To interpret these results, we observe that the equation b c' from which we obtain ax+by; comparing this with ax the equation a x+by-c, the left hand members, it will be perceived, are identical, while the right are essentially different; for if in the numerator c b' b c' b c', c b' be greater than and if cb' be less than b c', We conclude therefore, that the two equations proposed cannot in this case be satisfied, at the same time, by any system whatever of finite values for x and y. The question therefore in this case is impossible. Again let us suppose a b ba = 0, and at the same time 0 c b' -b c' 0 ; the value of x in this case is reduced to 0 To interpret this result, we remark that the equations proposed may, in consequence of the relation a b' — b a' = 0, be put under the form ax + by = c b c' ax+by=f equations, which are identical, since from the relation In order then to resolve the problem, we have in fact but one equation with two unknown quantities; the question therefore is indeterminate. by substitution in the equation cb'-bc'0, = or reducing, a c-ca' 0; we infer therefore that if the value of x be of the form, the value of y will be of the same form and the converse. PROBLEMS FOR SOLUTION AND DISCUSSION. 1. To find a number such, that if it be added to the numbers a and b respectively, the first sum will be m times the second. Putting for the number, we have How shall we interpret this result when m=1 ? How when m= = 1, and at the same time a=b? How when'm is greater than 1, and m b greater than a? What conditions are necessary in order that the question may be solved in the exact sense of the enunciation ? 2. The sum of two numbers is a, and the sum of their products by the numbers m and n respectively is b. What are the numbers? Putting x and y for the numbers, we have How shall we interpret these results, when m is greater than n, and n a greater than b? How when m =n? How when mn, and at the same time n a=b? What conditions are necessary in order that the question may be solved in the exact sense of the enunciation ? 3. It is required to make a mixture of gold and silver, the weight of which shall be a grains, in such proportions that the price of the mixture shall be b shillings, the value of a grain of gold being c shillings, and that of a grain of silver d shillings; how much must be taken of each to form the mixture required? Let x the quantity of gold, y that of the silver, we have a (b d) a (c—b) y= с d How shall we interpret these results when cd? How when bc and at the same time c = d? What conditions are necessary in order that the question may be solved in the exact sense of the enunciation ? 4. The sides of a rectangle are to each other in the proportion of m to n; but if the quantity a be added to the first and the quantity b subtracted from the second, the surface of the rectangle will be diminished by the quantity p. sides? What are the Let x and y represent the sides respectively, we have What conditions are necessary in order that this problem may be solved in the exact sense of the enunciation ? When will the values of x and y take the form. What will be the values of x and y if p = = 0, and how should the enunciation of the question be altered to correspond to these values? 5. A banker has two kinds of money, in the first there are a pieces to the crown, and in the second b pieces to the crown; how many pieces must be taken from each, in order that there may be c pieces to the crown? Putting x and y for the number of pieces of each respective ly, we have In what cases will the values of x and y become infinite or impossible? When will one of them become negative and how are we to interpret the result in this case? When will they take the form 9. When will either one or the other of the quantities become 0? What conditions are necessary in order that the problem may be solved in the exact sense of the enunciation ? SECTION X.-THEORY OF INEQUALITIES. 90. In the reasonings, which relate to the discussion of a problem, we have frequent occasion to make use of the expressions "greater than," "less than." In such cases we shall attain a greater degree of conciseness, by representing each of these expressions by a convenient sign. It is agreed to represent the expression "greater than" by the sign >; thus, a greater than b is expressed by a>b. The same sign by a change of position is made to represent the phrase "less than ;" thus, a less than b is expressed by ab. An equation of the form a=a is called an equality. An expression of the form ab or ab is called an inequality. The principles established for equations apply in general to inequalities. As there are some exceptions however, we shall state the principal transformations, which may be made upon inequalities, together with the exceptions which occur. 1o. We may always add the same quantity to both members of an inequality, or subtract the same quantity from both members, and the inequality will continue in the same sense. Thus, let 35; adding 8 to both sides, we have 8+358, or 11 <13 Again let-3-5; adding 8 to both sides we have 8-38-5, or 5>3 This principle enables us, as in the case of equations, to transpose a term from one member of an inequality to the other; thus, from the inequality a2 + b2 >3 c2 — a2, we obtain 2 a2 + b2 3 c2. 2o. We may in all cases add member to member two or more inequalities established in the same sense, and the inequality, which results, will exist in the sense of the proposed. Thus, let there be a> b, c>d, e>f; we have a+c+eb+d+f. But if we subtract member from member two or more inequalities established in the same sense, the inequality, which results, will not always exist in the sense of the proposed. Let there be the inequalities 4 <7, 2<3, we have by subtraction 4-27-3, or 2 <4. But let there be the inequalities 9 <10 and 68, subtracting the latter from the former, we have 9-610-8, or 32. 3o. We may multiply or divide the two members of an inequali ty by any positive or absolute number, and the inequality, which results, will exist in the sense of the proposed. Thus, if we have a <b, multiplying both sides by 5, we have 5 a5 b. By means of this principle, we may free an inequality from its denominators. Thus, let there be we have by multiplication (a2 — b2) 3 a > (a2 — b2) 2 d, and by division 3 a > 2 d. But if we multiply or divide the two members of an inequality by a negative quantity, the inequality, which results, will exist in the contrary sense. Thus, let 87; multiplying both sides by 3, we have -24-21. From this it follows, that if we change the sign of each term of an inequality, the inequality, which results, will exist in a sense contrary to that of the proposed; for this transformation will be equivalent to multiplying both members by 1. 91. Let there now be proposed the inequality Here 2 is the limit to the value of x, that is, if we substitute for x in the proposed any value greater than 2, the inequality will be satisfied. The process, by which the limit to the value of the unknown quantity is determined, is called resolving the inequality. |