The quantities a and b being dissimilar, we are obliged to represent the subtraction, as in the answer given. 8= 4, That the above expression, a—b, contains the true answer, may be proved by referring to numbers. Let α = 12, and b =8; then a b, that is, 12 which is the difference of the given quantities. Here, too, the sign of the number to be subtracted is changed 8bbey -4ake 31. From 8 a b c a take 4 abc., galey SECTION II. To subtract a Compound Quantity. 1. A man, who has 4 x dollars in his pocket, pays one debt of 3 x dollars, and another debt of y dollars. How many has he left? ANS. x y dollars. It is here required to take the whole value of the compound quantity, 3 x + y, the sums paid away, from 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 xy; which, reduced, gives x y. To illustrate this by figures, let x = 5, and y = 3; then 4 x = 20, and 3 x + y: = 15+ 3, or 18: now, 15 2, and 20 15 and 3 is also 2. In this 20 18= last expression, we may either subtract 15 from 20, and then subtract 3 from the remainder; or we may add 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 9. From 6 a take 3 a + b. 10. Subtract 2 x + y from 4 a. 19. From 5 x take 3 x y. In this example, the value of the is to be taken from 5 x. ANS. 2x+y. compound quantity The whole value of 3 x y 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, 5 x 3 x + 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, 4; that is, we are required to and 3 x take 18 y = 18 4 from 30. Now, 18 4 16, which is the true answer. 14, and But if we 30 14 = 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 18+ 4 = 16. We may either add 4 to 30, and subtract 18 from the sum; or we may subtract 18 from 30, and add 4 to 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. 40. From x (x y) take 2 x (xy) — x y. 41. Take a x + b from 3 a x. - According to the principles already explained, becomes +x, when it is subtracted from any quantity; we have, therefore, x + x = 2 x ; 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. 8, b 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 = 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 8611 + 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 quan tities, from which the subtraction is made, remain unchanged. From the several questions proposed in this chapter, |