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Examples of Elimination by the method of the Greatest Common Divisor.

2. Obtain one equation with one unknown quantity from the two equations

23 + y = a,

25 + y = b, by the elimination of x.

Ans. (73 — a) 5 – (35 b)3 = 0. 3. Obtain one equation with one unknown quantity from the two equations

+y = 2,

24 + 23 y + x2 y2 + 2y3 + y4 =1, by the elimination of x.

Ans. y8 4y6 + 14 y4 – 20 y2 +9=0. 4. Obtain one equation with one unknown quantity from the two equations

22 + xy + y2 = 1,

23 + y3 = 0, by the elimination of x.

Ans. 4 yo — 6 74 + 3 y2—1=0. 5. Obtain one equation with one unknown quantity from the two equations

23 + y 22 +1+y=4,

23 + 22 + yx= 3, dy the elimination of x.

Ans. Either y-1=0, or y? - 3y +21=0. 6. Obtain one equation with one unknown quantity from the three equations

1 + y +z = a,
a z + xy + yz=b,

x y z = C, by the elimination of x and y.

Ans. 23. a z2 + bz-C=0.

Examples of Elimination by the method of the Greatest Common Divisor.

7. Obtain one equation with one unknown quantity from the three equations

2 + y +z = a,
2 + y2 + z2 = b,

x y + 2z+yz = C, by the elimination of x and y.

Ans. These three equations involve an impossibility unless

al-b2c=0; and in case this equation is satisfied by the given values of a, b, and c, the three given equations are equivalent to but two, one of them being superfluous, and, by the elimination of x, they give the indeterminate equation with two unknown quantities

y + yz + 22-ay-az+c=0. 8. Obtain one equation with one unknown quantity from the three equations

x + y =4,

2,

z + x2 = 10, by the elimination of u and y.

Ans. 28 -8 26 + 16 24 +% -10=0. 9. Obtain one equation with one unknown quantity from the four equations

2 + y +ztura,
xy+xz+uu+yz+yu+zu=b,
xyz + xyu+xzu+yzu=c,
x y zu=

ee,
by the elimination of x, y, and z.

Ans. 24 - a u3 + b ul-cute=0.

y + z2

Elimination by Addition and Subtraction.

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23 + x

=3

10. Solve the two equations

3, y x (y 22 + 1) — 23 + x=6. Solution. The elimination of a gives

3 y3=0, or y=1; which, being substituted in the first of the given equations, produces

3. 11. Solve the two equations 2Py4_8y2 x2 +1622 - 90 xy +60 (2- y2)-720 (7-1), (y2 – 4 y + 4) x

12 5

Ans. X=4, y=2. 12. Solve the three equations

2 y +z=5,
x y z + z = 15,
x y + xy - 2x + 2z=8.

Ans. x=2, y=1, z=3. 159. Problem. To solve two equations of the first degree by Elimination by Addition and Subtraction.

Solution. The given equations may, as in art. 146, be reduced to the forms

A x +By+M= 0,

A r+ B'y + M = 0. The process of the preceding article, being applied to these equations in order to eliminate x, will be found to be the same as to

Multiply the first equation by A' the coefficient of x in the second, multiply the second by A the coefficient of x in the first, and subtract the first of these products from the second.

Examples of Elimination by Addition and Subtraction.

Thus, these products are

A Aa + A' By + A' M = 0,

A A'x + A B'y + A M = 0; and the difference is

(A B' A' B)y + AM A'M=0; whence

AM - AM

A B - AB In the same way y might have been eliminated by multiplying the first equation by B', and the second by B, and the difference of these products is

(A B' — A' B) x + B' M - BM=0; whence

В М. В М

A B - A' B' the values of x and

Y

thus obtained being the same as those given in art. 146.

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160. Corollary. This process may be applied with the same facility to any equations of the first degree.

161. EXAMPLES.

1. Solve, by the preceding process, the two equations

13 x +7 y — 341 = 71 y + 434 x,
2x+4y=1.

Ans. x=-12, y=50.

Solve, by the preceding process, the two equations

$2+4y=6,
4 = ++ y = 5.
Ans. I= 12, y=

=16.

Examples of Elimination by Addition and Subtraction.

3. Solve, by the preceding process, the three equations

x + y + z = 30, 82 + 4y + 2z= 50, 273 +9y+3z= 64. Ans. I=

-7,2=361.

y

31

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4. Solve, by the preceding process, the three equations

100 = 5 y + 360,
23 + 200 = 164 z — 610,
2 y + 3z= 548.
Ans. =

360, y= 124, z= 100.

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5. Solve, by the preceding process, the four equations

1—9y + 3- 10u= 21, 2x + 7y

u= 683, 32 + y + 5z + 2 u= 195, 42—6 y – 2z — 9u= 516.

Ans. x=100, y= 60, z=- 13, u= -50. 6. Solve, by the preceding process, the four equations

x + } y + z =

a + dy + }z= = 76, *2 + gz + šu=79,

z + = 248. Ans. x=

- 12, y=30, z=168, u=50. 7. Solve, by the preceding process, the six equations

x+y++ttu=
x+y+z+u+w=21,
2 + y +z+t+w= 22,
atytutt+w=
utztutt tw= 24,4
y +z

+++w= 25.
Ans. a
-2, y 3, 25

= 4, u=5,1=6,w=7.

y +

20,

23,

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