28. 4x2+5y=6+20xy — 25y2+2x, 7x-11y=17.
29. 9.x2+33x-12=12xy - 4y2+22y, x2-xy=18.
30. (x2-y2)(x-y)=16xy, (x4-y1) (x2 — y2)=640x2y2.
31. 2x2-xy+y2=2y, 2x2+4xy=5y.

33. y (y2-3xy-x2)+24=0, x(y2 - 4xy + 2x2)+8=0. 34. 3x3-8xy2+y3+21=0, x2 (y−x)=1.

35. y2(4x2-108)=x(x3-9y3), 2x2+9xy+y2=108. 36. 6x+x2y2+16=2x (12x+y3), x2+xy-y2= 4. 37. x (a+x)=y(b+y), ax+by=(x+y)2.

38. xy+ab=2ax, x2y2+a2b2=2b2y2.

137. Equations involving three or more unknown quantities can only be solved in special cases. We shall here consider some

of the most useful methods of solution.

Write u, v, w for y+z, z+x, x+y respectively; thus

vw=30, wu=15, uv=18

Multiplying these equations together, we have

u2v2w230 × 15 × 18=152 × 62;

... uvw= ± 90.

Combining this result with each of the equations in (1), we have u=3, v=6, w=5; or u=- − 3, v=-6, w=- -5;

.(1).

Solving these homogeneous equations as in Example 4, Art. 136, we obtain

Example 4. Solve x2-yz=a2, y2 — zx=b2, z2—xy = c2.

Multiply the equations by y, z, x respectively and add; then

c2x + a2y+b2z=0.

Multiply the equations by z, x, y respectively and add; then

b2x+c2y+a2z=0

From (1) and (2), by cross multiplication,

..(1).

(2).

Substitute in any one of the given equations; then

k2 (a+b+c6-3a2b2c2)=1;

9. x2y222u=12, x2y2zu2=8, x2yz2u2=1, 3xy222u2 = 4.

14.

15.

x3+y3+z3=a3, x2+y2+z2=a2, x+y+z=a.

x2+y2+22=yz+zx+xy=a2, 3x-y+z=a3.

16. x2+y2+z2=21a2, yz+zx−xy=6a2, 3x+y−2z=3a.

INDETERMINATE EQUATIONS.

138. Suppose the following problem were proposed for solution:

A person spends £461 in buying horses and cows; if each horse costs £23 and each cow £16, how many of each does he buy?

Let x, y be the number of horses and cows respectively; then

23x+16y= 461.

Here we have one equation involving two unknown quantities, and it is clear that by ascribing any value we please to x, we can obtain a corresponding value for y; thus it would appear at first sight that the problem admits of an infinite number of solutions. But it is clear from the nature of the question that x and y must be positive integers; and with this restriction, as we shall see later, the number of solutions is limited.

If the number of unknown quantities is greater than the number of independent equations, there will be an unlimited number of solutions, and the equations are said to be indeterminate. In the present section we shall only discuss the simplest kinds of indeterminate equations, confining our attention to positive integral values of the unknown quantities; it will be seen that this restriction enables us to express the solutions in a very simple form.

The general theory of indeterminate equations will be found in Chap. XXVI.

Example 1. Solve 7x+12y=220 in positive integers.

Divide throughout by 7, the smaller coefficient; thus

If in these results we give to p any integral value, we obtain corresponding integral values of x and y; but if p> 2, we see from (3) that x is negative; and if p is a negative integer, y is negative. Thus the only positive integral values of x and y are obtained by putting p=0, 1, 2.

The complete solution may be exhibited as follows:

to make the coefficient of y differ by unity from a multiple of 7. A similar artifice should always be employed before introducing a symbol to denote the integer.

Example 2. Solve in positive integers, 14x-11y=29..

Divide by 11, the smaller coefficient; thus

=2 −x+y=integer;

.(1).