x in one equation, and y in the other; also x and y are both multiplied by 8. (Art. 51.) All such circumstances enable us to resort to many pleasant expedients which go far to teach the true spirit of algebra. Add these two equations, and ++8(x+y)=325. 8 Or let s represent the sum of x+y, then 18+8s=325. Clear of fractions, and s+64s=325×8. Unite and divide by 65 and s=5×8. Or x+y=5a. (A) By returning to the value of s, and Divide by 63 and y=3a=24. Whence x=2a=16. Let the pupil take any one of the formal rules for the solution of the preceding equations, and mark the difference. 6. Given r+3y=21 and y+3x=29 to find x and y. Ans. x 9. y=6. 7. Given 4x+y=34 and 4y+x=16, to find x and y. Ans. x=8. y=2 8. Given x+y=14 and x+y=11, to find x and y. Ans. x 24. y=6. 9. Given x+y=8 and +x+y=7 to find x and y. Ans. x=6. y=4. 10. Given 4x-7y=99 and 4y+-7x=51 to find x and y. Ans. x=7. y=14. (x-12=4y+8 S_2y-x 11. Given +++xs=2 5. 4 +27 Ans. x 60. y=40 shall have + b=6 and y Multiply the first equation by c, the second by a, and we Putting this value in equation (C) and reducing we find 15. Given (x+150: y-50::3: 2) to find x and y. x-50: y+100::5:9) Ans. x=300. y=350. 16. Given 3x+by+1=6x2-24y2 +130 2x-4y+3 151-16х9ху-110 } to find x & y. Ans. x=9. y=2. NOTE. For solutions of examples 15 and 16, see Universal Key to the Science of Algebra, page 26. 17 Given (3x-y-3 to find the values of x and y -x+7y=33 19. Given x+y=8 and x2-y2=16 to find x and y. Ans. x=5. y=3. 20. Given 4(x+y)=9(x-y) and x2-y2=36 to find x & y. Ans. x=6. y=21. 21. Given x : y::4:3 and x-y3=37, to find x and y. Ans. x=4. y=3. 22. Given x+y=a and x2-y2=ab to find x and y. 24. Given (x+2)+8y=31 and (y+5)+10x=192 to Ans. x=4. y=15. find the values of x and y. Ans. x=19. у=3. 25. Given 3x+7y=79 and 2y+x=19 to find the values of x and y. Ans. x=10. y=7. 26. Given (x+y)+25=rand(x+y)-5=y to find the values of x and y. Ans. x=85. y=35. CHAPTER III. Solution of Equations involving three or more unknown quantities. (Art. 52.) No additional principles are requisite to those By the 1st method, transpose the terms containing y and z in each equation, and x= 9- y-z, x=16-2y-3z, x=21-3y-4z. Then putting the 1st and 2d values equal, and the 2d and Hence, 5-z-7-2z, or z=2. Having z=2, we have y=5-z=3, and having the values of both z and y, by the first equation we find x=4. 2. Given { 2x+4y-3z=22) 4x+2y+5z=18 to find of x, y (6x+7y- z=63 and z. Multiplying the first equation by 2, 4x+8y- 6z=44 And subtracting the second, The result is, (A) 4x-2y+5z=18 10y-112-26 Then multiply the first equation by 3, 6x+12y-9z=66 Substituting the value of z in equation (B) and we find y=7. Substituting these values in the first equation and we find x 3. 3. Given 3x-9y+82-41 5x-4y-2z-20 to find x, y and z. (11x-7y-62-37) To illustrate by a practical example we shall resolve this by the principles explained in (Art. 51.) 3mx-9my-+8mz=41m 5nx-4ny-2nz=20n Sum (3m+5n)—(9m+4n)y+(8m-2n)z=41m+20n -6z-37 Rem. (3m+5n-11)x+(7-9m-4n)y+(8m--2n+3)z= From equations (1) and (2) we find m=-1+ and n=1. These valucs substituted in equation (3) we have Multiply both numerator and denominator by 11, and we shall have z=-123-520-407-10 =1. -24-52-66 -10 |