200. The student should examine every problem, the result N of which appears under the form and endeavour to interpret that result. He may expect to find in such a case that the problem is impossible, but that by suitable modifications a new problem can be formed which has a very great number for its result, and that this result becomes greater the more closely the new problem approaches to the old problem. 201. Again, let us suppose that in Art. 193 we have a=b, 0 and also c=bn; then the value of x takes the form 0° On examining the problem we see that, in consequence of the suppositions just made, A and B are together at P, and are travelling with equal speed, so that they are always together. The question, when are A and B together, is in this case said to be indeterminate, since it does not admit of a single answer, or of a finite number of answers. 202. The student should also examine every problem in which the result appears under the form and endeavour to 0 0' interpret that result. In some cases he will find, as in the example considered above, that the problem is not restricted to a finite number of solutions, but admits of as many as he pleases. We do not assert here, or in Art. 200, that the interpretation of N Θ 0 the singularities and will always coincide with those given in the simple cases we have considered; the student must therefore consider separately each distinct class of examples that may occur. XV. ANOMALOUS FORMS WHICH OCCUR IN THE SOLUTION OF SIMPLE EQUATIONS. 203. We have in the preceding chapter referred to the forms and 0 which may occur in the solution of an equation of the first degree. We shall now examine the meaning of these forms when they occur in the solution of simultaneous equations of the first degree. We will first recall the results already obtained. 204. Every equation of the first degree with one unknown quantity may be reduced to the form ax = b. Now from this we case no finite value of x can satisfy the equation, for whatever finite value be assigned to x, since ax = 0, we have 0 = 6, which is impossible. If a=0 and b = 0, the value of x takes the form 0 0 in this case every finite value of x may be said to satisfy the equation, since whatever finite value be given to x we have 0 = 0. If b=0 and a is not = 0, then of course x = 0; this case calls for no remark. 205. Suppose now we have two equations with two unknown quantities; let them be ax + by = c and a'x+b'y = c. We will first make a remark on the notation we have here adopted. We use certain letters to denote the known quantities in the first equation, and then we use corresponding letters with accents to denote corresponding quantities in the second equation; here a and a have no necessary connexion as to value, although they have this common point, namely, that each is a coefficient of x, one in the first equation and the other in the second equation. Experience will establish the advantage of this notation. thus Instead of accents subscript numbers are sometimes used; α and a might be used instead of a and a' respectively. By solving the given equations we obtain X= b'c-be b'a-ba'' y = a'c-ac I. Suppose that b'a- ba' 0; then the values of x and the forms A B = 4 and; we should therefore recur to the given equa 0 tions to discover the meaning of these results. From the relation b=kb. By substituting these values of a' and b' we find that the second of the given equations may be written thus: Now if be different from c, the last equation is inconsistent k with the first of the given equations, because ax + by cannot be equal to two different quantities. We may therefore conclude that the appearance of the results under the forms and 0 Α B 0 indicates that the given equations are inconsistent, and therefore cannot be solved. II. Next suppose that b'a-ba' = 0, so that = and also α b merators in the values of x and y become zero as well as the denominators, so that the values of x and y take the form Now by what we have shewn above, the second of the given equations may be written c, so that the second given equation is only a repetition of the first; we have thus really only one equation involving two unknown quantities. We cannot then determine x and y, because we can find as many values as we please which will satisfy one equation involving two unknown quantities. In this case we say that the given equations are not independent, and that the values of x and y are indeterminate. 206. We have hitherto supposed that none of the quantities a, b, c, a', b', c'′ can be zero; and thus if the value of one of the 0 A or the value of the other 0 unknown quantities takes the form takes the same form. But if some of the above quantities are zero, the values of the two unknown quantities do not necessarily take the same form. For example, suppose a and u to be zero; A then the value of x takes the form and the value of y takes Thus we have two cases. First, if is not equal to two equations are inconsistent. Secondly, if is equal to b the two equations are equivalent to one only. In the second case, с since the relation= makes the numerator of x also vanish, b 0 the values of both x and y take the form in this case x is in C determinate but y is not, for it is really equal to q 207. Before we consider the peculiarities which may occur in the solution of three simultaneous simple equations involving three unknown quantities, we will indicate another method of solving such equations. |