In all these cases there is one general mode of procedure, namely, we exterminate all the unknown quantities from the operation, except one, the value of which is to be found by the foregoing methods; having obtained the value of this, the value of each of the other unknown quantities will be readily found by means of it, from some of the preceding equations. For two unknown quantities. 90. First Method. RULE 1. Find the value of one unknown quantity in each of the equations, by the foregoing rules; it may be either of the two quantities at pleasure, but must be the same unknown quantity in both equations. II. Put the two values thus found equal to each other '; this equation will then contain but one unknown quantity, the value of which is to be found by the preceding rules. III. Having thus found the value of one unknown quantity, substitute it for that quantity in either of the preceding equations, and the value of the other unknown quantity will be found. EXAMPLES. 1. Given x+y=13, and x-y=3, to find the value of x and y. First, to find the value of x in each equation. From the first, x=13-y; and from the second, x=3+y; these two values of x are evidently equal to each other; wherefore 13-y=3+y, whence 2y=10, and y=5; now substitute this value of y in either of the former equations, suppose in x=3+y, and it becomes x= (3+y=) 3+5=8. 2. Given 2x+3y=17, and 3 x-2y=6, to find x and y. From the first equation, 2x=17-3y, e These two values being equal to the same unknown quantity, are evidently equal to one another: the unknown quantity, whose two values (or rather, two different expressions of the same value) are thus found, is said to be exterminated, because it does not appear in the resulting equation, Making these two values of x equal to each other, we shall have this equation cleared of fractions, (Art. 79.) becomes 51-9y=12+4y. Whence 13 y=39, and y=(==) 3. 39 13 4. Given x+4y=18, and x-3y=4, to find x and y. Ans. x=10, y=2. 5. Given 4x+3y=25, and 5x-4y=8, to find x and y. 8. Given +3y=217, and x: √y::5: 4, to find x and y. x 4 Ans. x=100, y=64. 91. Second Method. RULE I. Find the value of either of the unknown quantities in one of the given equations, and substitute this value for that quantity in the other given equation; this equation will then contain only one unknown quantity, which may be found as before f. This rule is evident, for it is plain that in any expression whatever, we II. Find the value of the other unknown quantity, as directed in the preceding rule. 9. Given x+y=20, and x-y=8, to find x and y. From eq. 2.x=8+y; this value being substituted for x in the first equation, (x+y=20,) gives 8+y+y=20, or 2y+8=20, whence 2y=12 and y=6; this value of y being substituted for y in the equation, x=8+y, gives x=8+6=14. 10. Given 2x+3y=7, and 3 x-2y=4, to find x and y. 21-9y stituted for 3x in the second equation, it becomes -2y=4, 2 which by multiplication and transposition, becomes 13 y=13, whence y=1; this value being substituted for y in the equation it becomes 7-3 -=2, for the value of x. being substituted in the given equation, we have, +z 2 5 5 + 3 4 whence 59 z=59 × 15, or z=15; this value substituted 15 12. Given x+y=10, and x-y=2, to find x and y. Ans. x=6, y=4. 13. Given 2x+32=38, and 6x+5 z=82, to find a and z. Ans. x=7, z=8. 14. Given y-2 z=7, and 2 y―z=20, to find y and z. Ans. y=11, z=2. can substitute their equals, instead of any of the quantities which compose it, without altering the value of that expression. 15. Given 2x+3y=34, and x+y: x−y :: 7 : 1, to find and y. Ans. x=8, y=6. 92. Third Method. RULE I. Multiply the first equation by the coefficient of one of the quantities in the second, and the second equation by the coefficient of the same quantity, in the first; the products will be two new equations, in both which the coefficients of that quantity will be the same. II. If the terms, with equal coefficients, have like signs, subtract one of the new equations from the other; but if they have unlike signs, add the new equations together: the result will be an equation with only one unknown quantity, which may be found as before. 16. Given 2x+5y=23, and 7 x-3y=19, to find x and y. To exterminate x, multiply the first equation by 7, and the second by 2, and the products (or new equations) will be 14x+35y=161. New equations. And 14x-6y= 39.} Whence 41 y=123 by subtracting the lower from the Now to exterminate y, multiply the first given equation by 3, and the second by 5, and the new equations will be 6x+15y=69. And 35x-15=95. New equations. Whence 41 x = 164 (by adding) and x=(- 41 We are at liberty to employ any process, where equals operate in a similar manner upon equals; under this restriction, we are authorised to make use of addition, subtraction, multiplication, division, involution, and evolution, according as it suits our purpose; in this rule equal multiplication is used, but sometimes equal division, when it can be used, makes the work shorter. This equation multiplied by (the least common multiple of its denominators, viz.) 15, gives (12 y+5y=) 17 y=272; whence y= 272 17 =) 16; this value substituted for y in the first given equation, 2 equations together, we get 2 x=(1—_—_=)—~~, whence x=; and by 4 3 subtracting the second from the first, 2y=(1 whence 19. Given 2x+3y=13, and 5x-2y=4, to find x and y. Ans. x=2, y=3. 20. Given x+y=5, and x-y=1, to find x and y. x=3, y=2. 21. Given 7y+9 z=169, and 8z-9y=−2, to find y Ans. y=10, z=11.' 22. Given 2x+9=5z+8, and 3 z−x=x+ Answer and z. to find x and x. For three unknown quantities. 93. First Method. RULE I. Let x, y, and z, be the three unknown quantities, whose values are sought; first find the value of x in each of the three given equations. |