SECTION II. To subtract a Compound Quantity. dollars 1. A man, who has 4 x dollars in his pocket, pays one debt of 3 x dollars, and another debt of y How many has he left? ANS. xy dollars. It is here required to take the whole value of the compound quantity, 3 x + y, the sums paid away, frcm 4 x. Now, if only 3 x be taken, it is evident that not enough is subtracted by the value of y, whatever that may be. The work may, therefore, be expressed thus, 4 x 3 x-y; which, reduced, gives xy. To illustrate this by figures, let x = 5, and y = 3: then 4 x 20, and 3 x + y = 15 + 3, or 18: now, 2018= 2, and 20 15 -3 is also 2. In this last expression, we may either subtract 15 from 20, and then subtract 3 from the remainder; or we may add 15 and 3 together, and subtract their sum from 20. The result is the same both ways. Here, the signs of both the quantities to be subtracted are changed from + to. 2.) Subtract 3 a + b from 5 x. ANS. 5x-3a-b 5 x y is to be taken from 5 x. ANS. 2x+y. compound quantity The whole value of 3 x is not to be subtracted, but the difference between that value and the value of y. If, therefore, we subtract the whole of 3 x, we subtract too much by the value of y, which must afterwards be added, to give the true answer. The work may be expressed thus, -3x+y; which, reduced, is 2 x + y. Perhaps this will be better understood, if illustrated by figures. Let x 6, and y = 4; then 5 x = 30, and 3 x - y = 184; that is, we are required to 4 from 30. Now, 18 4 14, and 1416, which is the true answer. But if we take the whole of 18 from 30, we take too much by 4, as we are required to subtract only the excess of 18 over 4; we must, therefore, add 4 to the remainder, to obtain the true answer; thus, 30 - 18416. We may either add 4 to 30, and subtract 18 from the sum; or we may subtract 18 from 30, and add 4 to take 18 30 = the remainder. Here, too, both the signs of the quantities to be subtracted are changed, the to- and 37. Subtract 12 + 4 a from 27. 38. Subtract a + 12 from 19. 39. From 5 (a+b) take 2 (a + b) — x. — x y * ક્ According to the principles already explained, — X becomes+x, when it is subtracted from any quantity; we have, therefore, x+x=2x; that is, subtracting a negative quantity is the same thing as adding a positive quantity of the same value. If A is in debt 1000 dollars, we should subtract that sum in forming an estimate of his property; but if B cancels that debt for him, that is, subtracts that - quantity, he evidently increases or adds to the amount of his property as much as if he had actually given 1000 dollars into his hand. 2. From a + b subtract x ・y. = It is here required to subtract the difference of two quantities, x and y, from the sum of two other quantities, a and b. Suppose a 8, b = 6, x 11, and y = 2: we then have 8+ 6, from which we are to subtract 112; that is, 149, or 5, which is the answer. But 8 + 6 — 11+2 is also equal to 5, which corresponds with the answer as expressed above. The signs of the quantities subtracted are changed as before; but in all cases, the signs of the other quantities, from which the subtraction is made, remain unchanged. From the several questions proposed in this chapter, and the reasoning which follows them, we derive the following general RULE for Subtraction in Algebra: Change all the signs in the quantity to be subtracted, each+to and each to +; and unite the terms that are similar, as in Addition. The subtraction is, in fact, performed, when the signs of the terms to be subtracted are changed. The object of the remaining part of the operation, is, to reduce the number of terms, by uniting or cancelling such as are similar, that the answer may be presented in its simplest form. 3. What is the value of x (a + b c) ? ANS. x- α b + c. The expression used in this question implies, that the whole quantity included in the parenthetical marks (), namely, a + bc, is to be subtracted from x; of course, all the signs must be changed. 4. What is the value of a b-(-cx+d—16)? ANS. ab cxd 16. abc + It is recommended to the student, in performing these examples, actually to change the signs; at least, until he becomes perfectly familiar with the operation. The last two examples, thus prepared, will stand |