XX. ON SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS. 249. FOR the solution of Simultaneous Equations of a degree higher than the first no fixed rules can be laid down. We shall point out the methods of solution which may be adopted with advantage in particular cases. 250. If the simple power of one of the unknown symbols can be expressed in terms of the other symbol by means of one of the given equations, the Method of Substitution, explained in Art. 217, may be employed, thus: Ex. To solve the equations get Substitute this value for a in the second equation, and we From which we find the values of y to be 30 and 20. And we may then find the corresponding values of x to be 20 and 30. 251. But it is better that the student should accustom himself to work such equations symmetrically, thus: Then from this equation and (1) we find x= =30 or 20 and y=20 or 30. Then from this equation and (1) we get 27 or 5 and y=-5 or -7. and from this equation and (1) we find x=2 or 3 and y=3 or 2. Taking the square root of each side, and taking only the positive root of the right-hand side into account, x + y = 5; .. x=5-y. Substituting this value for x in (2) we get (5-y) y+4y=18, an equation by which y may be determined. NOTE. In some examples we must subtract the second equation from the first in order to get a perfect square. Adding this to (4), we get x2+2xy + y2=16; :: x+y=±4. Then from this equation and (3) we find x=3 or -1, and y=1 or −3. The equations A and B furnish four pairs of simple −7 and −4, from which we find the values of x to be 7, 4, and the corresponding values of y to be 4, 7, -4 and -7. 258. The artifice by which the solution of the equations given in this article is effected is applicable to cases in which the equations are homogeneous and of the same order. |