that A had got a certain sum, and that B had $30 more than A, but C $50 less than A. How much did each obtain? Ans. A $60; B $90; C $10. III. TRANSPOSITION OF THE UNKNOWN QUANTITY. may §51. We have found that when any term has the sign + it be removed from one member of the equation to the other, if we take care to change the sign to -; for this has been done every time we have subtracted a term from both sides. Thus, in the equation [x+5=20;] if we subtract 5 from both sides, it is plain that the first member becomes x, and the last member becomes 20-5; so that the equation would become x=20-5. §52. So also any term that has the sign may be removed from one member to the other, if we take care to change the sign to +. Because this is the same as adding that term to both sides. Thus, in the equation x—5—20, if we add 5 to both sides, the first member becomes x, and· the last member becomes 20+5. So that the equation becomes x=20+5. §53. When we remove a term from one member of an equation to the other member, we say that we transpose that term; and the operation of doing it, is called transposition. §54. It was stated in §32, that an equation must be brought so that the unknown quantity will occupy one member of the equation, and the known quantities embrace the other member. And, as it frequently happens that the unknown quantities are on both sides, we are obliged to resort to transposition in order to make one side free from them. And likewise, it is often necessary to transpose known quantities from the member which contains the unknown quantity. $55. Any term may be transposed from one member of an equation to the other, care being taken to change the sign when we change the side. EQUATIONS.-SECTION 4. 1. Reduce the equation 4x-14-3x+12 §56. In transposing, it is generally best to write first the unknown quantity that is on the left; and then bring over those which are on the right, if there are any there. Then write those known quantities that are already in the right hand member, and then transpose after them what known quantities there are in the left. 2. Given 21-7x-40-11x, to find x. 3. Given 40-6z=136-14z, to find z. 4. Given y+12=3y-4, to find y. 5. Given 5x-15=2x+6, to find x. Ans. x=4. Ans. z=12. Ans. y=8. Ans. x=7. Ans. x= 12. 7. Given 4-9y-14-11y, to find y. Ans. y=5. 6. Given 40-6x-16-120-14x, to find x. 8. Given x+18=3x-5, to find x. Solution. Transposing 3x, x-3x+18=-5 Uniting terms, Dividing by 2, -2x=-23 X = -11. $57. It is of no consequence what sign accompanies the final result; as the magnitude of the quantity is not affected by the sign. If we remember that is understood and may be written with every positive quantity, it will be very evident that the equation -x-11 is just as good as the equation +x=+11. In both cases, the quantity x is equal to the number 112. §58. In the result of the last question, 11 may be transposed to the first member; and x may be transposed to the Jast member. Of course, this will change the signs; and the equation will become 11. And if 11-x, it is evident that x=114. This coincides with what was shown in §57. $59. From what has just been said, we see that all the terms of each member may be transposed, so that the sign of each term may be changed; and still the equation shall retain the same members as at first; and that it is also immaterial which member is written first. And hence, in any equation the signs of all the terms may be changed without affecting the equality. §60. It is evident that all the terms of one member may be transposed to the other member. When this has been done, the member from which the terms have been transposed becomes, 0. Thus, the equation x-3y-xy, may be made x+xy-3y=0; where-3y balances x+xy. PROBLEMS. 1. A man has six sons, whose successive ages differ by 4 years; and the eldest is three times as old as the youngest. What are their ages? Forming the equation, x+4+4+4+4+4=3x 2. A person bought two horses, and also a hundred dollar harness. The first horse, with the harness, was of equal value with the second horse. But the second horse with the harness cost twice as much as the first. What was the price of each horse? Forming the equation, x+100+100=2x Transposing from both members, x-2x-100-100 Uniting terms, Or -200 3. A privateer running at the rate of 10 miles an hour, discovers a ship 18 miles off sailing at the rate of 8 miles an hour. How many hours can the ship run before she will be overtaken by the privateer? The equation will be 10x=8x+18. Ans. 9 hours. 4. A gentleman distributing money among "some poor people, found that he would lack 10 shillings if he undertook to give 5s to each. Therefore, he gave only 4s to each, and finds that he has 5s left. How many persons were there. It will be found that his money by the first supposition =5x-10; and by the last supposition, it =4x+5. Ans. 15. 5. I once had $84 in my possession; and I gave away so much of it, that I have now three times as much as I gave away. How much did I give away? mains. If I gave away $x, then $84-x will be what re- 6. A certain sum of money was shared among five persons, A, B, C, D, and E. Now, B received $10 less than A; C$16 more than B; than D. And it was found that the shares of the last two put together, were equal to the sum of the shares of the other three. How much did each man receive? D $5 less than C; E $15 more Ans. A $21; B $11; C $27; D $22; E $37. 7. A person wishes to give 3 cents apiece to some beg. gars, but finds that he has not money enough by 8 cents. He gives them 2 cents apiece and has 3 cents left. How many beggars were there? Ans. 11. 8. A courier who had started from a certain place 10 hours ago, is pursued by another from the same place, and on the same road. The first goes 4 miles an hour, and the second 9. In how many hours will the second overtake the first? In the operation, it must be remembered how far the first had the start, before the equal time for both began. Ans. 8 hours. |