ON THE SOLUTION OF QUADRATIC EQUATIONS CONTAINING TWO UNKNOWN QUANTITIES.

189. An equation containing two unknown quantities is said to be of the second degree when it involves terms in which the sum of the exponents of the unknown quantities is equal to 2, but never exceeds 2. Thus,

3x2-4x+y-xy-5y+6=0, 7xy-4x+y=0,

are equations of the second degree.

It follows from this that every equation of the second degree containing two unknown quantities is of the form

ay+bry+c+dy+ex+f=0,

where a, b, c,..... represent known quantities, either numerical or algebraical; i. e., the equation contains the second power of each of the unknown quantities, the first power of each, and the product of the two. Not that every equation of the second degree contains all these, but when any one of them is wanting the coefficient of that term, in the general form, is said to be

zero.

Let it be required to determine the values of x and y, which satisfy the equations.

a y2+bxy+cx2+dy+ex+ƒ =0
a'y3+b'xy+c'x2+d'y+e'x+f'=0...

(1)
(2)

Arranging these two equations according to the powers of y, they become a y2+(b x+d)y+(c x2+ex+ƒ )=0 .

Multiply (3) and (4) by a' and a respectively, and also by k' and k; then

Subtracting (6) from (5), and also (7) from (8), we have

Multiplying (9) by h’k—hk', and (10) by a'k-ak', we have (a'h — ah') (h'k — hk') y+(a'k—ak')(h'k—hk')=0 . . (11) (a'k-ak')y+(a'k—ak') (h'k—hk')=0.. (12)

(13)

.. (ah-ah') (h'k-hk')=(a'k-ak')2. . . Substituting the values of h, h', k, k' in equation (13), we have {(ab—ab)x+a'd—aď}. {(vo-be)x+(b'e—bé'—c'd+cd)x2+(b'ƒ—bƒ ̃'+de—de)x+d'ƒ—dƒ" }

`= {(a ́o—–—ac)x2+(a ́e—ac)x+a'ƒ—aƒ'}2

Hence, by multiplying and expanding, the final equation in x is of the fourth degree, which will, in general, be the degree of the equation obtained by eliminating between the two equations of the second degree; but the general form includes a variety of equations, according to the values of the coefficients a, b, c, &c.; when d, e, f, d', e', f' are each =0, the solution may be obtained by quadratics, the resulting equation in x being

{(a'b-ab')x+a'd-ad'}. {(b'c-bc')x-(c'd-cd')}=(a'c—ac')2x2.

Although the principles already established will not enable us to solve equations of this description generally, yet there are many particular cases in which they may be reduced either to pure or adfected quadratics, and the roots determined in the ordinary manner.

EXAMPLE I.

Required the values of x and y, which satisfy the equations,

3 3

3 3

EXAMPLE IV.

3

+x*y*+y2=a

x3+x2y2+y3=b

3

3 3

Square (1), x2+xay2+y3+2x} .x}y}+2x}y}+2y} .x}y}=a2.

3 3

=b.

2x2. x*y*+2x2y2+2y3x*y*=a2—b,

3 3 3

3 3

3

Qx*y*(x2+x*y*+y3)=a2—b

Equations (3) and (4) are common adfected quadratics; solving these in the usual manner, we find

Hence, in the above equations, two of the roots of x and y are possible, and

The following examples, though a valuable exercise, are likely to detain the student

long, and may, if necessary, be omitted.