7th. If we suppose their rate of travel has a given ratio, as m n- ; then, x= 2 2ma 2a; that is, the first travels twice the dis tance from A to B before overtaking the second. The results in the last two cases evidently correspond to the circumstances of the problem. IV. CASES OF INDETERMINATION IN SIMPLE EQUATIONS AND IMPOSSIBLE PROBLEMS. 167. An Independent Equation is one in which the relation of the quantities which it contains can not be obtained directly from others with which it is compared. are equations which are independent of each other, since the one can not be obtained from the other in a direct manner. x+3y=19, 2x+6y=38, are not independent of each other, the second being derived directly from the first, by multiplying both sides by 2. 168. An Indeterminate Equation is one that can be verified by different values of the same unknown quantity. Thus, if we have, By transposition, x-y=3, If we make y=1; then, x=4. If we make y=2; then, x=5, and so on; from which it is evident that an unlimited number of values may be given to x and y, that will verify the equation. If we have two equations containing three unknown quantities, we may eliminate one of them; this will leave one equation containing two unknown quantities, which, as in the preceding example, will be indeterminate. If we eliminate x, we have, after reducing, y-22-1; whence, y=1+2%. If we make z =1; then, y=3, and x=20+52-3y=16. If we make z=2; then, y=5, and x=15. So, any number of values of the three unknown quantities may be found that will verify both equations. These examples are sufficient to establish the following General Principle. When the number of unknown quantities exceeds the number of independent equations, the problem is indeterminate. A question that involves only one unknown quantity is sometimes indeterminate; the equation deduced from the conditions being identical. (Art. 145.) The following is an example: What number is that, whose increased by the is equal to the diminished by the? Clearing of fractions, 15x+10x=33x-8x; or, 25x=25x; which will be verified by any value whatever of x. 169. The reverse of the preceding case requires to be considered; that is, when the number of equations is greater than the number of unknown quantities. Each of these equations being independent of the other two, one of them is unnecessary, since the values of x and y, may be found from either two of them. When a problem contains more conditions than are necessary for determining the values of the unknown quantities, those that are unnecessary are termed redundant. The number of equations may exceed the number of unknown quantities, so that the values of the unknown quantities shall be incompatible with each other. The values of x and y, found from equations (1) and (2), are x=5, y=7; from (1) and (3), x=6, and y=6; and from (2) and (3), x=4, and y=9. It is manifest that only two of these equations can be true at the same time. EXAMPLES TO ILLUSTRATE THE PRECEDING PRINCIPLES. 1. What number is that, which being divided successively by m and n, and the first quotient subtracted from the second, the remainder shall be q? Ans. x= mnq m-n What supposition will give a negative solution? An infinite solution? An indeterminate solution? Illustrate by numbers. 2. Two boats, A and B, set out at the same time, one from C to L, and the other from L to C; the boat A runs m miles, and the boat B, n miles per hour. Where will they meet, supposing it to be a miles from C to L? Ans. ma m+n na mi. from C, or -mi. from L. m+ n Under what circumstances will the boats meet half way between C and L? Under what will they meet at C? At L? Above C? Below L? Under what circumstances will they never meet? Under what will they sail together? Illustrate by numbers. 3. Given 2x-y-2, 5x-3y=3, 3x+2y=17, 4x+3y 24; to find x and y, and show how many equations are redundant. (Art. 169.) Ans. x 3, y=4. x=3, 4. Given x+2y=11, 2x-y-7, 3x-y=17, x+3y=19; to show that the equations are incompatible. (Art. 169.) V. A SIMPLE EQUATION HAS BUT ONE ROOT. 170. Any simple equation involving only one unknown quantity, (x), may be reduced to the form mx=n; for all the terms containing x may be reduced to one term, and all the known quantities to one term; whence, x= n m Now, since n divided by m can give but one quotient, we infer that a simple equation has but one root; that is, there is but one value that will verify the equation. VI. EXAMPLES INVOLVING THE SECOND POWER OF THE UNKNOWN QUANTITY. 171. It sometimes happens in the solution of an equation, that the second or some higher power of the unknown quantity occurs, but in such a manner that it is easily removed. The following equations and problems belong to this class: 1. (4+x)(x-5)=(x—2)2. Performing the operations indicated, we have x2-x-20-x2—4x+4. Omitting x2'on each side, and transposing, we have 7. It is required to find a number which being divided into 2 and into 3 equal parts, 4 times the product of the 2 equal parts shall be equal to the continued product of the 3 equal parts. Ans. 27. 8. A rectangular floor is of a certain size. If it were 5 feet broader and 4 feet longer, it would contain 116 feet more; but if it were 4 feet broader and 5 feet longer, it would contain 113 feet more. Required its length and breadth. Ans. Length, 12 feet; breadth, 9 feet. VI. OF POWERS, ROOTS, RADICALS, AND INEQUALITIES. I. INVOLUTION, OR FORMATION OF POWERS. 172. The Power of a number is the product obtained by multiplying it a certain number of times by itself. Any number is the first power of itself. When the number is taken twice as a factor, the product is called the second power or square of the number. When the number is taken three times as a factor, the product is called the third power or cube of the number. In like manner, the fourth, fifth, etc., powers of a number are the products arising from taking the number, as a factor, four times, five times, etc. The Index or Exponent of the power is the number which denotes the power. It is written to the right of the number, and a little above it. |