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EXAMPLES.

Given {7x+90=59} 3x+8y=152} to find the values of x and y.

Multiply the second equation by the indefinite quantity m, and we shall have 3mx+8my=132m.

Adding this equation to the first,

3mx+7x+8my-9y=132m+59.

7

Assume 3m+7=0, Whence m=

3'

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4. Given {18-3x=75, to determine x and y.

Result, x=15, y=12.

5. Given

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*+*+2+16=2x+27. Required the

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values of x and y?

4

Result, x=60, y=20.

Examples to exercise the foregoing rules.*

x-y=d,

Given {xy=p, to find x and y.

Squaring the first equation, x-2xy+y2=da.
Multiplying the second by 4, 4xy=4p.
Taking the sum, x2+2xy+y2=d3+4p.
Extracting the root, x+y=√d2+4p, (A.)

Adding the first, 2x=d+d+4p, or x=

d+do+4p

2

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n

Dividing the second by the first, x+y=.

m

n

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+m

Hence, adding and subtracting the first, 2x=

m

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In the solution of these and other similar problems, the ingenious student will find expedients which are not clearly indicated by any of the preceding rules, by which his labour may be frequently abridged.

3. Given{b, to find x and y.

Subtracting three times the first equation from the second, x3-3x2y+3xy2-y3-b3α..

Hence, extracting the root, x-y=3b-3a, which put=c, (A.)

α

Divide the first equation by this, whence xy== (B.)

To the square of equation (A,) add four times equation (B,) and we shall have

x2+2xy+y3=c3+·

Whence by evolution, x+y=√(c°+1a)

Adding and subtracting equation A, we find

2x=√(c2+1a)+c, and 2y=√(c2+1a)—

-C;

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4. Given {x+y=a,

2x+y3=b, to find x and y.

From the first x2+3x2y+3xy2+y3=a3.

By subtraction, 3x2y+3xy-a3—b.

Dividing by three times the first, xy=

put =c.

a3

3a

which

Subtracting four times this equation from the square of the first, x3-2xy+y2—a2-4c. ·

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Otherwise, dividing the second equation by the first,

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b

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Squaring the first equation, x+2xy+y2=a2.
Subtracting the latter from the former, we have

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And, by extracting the root, x-y=√2

4ba3

3a

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5. Given, to find x and y in terms of s and d.

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7. Given {Quere the values of x and y.

8. Given

Sxy:: 3:2,

xy-20, to find x and y.

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Ans. x=4, y=2.

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13. Given

[x2y+xy2=120,

and y?

x+y3=152. What are the values of x

14. Given 468, to find x and y.

Result, x=7, y=5.

15. Given {xy+84, to find x and

y.

Ans. x=5, y=3.

Result, x=8, y=6.

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* When we have two equations, and two unknown quantities, neither of which rises above the first power, the required values may be obtained, from the results of the 16th example, by simple substitution.

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