and therefore, (Art. 147), multiplying by 18, 2y---4x+1=18-24-6y + 3x – 3y ; .' by transposition, 7=7x-lly. But from the second equation, 7x=12y. Substituting therefore this value in the preceding equation, it becomes 12y-117=7, or y=7. 12y 84 and .. x= =12, 7 7 154+ 47 3 33 11x +152 3y +1 2 4y 33x-9y+-6--3x=33+15x+ ; 3 multiplying again by 3, and transposing, we shall have 45x-319=31. Multiplying the second equation by 12, 6x+4y-3y +15=114+152-18y-6; .. by transposition, 19y-5x=131. Multiplying this by 9, 1714--45x=1179; but 45x - 3y= 81; .. by addition, 140y=1260; and by division, y=9. Now, 5x=197 - 131=171—131=40; .. by division, x=8. =18 43 80+-3x 40+3y-8 7 5 to find the values of x and y. Multiplying the first equation by 105, the least common multiple of 3, 7, and 15, 560+212=1925-60x—45y + 120; and dividing by 9, 9x+ 5y=165. From the second equation, 50y +6x-35=275+50x ; and dividing by 2, 25y-22x=155 ; but multiplying the equation found above, by 5, 25y+45x=825; • by subtraction, 67x=670, and by division, x=10. Now 5y=165-9x=165--90=75, ..y=15. 4x, 5y_9 Ex. 5. Given -- + 1 ya y to find the values 5 4 7 3 of x and and at + y. y 2 Reducing the first equation to lower terms, 4.5 9 1; X у у morf eht 2nd equation, by trans", + a 2 1 .. by addition, 2 and, consequently, x=4. Now 4+1=2; 4 .. 2y=4, and y=2. y 6 y to find the values of d tes -=R, y Multiplying the first equation by c, and the second by a, we shall have ac bc y ad and + =na, y x and y.. с ac 1 ... by subtraction, (bc-ad).-=mc--na; y bc-ad • Y тс — па a b mbc-nab And - m bc-ad mbc-mad-mbct nab_nab-mad bc-ad bc-ad be ad nb-md 5x +9 Ex. 17. Given 3 -5 5 Зу 107 2xy 4+15y. 8 6x-2 2c +5 to find the values of c and y. and y OC -2wy 8 Multiplying the first equation by 15y, .. 457--217-6x=754-25x--45; and by transposition, 517–19x=45. Multiplying the second equation by 2x+5, 8x +20+30xy+75y 107 2xy +5y ; 6.-2 8 107_8x +20+30xy+75y. ::. (Art. 186) 5y+ 6x-2 and multiplying by 6x-2, we shall have 321x107 30xy-10y+ =8x+20+30xy +-75y 4 3213-107 c. (Art. 186), =8x +85y + 20, 4 and 3213-107=32x+340y +30; .. by transposition, 340y-289x=-187. The coefficients of y in this case, having aliquot parts ; multiplying the first by 20, and the last by 3, 1020y--380x= 900, and 1020y-867x=-561; = ... by subtraction, 487x=1461, and x=3; consequently, 5ly=45+19x=45+57=102; ..y=2 16+60x _ 16xy-107 Ex. 8. Given Ex 3y-1 5+2y 27x2-12y +38 and 2+by+9x= 3x-2y+1 to find the values of u and y. Multiplying the first equation by 5+2y, 10x +-16xy 80+300x+32y+120xy =16xy-107; 3y-1 80+300x +32y+120xy ... trans" 40x+107 = 3y-1 and multiplying by 3y-1, we shall have 120xy~40x +3217-107=80+300x * 32y + 120xy ; .. by transposition, 289y— 340x=187. And from the second equation, 27x% -12y +15x+2y+2=27x02 - 12y2 +38; ... by transposition, 15x+2y=36; whence, the coefficients of x having aliquot parts, multiplying the first equation by 3, and the second by 68, 867y-1020x=561, and 136y+1020x=2448 ; .. by addition, 1003y=3009, and y=3; consequently, 15x=36 - 2y=36-6=30; and .. by division, x= 2. 2y— 59- 2x Ex. 9. Given x -20 23-3 2 73— 3y 8-18 3 to find the values of x and y. Ans. x=21, and y=20* and yt Y-3 = 30 33 -1 5 = 6 to find the values of x and y. Ans. x=7, and y=5. |