and therefore x, y and z are known from (3). The same artifice solves the equations x² − yz = a, y² − xz = b, z² − xy = c.
2. Equations which are homogeneous and of the same degree.—These equations can be solved by substituting y = mx. We proceed to explain the method by an example.
Ex. To solve 3x² + xy + y² = 15, 31xy − 3x² − 5y² = 45. Substituting y = mx in both these equations, and then dividing, we obtain 31m − 3 − 5m² = 3 (3 + m + m²) or 8m² − 28m + 12 = 0. The roots of this quadratic are m = ½ or 3, and therefore 2y = x, or y = 3x.
Taking 2y = x and substituting in 3x² + xy + y² = 0, we obtain y² (12 + 2 + 1) = 15; ∴ y² = 1, which gives y = ±1, x = ±2. Taking the second value, y = 3x, and substituting for y, we obtain x² (3 + 3 + 9) = 15; ∴ x² = 1, which gives x = ±1, y = ±3. Therefore the solutions are x = ±2, y = ±1 and x = ±1, y = ±3. Other artifices have to be adopted to solve other forms of simultaneous equations, for which the reader is referred to J.J. Milne, Companion to Weekly Problem Papers.
II. Indeterminate Equations.
1. When the number of unknown quantities exceeds the number of equations, the equations will admit of innumerable solutions, and are therefore said to be indeterminate. Thus if it be required to find two numbers such that their sum be 10, we have two unknown quantities x and y, and only one equation, viz. x + y = 10, which may evidently be satisfied by innumerable different values of x and y, if fractional solutions be admitted. It is, however, usual, in such questions as this, to restrict values of the numbers sought to positive integers, and therefore, in this case, we can have only these nine solutions,
| x = 1, 2, 3, 4, 5, 6, 7, 8, 9; y = 9, 8, 7, 6, 5, 4, 3, 2, 1; |
which indeed may be reduced to five; for the first four become the same as the last four, by simply changing x into y, and the contrary. This branch of analysis was extensively studied by Diophantus, and is sometimes termed the Diophantine Analysis.
2. Indeterminate problems are of different orders, according to the dimensions of the equation which is obtained after all the unknown quantities but two have been eliminated by means of the given equations. Those of the first order lead always to equations of the form
ax ± by = ±c,