Example 8. It is required to divide $300 among A, B, and C; so that A may have twice as much as B, and C may have as much as A and B together. Let us represent B's share by x. Then A's share will be 2x; and C will have as much as both put together, which is 3x. Then, §10. As we sometimes wish to speak of particular parts of our calculations, mathematicians have given the name term to any quantity that is separated from others by one of the SIGNS or. Thus, in the last Example, and first line of the operation, the first x is the first term, the 2x is the next term, and the 3x is the next term; and the 300 is the last term. §11. When a figure is put before a letter to denote how many times we take the quantity which that letter stands for, the figure is called a co-efficient.* Thus, in the term 2x, 2 is a co-efficient of x; in the term 3A, 3 is a co-effi cient of A. §12. It must also be understood, that a letter without any number before it, has 1 for its co-efficient. Thus, a represents 1; a=1a; &c. The 1 is omitted because it is plainly to be understood. * This name was given by Franciscus Vieta, about 1573. §13. Any number, or letter, or any thing else, used to denote a quantity, when it is not united to any other quantity by either the sign + or, is called a simple quantity. Thus, x is a simple quantity; 2x is a simple quantity; 12 is a simple quantity; &c. ADDITION AND SUBTRACTION OF SIMPLE QUANTITIES. §14. In Algebra, simple quantities are added by writing them down, one after another; being careful to put the sign+between them. Thus we add 9 to x, by writing them 9+x, or, x+9. The pupil must understand that x stands for some number; but it is often the case that we do not know what that number is. §15. It will readily be seen, that it is of no consequence which quantity is put first; for 9+7 is the same amount as 7+9. §16. One simple quantity is subtracted from another simple quantity, by writing down both quantities, one after another, and putting the sign — before the quantity which we subtract. Thus, we subtract 4 from x, by writing them a 4. §17. When two simple quantities have been added, or one simple quantity subtracted from another, they then will consist of more than one term. §18. Quantities that consist of more than one term, are called compound quantities. Thus, a+b, is a compound quantity. So is a-b, and x+7, and x-7, &c. §19. It sometimes happens, that in those compound quan. tities which are made by adding or subtracting, there are two or more terms of the same kind; such as x+2x, 4a-2a, 5x-3x+x. Such quantities are called like quantities. §20. When in any compound quantity, there are two or more terms of the same kind, they may be united by performing the operation which is expressed by the sign. Thus, x+2x is united into 3x. 4a-2a is united into 2a. 5x-3x +x is united into 3x. §21. Numeral quantities* may be united in the same manner. Thus, 4+3 may be united into 7. 9-4 may be united into 5. 6—2+5 may be united into 9. EXAMPLES. Unite, as much as possible, the terms in each of the following compound quantities. 1. a+b+8—4+2a+6+3b-b+3a. Take one letter and go through the whole quantity with that first; and then take another letter and do the same. And then another, &c. Ans. Uniting the a's, they equal +6a; uniting the b's, they equal+3b; uniting the numeral quantities, they equal +10. Therefore the answer is, 6a+3b+10. 2. 2a+b+4a+2b+8+6-3a-b-4. 3. 3x+y+z+3x+3y+z+4x-2y+z. 4. 18-9+5+8a+4x+7x+5a-6x-7a. * Numeral quantities are expressed by figures; and literal quantities, by letters. 5. 4y+7%+9a-5a-4z-y+2z+3a+2y. 6. 7a-a-a+5b+4z+a-2z+a-z+4a. 7. 8—4+6+5—2+8+x+3x—2x+4x. §22. When any simple quantity begins, with the sign +, it is called a positive quantity; as+a,+3a. §23. When any simple quantity begins with the sign —, it is called a negative quantity; as, —6, —5a. §24. In algebra, the perfect representation of any simple quantity requires both the specified sum, and either the sign+, or the sign —; as, +5, 40, +x, −3x. §25. But, when a positive quantity stands by itself, or when it is the first term of a compound quantity; the sign that belongs to it, is generally omitted on paper, and also in our reading; as, x, 2, a+b, x—y. §26. Therefore, when a simple quantity, or the first term of a compound quantity, does not begin with a sign, we say that the sign + is understood. That is, we think of the quantity the same as if + was before it. §27. In reading compound quantities, the pupil must be careful to join the sign to the term that is immediately after it. Thus, the first example under §21, must be read a ; plus b; plus 8; minus 4; plus 2a; &c. We are now ready for the following GENERAL RULE FOR UNITING TERMS. §28. Select one kind of like quantites, and add into one sum all the positive co-efficients that belong to them. Then add into another sum all the negative co-efficients that belong to them. Then subtract the less sum from the greater; and prefix the sign of the greater to the difference, annexing the common letter. NOTE. It sometimes happens that the negative quantity is greater than the positive quantity. In such cases, the difference will have the sign Examples. Unite the terms in the following. 1. 3a+4b+2b-3c+3c-b+a. Now +66-b-+5b; and +3c-3c balance each other, so as to be equal to 0. The final answer is 4a+5b. 2. a+b+3a-c+4a-3c+b. 3. 5x+5y+3x-2y. Ans. 8a2b-4c. Ans. 8x+3y. . 4. b+a+3a-5b+4x-a+3a-x. Ans. 6a-4b+3x. 5. 4a-5b-10+2y-x+4. Ans. 4a-5b+2y-x-6. 6. 2x+3y-3x+5y-x-z. 7. x+z+x−x+x+3y. 8. 3a-5-4b+6—2a-3+6b. 9. 4x+4+5y+6-2+3x-2y. Ans. -2x+8y-z. Ans. 3x+3y. Ans. a+2b-2. Ans. 7x+3y+8. Ans. 5c-2a-2b. 10. 4c+3a-b-3a+c-2a-b. 20. a-b-a-b-67-42+7a+3b. |