+5)=) y2+10y+25: substitute these values respectively for x and x2 in the second equation, and it becomes 2 y2+10y+25= 6y+55.

By transposition 2 y2+4y=30.

By division y+2y=15.

By comp. the sq. y2+2y+1=16.

By evolution y+1=+4.

By transposition y=(+4-1) 3, or -5. *

Whence x=(y+5=3+5, or—5+5=) 8, or O.

11. Given x2+xy=35, and xy-2 y2=2, to find x and y. Let vy=x: this value substituted for x in both equations,

they become vy+ry=35, whence y=

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35

v2 + v

2

35
v2 + v v-2

By multiplication 35 v—70=2 v2 + 2 v.
By transposition 2 v2—33 v=—70.

By division v— 16.5 v=-35.

and vy2 —2 y2

when ce

By comp. the sq. v2 — 16.5 v+68.0625=33.0625.
By evolution v-8.25=(±√/33.0625=) ±575.
By transposition v=(S.25 +5.75=) 14, or 2.5.

2

~ᄒ

Whence by taking v=14, we have y=(√2-2
1 — 2 =) √ ; and

1

x= (ry=) 14√ and by taking v=2.5, we shall have y=

2

2

(√__y=√_—-—=) √4=2, and x= (ry=) 2.5 × 2=5.

-

2

12. Given x+y+xy=19, and r3y+ry2=84, to find x and y. Let s=x+y, and p=xy, then the given equations will become (x+y+xy=) s+p=19; and x2y+xy2=(x+y×xy=) sp= 84: from the square of the equation s+p 19, take four times

sp=84, and you will have s2-2 sp+p2=25; whence by evolution s-p=5: this equation added to, and subtracted from s+p= 19, gives 2s= (19+5=) 24 or 14, and s=12 or 7; and likewise 2p= (195) 14 or 24, and p=7 or 12; therefore x+y= (s) 12 or 7, and xy= (p=) 7 or 12. Subtract four times the last from the square of the last but one, and there remains x22 xy + y2 = (172 —28, or 7]2—48=) 116 or 1; whence, by evolution, x-y=±√116, or±1; add this to, and subtract it from x+y=12 or 7, and we shall have 2x= ±√116+12, or +1+ 7= (by taking the latter value only) 8 or 6, whence x=4 or 3; likewise 2y= (7F1=) 6 or 8, whence y=3 or 4; if we make x =4, then y=3; if x=3, then y=4 °.

2

x2 y2

13. Given +

y x

2

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Let x=z+v, and y=z-v, then by adding these two equations together, (x+y=) 2 z=6, and z=3; whence x=3+v, and y= 3-v. Multiply the first given equation by xy, and x3+y3=9xy; which by substituting 3+v for x, and 3-v for y, becomes 3+v3 +3-v)3=9x3+v×3—v; this by involution, multiplication, and addition, becomes 54+18 v2=81–9 v2, whence by transposition 27 v2=27; therefore v2=1, and v≈+1, whence x= (z+0 =3+1=) 4 or 2; and y= (z−v=371=) 2 or 4; if x=4, then y=2; but if x=2, then y=4.

13.

14. Given x2+8x=65, to find x. Ans. x=5, or 15. Given y2-12 y=540, required the value of y? Ans. y= 30, or -18..

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16. Given z2 – 20 z=-91, what is z equal to? 13, or 7.

17. Given 3x2-21x-450-6000, to find x. or-43.

Ans. z=

Ans. x=50,

18. Given z2+z=2, required the value of z? Ans. z=1,

or -2.

• If the former values of x-y, namely, ±√116 or 10.77 be taken, then for the affirmative values = 11.385, &c. and y =.615, &c. and for the negative values r = .615, &c. and y=11.385, &c. both of which values answer the conditions of the question equally with those given above. It appears from the solution, that this example (which was inserted by mistake) is not an adfected, but a pure quadratic, and therefore is misplaced; the same may be said of the thirteenth example.

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22. Given ar2+br=c, to find x. Ans. x=- +

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24. Given 2x2+x−y=3y2+2y, and x+y=7, to find x and y. Ans. x=42 or 4, y=−35 or 3.

98. By this rule may be solved all equations whatever, wherein there are only two different dimensions of the unknown quantity, provided the index of the one be exactly double that of the other.

RULE. Having completed the square and extracted the root as before, transpose the known quantities which are on the same side with the unknown one, and then extract the root implied by the index of the unknown quantity, from both sides of the equation P.

25. Given a +6 x2=72, to find the values of x. By completing the square x2+6x2+9=81.

By evolution x2+3=±9.

By transposition x2=(±9−3=)6 or — 12.

By evolution x= (± √/6=) ±2.4494897428, or±√-12, the latter of which are impossible.

P Every equation will have as many roots as the unknown quantity has dimensions; thus, in the 25th example, ☛ being in the fourth power, the equation will have four roots, as appears by the solution. In example 26, y comes out equal to 49 or 25, the latter of which being substituted in the equation, will not answer, except -5 be substituted for the square root instead of +5, the reason of which is obvious, since the 25 arose from -5 X -5. One root only is required in the following examples, as finding the rest would, in many cases, require rules which have not yet been given.

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By transposition √y=(+7+2=) 9, or −5.
By involution y= (9)2=) 81, or (-5)2=) 25.

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Ans. y=2.
Ans. z=4.

28. Given x+2x2=24, to find one value of x.
29. Given y6-4 y3-32, to find y.
30. Given z-2/z=0, to find z.
31. Given 2x4x2-496, to find x.

Ans. x=4.

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Every problem proposed to be solved algebraically, contains some conditions laid down, which are called the data; from whence one or more quantities are required to be found, called the quæsita. The first thing necessary to be done preparatory to the solution, is to understand clearly the import and signification of the problem: it must be freed from every thing ambiguous and un

a The word problem is derived from the Greek gobλnux. An algebraic problem is a proposition wherein some unknown truth is required to be investigated or discovered, and the truth of the discovery demonstrated.

necessary; the conditions, and the manner of their dependance on each other, must be clearly ascertained and stated, and they must be carefully distinguished from the quantities proposed to be found: when this is accomplished, the conditions of the proposed problem will be exhibited under the form of one or more equations; namely, as many equations as there are unknown quantities, the solution of which is the subject of the preceding rules.

Much depends on a proper substitution for the quantities required: no general rule for this can be given; sometimes a letter must be put for each; frequently, having substituted a letter for one of the unknown quantities, expressions for the others may be derived by means of this and the conditions proposed, without the aid of new letters; sometimes a substitution for the sum, difference, product, quotient, roots, powers, &c. of the unknown quantities, may be conveniently made; but the proper application of these must be learned by experience and practice.

The following modes of substitution will apply in many cases. For one unknown quantity put x, for two put r and y, x being the greater, y the less; for their sum x+y, for their difference x-y, for the square of the greater x2, for the cube root of the less √y, for the sum of their squares x2+y2, for the difference of their squares x2-y2, for the square of the sum r+y2, for the cube root of their difference 3√x-y, for their product xy, their , where the greater is proposed to be divided by the

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less, or where the less is proposed to be divided by the greater.

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In general, the sum of any two quantities is represented by interposing the sign + between them; the difference by the sign, the product by the sign x, or by placing them as coefficients to each other, and the quotient by placing the dividend above the divisor; following in every case the method applicable to it, as proposed in algebraic notation.

1. What number is that, to which 9 being added, the sum will be 23?

Let the required number be represented by x.

To which adding 9, the sum will be x+9.

This, by the problem, equals 23, whence x+9=23.

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