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16. x2 - 2 (a + b) x2 + a2 + 2ab + b2 = 0.

17. x1- 2x2a2 - 2x2b2 + a* + ba — 2a2b2 = 0.

18.

19.

4x-4x3- 7x2 - 4x + 4 = 0.

9x-24x3-2x2 - 24x + 9 = 0.

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25.

(x − a)3 (b -- c)3 + (x − b)3 (c − a)3 + (x − c)3 (a − b)3 = 0.

26. x(x-1) (x-2)=9.8.7.

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37.

(x + α)*+ (x + b)* = 17 (a — b)*.

38. /x+a-x= /b.

39. abx (x + a + b)3 − (ax + bx + ab)3 = 0.

40. abcx (x + a + b + c)3 − (xbc + xca + xab + abc)2 = 0.

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42. x+b (a + b) x3 + (ab - 2) b2x2 - (a+b) b3x + b = 0.

43. (x+6)=2ax2+2ab3x - a2x2.

44.

(x − a)

(x-6)9

+

(x - c)3

=1.

(x − a)3 − (b −c)2 + (x − b)3 — (c − a)2 + (x −c)3 — (a — b)3 — '

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CHAPTER X.

SIMULTANEOUS EQUATIONS.

140. A SINGLE equation which contains two or more unknown quantities can be satisfied by an indefinite number of values of the unknown quantities. For we can give any values whatever to all but one of the unknown quantities, and we shall then have an equation to determine the remaining unknown quantity.

If there are two equations containing two unknown quantities (or as many equations as there are unknown quantities), each equation taken by itself can be satisfied in an indefinite number of ways, but this is not the case when both (or all) the equations are to be satisfied by the same values of the unknown quantities.

Two or more equations which are to be satisfied by the same values of the unknown quantities contained in them are called a system of simultaneous equations.

The degree of an equation which contains the unknown quantities x, y, z... is the degree of that term which is of the highest dimensions in x, y, z....

Thus the equations

ax + a3y + a3z = a*,

xy + x + y + z = 0,

x2 + y2+ z2 - 3xyz = 0,

are of the first, second and third degrees respectively.

141. Equations of the First Degree. We proceed to consider equations of the first degree, beginning with those which contain only two unknown quantities x and y. Every equation of the first degree in x, y, z,... can by transformation be reduced to the form

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where a, b, c,... k are supposed to represent known quantities.

NOTE. When there are several equations of the same type it is convenient and usual to employ the same letters in all, but with marks of distinction for the different equations.

Thus we use a, b, c... for one equation; a', b', c'... for a second; a", b", c"... for a third; and so on. Or, we use a, b, c, for one equation; a, b, c, for a second; and so

on.

Hence two equations containing x and y are in their most general forms

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142. Equations with two unknown quantities. Suppose that we have the two equations

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Multiply both members of the first equation by b', the coefficient of y in the second; and multiply both members of the second equation by b, the coefficient of y in the first. We thus obtain the equivalent system

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Substitute this value of x in the first of the given equations; then

cb'-c'b

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+ by = c,

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The value of y may be found independently of x by multiplying the first equation by a' and the second by a; we thus obtain the equivalent system

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which is equal to the value of y obtained by substitution.

NOTE. It is important to notice that when the value either of x or of y is obtained, the value of the other can be written down.

For a and a' have the same relation to x that b and b' have to y; we may therefore change x into y provided that we at the same time change a into b, b into a, a' into b', and b' into a'. Thus from

X =

cb'-c'b

ab' - a'b

ca'-c'a we have y = ba-b'a'

S. A.

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