ALGEBRA 



Addition. The processes by which problems 

 in addition are solved algebraically are much 

 like those employed in arithmetical addition. 

 n in arithmetic we add 4 and 6 we get a 

 term expressing the result of this addition; it 

 is 10. In algebra if we add a and a we obtain 

 the term 2a for a sum. If, however, we add a 

 and b, we obtain no single term which will 

 express this sum. To express the addition of 

 algebraic quantities which are unlike we con- 

 nect the quantities with the sign +. To ex- 

 press the addition of algebraic quantities which 

 tike, or similar, we add their coefficients. 

 For example, 2.r plus 3x plus 4j=(2+3+4)x 

 =Qx. But x plus y plus z=x+y+z. When- 

 ever two or more unlike quantities are added 

 the operation is algebraically complete when 

 the quantities are connected by the sign +. 



Add 3a, 46, 6a and b. In this problem like 

 terms are used twice. We must combine these 

 before completing our addition that we may 

 have the expression in its simplest form. Com- 

 bining similar terms: 



46+ 6=56. 

 The addition completed is expressed: 



3a+46+6a+6=9a+56. 

 The problem may be given this form : 

 3a+46 

 6a+ b 



9a+56 



In the illustrative problems given above, all 

 the terms have the plus sign expressed or un- 

 ood. cWhen no sign is expressed the sign 

 + is always understood.) In the following 

 problems note that some of the terms have 

 minus signs. In each case arrange like terms 

 under each other in columns. Add like terms 

 having plus and minus signs separately, then 

 subtract the quantity representing the larger 

 sum from that representing the lesser. 



Add: 20 s 6 2 c+66d 2 +2d 3 ; 4a 3 +36 2 c 46d 2 

 3d 3 ; 3a 3 +26 2 c+26d 2 4cP ; 2a 3 86 2 c-h66d 2 



Arranging the terms m columns and adding: 

 20 3 6 2 c+ 66d 2 +2d 3 

 4a 3 +36 2 c 4M 2 3d 3 

 3a 3 -i-26 2 c+ 2bd* 4rf 3 

 20 3 86 2 c+ 



7C 3 46 2 c+106<f 2 +d 3 



An explanation of any column will make 

 these operations clear. Suppose we take the 

 second, which contains the term 6 2 c. Adding 

 the terms having the plus sign, we have 36 2 c 



190 ALGEBRA 



plus 26 2 c; their sum is equal to 56 2 c. Adding 

 6 2 c and 86 2 c, we obtain 96 2 c; 9b 2 c 

 plus 56 2 c equals 46 2 c. The quantity obtained 

 in adding two like terms having unlike signs 

 always takes the .s/r/7i oj the greater. 

 Solve the following: 



1. Add: 4x 3 +3?/+5z; 2x*+2y4z; Sx 3 8y 

 z. 



2. Add: 3a 26 c; a+36 2c; 3a 66+c. 



3. Add: 3a-f46+7?/; 26 3a+2y; 2a 56 

 7y; 2a+26+2y. 



Subtraction. It is sometimes difficult for the 

 beginner in algebra to understand the reason 

 for the rule for algebraic subtraction. We will 

 state it here as it is usually given, and then 

 explain it step by step, using practical problems 

 for illustration. The rule is: 



Set the like terms one under the other in the 

 minuend and subtrahend, then change all the 

 signs of the subtrahend and proceed as in ad- 

 dition. 



We have learned the principles underlying 

 addition, and know that the algebraic sum of 

 8a and 4a equals 4a. 



8a first quantity 

 4a second quantity 



4a sum. 



In adding in arithmetic we know that if 

 either of two numbers be subtracted from their 

 sum, the difference must be the other number. 

 Here, then, if 4a is subtracted from 4a, the 

 remainder must equal the first number, which 

 is 8a. This is simply an application of an 

 arithmetical truth, that in addition, if either of 

 two terms is subtracted from their sum, the 

 result, or remainder, is the other term ; though 

 the result may look unreal, it must be correct 

 because the above rule is correct. 



Let us show further proof: If we add 8a 

 and 4a the sum is 4a: 



8a first number 

 4a second number 



4a sum. 



Subtract 4a from 4a, and the remainder, if 

 the rule of arithmetic is correct, must be 8a, 

 for 8a is the other number. 



Again, the sum of 8a and 4a is 12a, 

 and the remainder must be the first term, 8a. 

 This will be made clearer if these last three 

 problems in subtraction are placed side by 

 side: 



Minuend 4a 4a 12a 



Subtrahend 4a 4a 4a 



Remainder 



8a 8a 8a 



