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3*

Let

-10x=8x+x* be given ; to find t.
First, 34-10=8+*
And then 3*-x=8+10
Therefore 2x=18, and x==9.

5. Given 6ax 3 -12abx* =30x3 +60%"; to find x. First, dividing the whole by zaxo, we shall have

2x-46=x+2
And then 2x==2+46
Whence x=2+46.

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2x

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First, X-3+ 40-19

3
And then 3x–9+2x=120-—-34-57
And therefore 3x+2x+3x=120—57+7

"That is, 8x=72, or x= -9.
8. Let Vx+=7 be given ; to find x.

First, V x=-5=2
And then şx=2* =4
Ånd 2x=12, or x=6.

9. Les

2a 9. Let xtva txa

be given; to find x.

Va’+**
First, Xv/6* +** ta****=2a*
And then xvá* *x*=a.

+=***
And ** X^*+**=a*--**=a4—2a*x****
Or a'x' +*+=84-20*** tx
Whence a’x* +2a*x* =a*
Or zaʻx'=a*
And consequently x =

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a."

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1. Given x-t-18=3x-5; to find x. Ans. x=1

cd ta-6. 2. Given 3y-a+b=cd; to find y. Ans. y=

3 3. Given (2x+10=20—3—2; to find X.

Ans. X=2

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7. Given ✓ 12+x=2*/*; to find x.

Ans. x4.

8. Given

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8. Given Va*t**=6****; to find x.

Ans. *

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9. Given xta=Va*+xvb**** ; to find x.

62 Ans. x

4a

REDUCTION OF TWO, THREE, OR MORE, SIMPLE

EQUATIONS, CONTAINING TWO, THREE, OR MORE, UNKNOWN QUANTITIES.

PROBLEM I.

To exterminate two unknown quantities, or to reduce the two

simple equations containing them to one.

RULE I.

1. Observe which of the unknown quantities is the least involved, and find its value in each of the equations, by the methods already explained.

2. Let the two values thus found be made equal to each other, and there will arise a new equation with only one unknown quantity in it, whose value may be found as before.

EXAMPLES.

1. Given

2x+3y=23

; to find x and sa 5*2=IO

23-3Y From the first equation =

2.

10+2; And from the second X

5 23-3Y

10-4-27 And consequently

2.

5

O

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2x

3*

5. Given

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= and

4. Given 4x+y=34, and 43**=16; to find x and y.

Ans. =8, and y=2. 39

2y +

+2=*; to find 5

4. Ş * and

Ans. x=, and = yo 6. Given xt=s, and x-y=d; to find x and y.

so tod

sa. · Ans.

and

JF

4

25

25

RULE 2.

1. Consider which of the unknown quantities you would first exterminate, and let its value be found in that equation, where it is least involved.

2. Substitute the value thus found for its equal in the other equation, and there will arise a new equation with only one unknown quantity, whose value may be found as before.

EXAMPLES.

x+2y=17? 1. Given

i to find x and 3: 34y= 2 From the first equation x=17--23, And this value, substituted for x in the second, gives

17–2y X3=2,
Or 51.6=2, or 51---7)=2 ;

That is, 7y=51-2549 ;
Whence y=4<=7, and x=17-2y=17--14=3.

x+y=13? 2. Given

; to find x and xn--y= 3S From the first equation x=13-y, And this value, being substituted for x in the second,

Gives 13-y=3, or 13-2y=3;
That is, 2y=13--3= 10,
Or g='=5, and x=13-y=13--5=8.

y.

3. Given

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