LESSONS IN ALCJKHKA. 





contains only one unknown quantity. Whence, y = ah + bh, 

 and x = ah 1 -f- bh-. 



This process is called extermination by substitution. 



CASE II. To exterminate an unknown quantity by sub- 

 stitution. 



I; i I,K. Find the value of one oftJte unknown quantities, in one 

 of the equations, in terms of the other unknown; and then in the 

 other equation SUBSTITUTE this value for the former unknown 

 quantity, h'rom this equation, find the value of this unknown 

 quantity, <i,< ' 



MPLE (5). Given * + 3y = 15, and 4-f 5y = 32; to find 

 the values of * and y. 



Hero, transposing 3y in the first equation, we have. 

 = 15 :fi/. 



Substituting the value of x in the second equation, we have, 

 60 12y + 5y = 32 ; 



Whence, by transposition, etc., 



y = 4. 



And, from the first equation, 



*= 15 12 = 3. 



There is a third method of exterminating an unknown quantity 

 from an equation, which, in many cases, is preferable to either 

 of the preceding. 



EXAMPLE (6). Given x -f 3y = a, and * 3y = b ; to find the 

 values of * and y. 



Here, if we add togetlier the first members of those two equa- 

 tions, and also the second members, we shall have, 



2x = a -f- b, 



an equation which contains only the unknown quantity x. The 

 other, having equal co-efficients with contrary signs, has disap- 

 peared. Still the equality of the sides is preserved, because we 

 have only added equal quantities to equal quantities. 



Whence, x = ^4 > 



i j 

 And 



x a b 



EXAMPLE (7). Given 3a;-f-y = ft, and 2*-{-y =d; to find the 

 values of x and y. 



Here, if we subtract the second equation from the first, we 

 shall have x h d, where y is exterminated, without affecting 

 the equality of the sides. Whence, y = 3d 2h. 



EXAMPLE (8). Given x 2y = a, and x -\-4y = b ; to find 

 the values of x and y. 



Here, multiplying the first equation by 2, we have, 



2* 4y 2o ; 

 Then, adding the second and third equations, we have, 



3* = b + 2o ; 

 Whence, x = J (fr -f 2a), 

 And y = -J (6 a). 



This process is called extermination by addition and sub' 

 traction, 



EXERCISE 37. 



1. Given 8 + y = 42, and 2* + 4y = 18 ; to find the values of x and y. 



2. Oiven 2z + 8y = 84, and 4* + 6y ' = 08 ; to find the values of x and j. 



3. Given 3* + 3y = 72, and 4* + 5y = 116 ; to find the values of z 



and y. 



4. Given J* + lOy = 124, and 2* + 9y = 124 ; to find the values of x 



and y. 



5. A privateer in chase of a ship 20 miles distant, sails 8 miles, while 



the ship sails 7. How far will each sail before the privateer will 

 overtake the ship ? 



6. The age* of two persons, A and B, are such that seven yean ago A 



was three times as old as B ; and seven years hence, A will be 

 twice as old as B. What is the age of each ? 



7. There are two numbers, of which the greater is to the less as 3 to 



2 ; and their sum is the sixth part of their product. What are 

 the numbers ? 



CABE III. To exterminate an unknown quantity by addition 

 and subtraction. 



RULE. Multiply or divide the equations, if necessary, by such 

 factors that the term which contains one of the unknoivn quantities 

 shall be the same in both equations. Then subtract one equation 

 the other, if the signs of this unknown quantity are alike, or 



add them together if the tiyns are unlike ; the result will be an 

 equation containing only one unknown, quantity, which ie to be 

 resolved as before. 



It mutt be kept in mind that both members of an equation 

 are always to bo increased or diminished alike, in order to pre- 

 serve their equality. 



EXAMPLE (9). Oiven 2* -f- 4y = 20, and 4* + 5y = 28 ; to 

 find the values of z and y. 



Here, multiplying the first equation by 2, we hare, 

 4t> + 8y = 40. 



Subtracting the second equation from this, we hare, 



Whence, y = 4, and = 2. 



In the solution of the succeeding problems, either of the three 

 rules for exterminating unknown quantities may be used at 

 pleasure. That quantity which is the least involved should be the 

 one chosen to be first exterminated. 



The student will find it a useful exercise to solve every 

 example by each of the several methods, and carefully to observe 

 which ia the most comprehensive, and the best adapted to 

 different classes of problems. 



EXAMPLE (10). To find a fraction such that, if a unit be 

 added to the numerator, the fraction will be equal to $ ; but if a 

 unit be added to the denominator, the fraction will be equal to J. 



Let x = the numerator, and y = the denominator. 



Here, by the first condition, we have * = J ; 



And by the second, we have 



Whence, * = 4, the numerator ; 

 And y = 15, the denominator. 

 Therefore, ^ is the required fraction. 



EXERCISE 38. 



1. Given 2r + y = 16, and 3r 3y = 6 ; to find the values of z and y. 



2. Given 4r + 3>j = 50, and 3z 3y = 6 ; to find the value* of z and y. 



3. Given 3* + y = 38, and 5i + 4y = 68 ; to find the values of r and y. 



4. Given 4z 40 = ly, and 6z 63 = 7y ; to find the value* of 



and y. 



5. The number* of two opposing armies are such, that the sum of 



both is 21,110; and twice the number in the greater army, 

 added to three times the number in the less, i* 52,219. What 

 is the number in each army ? 



6. The sum of two numbers is 220, and if three times the lea* be 



taken from four times the greater, the remainder will be 180. 

 What are the numbers ? 



7. The mast of a ship consist* of two parts ; one-third of the lower 



part added to one-sixth of the upper part, ia equal to 28 feet ; 

 and five times the lower part, diminished by six time* the upper 

 part, i* equal to 12 feet. What is the height of the mast ? 



8. What two number* ore those, whose di/trence is to their sum a* 2 



to 3 ; and whose sum is to their product a* 3 to 5 ? 



9. To find two numbers such that the product of their sum and 



difference shall be 5, and the product of the sum of their 

 squares and the difference of their square* shall be 65. 



10. To find two numbers whose sum ia 32, and whoae product i* 240. 



11. To find two numbers whose sum is 52, and the sum of their 



square* 1,424. 



12. A certain number consists of two digits or figure*, the sum of 



which i* 8. If 36 be added to the number, the digit* will be 

 inverted. What is the number ? 



The united age* of A and B amount to a certain number of yean, 

 consisting of two digits, the sum of which is 9. If 27 year* 

 be subtracted from the amount of their ages, the digit* will 

 be inverted. What is the sum of their ages f 



A merchant having mixed a quantity of brandy and gin, found if 

 he had put in 6 gallon* more of each, the compound would have 

 contained 7 gallons of brandy for every 6 of gin ; but if he had 

 put in 6 gallons less of each, the proportion* would have been 

 a* 6 to 5. How many gallon* did he mix of each ? 



13. 



It. 



KEY TO EXERCISES IN LESSONS IN ALGEBRA. 

 EXERCISE 34. 



1. d' + 5d*?i + lOd'h* + 10dfc + 5<U -f h. 



2. b" +j4l'- l y + B6*-V + Cf - V + Do* V + etc., in which 



the co-efficients which are here represented by A, B, C, etc., are 



1 n I 2 . 

 respective!/ n.n. , . . -^~, tc. 



