and therefore, (Art. 147), multiplying by 18, 2y-4x+1-18-24-6y+3x-3y;

.. by transposition, 7-7x-11y.

But from the second equation, 7x=12y. Substituting therefore this value in the preceding

equation, it becomes

and

3x+2y y-5 11x+152_3y+1

to find the values of x and y.

Multiplying the first equation by 33,

33x-9y6-3x=33+15x+.

4y

; 3

multiplying again by 3, and transposing, we shall

have 45x-31y=81.

Multiplying the second equation by 12,
6x+4y-3y+15=11x+152-18y-6;

.. by transposition, 19y-5x=131.

Multiplying this by 9, 171y-45x=1179; but 45x-31y= 81;

.. by addition, 140y=1280; and by division, y=9.

Now, 5x-19y-131-171-131=40;

.. by division, x=8.

to find the values of x and y.

Multiplying the first equation by 105, the least common multiple of 3, 7, and 15, 560+21=1925-60x-45y+120;

.. by transposition, 81x+55y=1485; and dividing by 9, 9x+5y=165.

From the second equation,

50y+6x-35=275+50x;

.. by transposition, 50y-44x=310; and dividing by 2, 25y-22x=155;

but multiplying the equation found above, by 5,

25y+45x=825;

y

Multiplying the first equation by c, and the se

cond by a, we shall have

ac

bc

+

=mc,

... by subtraction, (bc-ad).=mc―na ;

y

Multiplying the first equation by 15y,

... 45y-21y-6x=75y-25x-45; and by transposition, 51y-19x=45.

Multiplying the second equation by 2x+5, 8x+20+30xy+75y

2xy+5y

6x--2

107

=2xy·

8

6x-2

107_8x+20+30xy+75y,

.. (Art. 186) 5y+ 8

and multiplying by 6x-2, we shall have

and 321x-107=32x+340y+80;

by transposition, 340y-289x187. 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,

consequently, 51y=45+19x=45+57=102;

and x=3;

3x-2y+1

to find the values of x and y.

Multiplying the first equation by 5+2y,

40x+16xy

80+300x+32y+120xy
3y-1

=16xy-107;

... trans" 40x+107=

80+300x+32y+120xy

3y-1

and multiplying by 3y-1, we shall have 120xy-40x+321y-107=80+300x+32y+

120xy;

.. by transposition, 289y-340x=187.

And from the second equation,

27x-12y +15x+2y+2=27x2-12y2+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.