Thus, if x+5= 8; then transposing 5 gives x = 8—5—3, And, if x- -3+7=9; then transposing the 3 and 7, gives x=9+3-7= 5.

Also, if xa+b= cd: then by transposing a and b, it is xa- b+cd.

In like manner, if 5x

6=4x+10, then by transposing

6 and 4x, it is 5x-4x= 10 + 6, or x = 16.

RULE II.

WHEN the unknown term is multiplied by any quantity; divide all the terms of the equation by it.

Thus, if axab-4a; then dividing by a, gives x — b— 4. And, if 3x+5= 20; then first transposing 5 gives 3r 15; and then by dividing by 3, it is x = 5.

In like manner, if ax + 3ab4c2; then by dividing by a, it

402

is x+36=

a

4c2

-; and then transposing 36, gives ≈ — —- — .36.

a

RULE III.

WHEN the unknown term is divided by any quantity; we must then multiply all the terms of the equation by that divisor; which takes it away.

Thus, if 3+ 2; then mult. by 4, gives x12+8=20.

4

x

And, if 3b + 2c - d:

a

then by mult. a, it gives x = 3ab + 2ac -ad.

order in which they are here placed; and beginning every line with the words Then by, as in the following specimens of Examples; which two words will always bring to his recollection, that he is to pronounce what particular operation he is to perform on the last line, in order to give the next; allotting always a single line for each operation, and ranging the equations neatly just under each other, in the several lines, as they are successively produced.

RULE

RULE IV.

WHEN the unknown quantity is included in any root or surd; transpose the rest of the terms, if there be any, by Rule 1; then raise each side to such a power as is denoted by the index of the surd viz. square each side when it is the square root; cube each side when it is the cube root; &c. which clears that radical.

Thus, if x-3=4; then transposing 3, gives √x=7; And squaring both sides gives x = 49.

And, if 2x + 10 = 8:

Then by squaring, it becomes 2x + 10 ≈ 64 ;
And by transposing to it is 2x =

54;

Lastly, dividing by 2, gives x = 27.

WHEN that side of the equation which contains the unknown quantity is a complete power, or can easily be reduced to one, by rule 1 2, or 3 then extract the root of the said power on both sides of the equation; that is, extract the square root when it is a square power, or the cube root when it is a cube, &c.

=36:
Thus, if x 8x + 16 = 36, or (x + 4)2
Then by extracting the roots, it is x + 4 = 6;
And by transposing 4, it is x = 6

And if 3x2 1921 35.

25;

4 = 2.

Then, by transposing 19, it is 3x2 = 75;
And dividing by 3, gives x2 =
And extracting the root, gives x = 5.

Also, if 3x2 - 6 = 24.

Then transposing 6, gives r = 30;
And multiplying by 4. gives 3r2 = 120;

Then dividing by 3, gives r2 = 40;

Lastly, Extracting the root, gives x = √ 40 = 6·324555.

VOL. I.

Hh

RULE

RULE VI.

WHEN there is any analogy or proportion, it is to he changed into an equation, by multiplying the wo extreme terms together, and the two means together, and making the one product equal to the other.

Thus, if 2x9:: 3:5.

Then, mult. the extremes and means, gives 10x = 27;
And dividing by 10, gives x = 2,170.

And if xa: 56: 2c.

Then mult extremes and means gives cx = 5ab ;

And multiplying by 2, gives 3cx = 10ab;

Then mult extremes and means, gives 10 - x=2x ;
And transposing x. gives 10 = 3x ;

Lastly, dividing by 3, gives 3 = x.

RULE VIL

WHEN the same quantity is found on both sides of an equation, with the same sign, either plus or minus, it may be left out of both and when every term in an equation is either multiplied or divided by the same quantity it may be struck out of them all.

Also if there be 4ax + 6ub = 7ac.

Then striking out or dividing by a, gives 4x+ 66 = 7c. Then, by transposing 66, it becomes 4x=7c6b;

And then dividing by 4 gives x = Zc — 36.

Again, if x = }}·

-

Then, taking away the 7, it becomes 3x=;
And taking away the 3's, it is 2x = 10;
Lastly, dividing by 2 gives x

5.

MISCELLANEOUS EXAMPLES.

I. Given 7r

18=4x+6; to find the value of x. First, transposing 18 and 5x gives 3x — 24; Then dividing by 3, gives

8.

2. Gives

2. Given 20 4x 12 = 92 10x; to find .

--

First transposing 20 and 12 and 10x, gives 6x=84;
Then dividing by 6, gives x = 14.

3. Let 4ax - 56 - 3dx + 2c be given; to find x.
First, by trans. 56 and 3dx, it is 4ax 3dx

-

3d, gives x =

56 +2c

--

4a-3d

56 + 2c;

4. Let 5x2 12x = 9x + 2x be given; to find x.
First, by dividing by x, it is 5x 12 = 9+2x;
Then transposing 12 and 2x, gives 3x = 21;
Lastly, dividing by 3, gives r=7.

3

3. Given 9ax3 15abx2 = 6×2 + 12ax2; to find x. First, dividing by 3ax2 gives 3x 56 2x + 4; Then transposing 56 and 2x, gives x = 56 + 4.

First, multiplying by 3, gives x − {x+}x=6;
Then multiplying by 4, gives x + x = 24.
Also multiplying by 5 gives 17x = 120,
Lastly, dividing by 17, gives x = 77

x + 10. Then transposing 5 and x, gives 2x + 3x = 51; And multiplying by 2, gives 7x=102. Lastly, dividing by 7, gives x 144.

3x

8. Let +7 = 10, be given; to find x.

4

First, transposing 7, gives ✔ x = 3;
Then squaring the equation, gives 3x = 9;
Then dividing by 3, gives x = 3;
Lastly, multiplying by 4, gives x = 12.

9. Let 2x + 2√/a2 + x2 =

5a2

a2 +

be given; to find x.

First, mult. by✔a3 + x2, gives 2x √ a2 + x2+2a2 +2x2

= 5a2.

Then transp. 2a3 and 2x2, gives 2x √a2+x2=3a2—2xa ;

Then

2

2

2x2

2

;

Then by squaring, it is 4xa ×â2+x2 = 3a2
That is, 4a2 + 4x4 = 9a+ 12u2x2 + 4x* ;
By taking 4* from both sides it is 42x
Then transposing 12a2x2, gives 16a2x2,
Dividing by a2, gives 1· æ2
402;
And dividing by 16. gives æ2 = 2
Lastly extracung the root, gives x =

24.

= 4a4;

EXAMPLES FOR PRACTICE.

Ans. x5. Aus. a =

1. Given 2x-5 + 16 = 21; to find x. 2 Given 9x- 15 = x + 6; to find x. 3 Given 8-3x + 12=30—5x+4; to find x. Ans. x = 7.

Ans x 12.

4. Given x+}x−‡a

=

5. Given 3x + x + 2 = 6. Given 4ax + za− 2

13; to find x.
5.x-4; to find x.
= ax-bx; to find x.

Ans. x4.

6

13. Given a + x = √ a2 + x √ 462 + x2; to find x.