x2 + 21 = x2 + 2x + 1;

and (19) by squaring each side, x2 +

.. (17. Cor. 3.) x2 + 21 = 2x + 1,

and (19) by squaring each side, x2 + 21 = 4x2 + 4x + 1, which is free from surds.

And, if a'x + √a2x3 = c,

(19) by cubing each side, ax + √ax3 = c2,

and (17) by transposition, a33 = c

.. (19) by squaring each side, ax = c - 2a2cx + a*x2, which is free from surds.

(21.) Any proportion may be converted into an equation; for the product of the extremes is equal to the product of the

means.

Let a : b :: c:d, by the nature of proportion= .. (18. Cor. 1.) ad = bc.

C

(22.) EXAMPLES in which the preceding Rules are applied, in the Solution of Equations.

1. Given 4x + 36 = 5x + 34, to find the value of x.

(17) By transposition, 36 - 34 = 5x

- 4x,

Here 15, the product of 3 and 5, being their least common multiple, every term must be multiplied by it (18. Cor. 1.), and 15x - 105 = 3x + 5x;

3. Given зах - 4ab = 2ax - 6ac, to find the value of æ in terms of b and c.

(18. Cor. 2.) dividing every term by a, 3x —

.. (17) by transposition, 3x - 2x = 46 - 6c,
or x = 46 6c.

2

4. Given 3x2 - 10x = 8x + x2, to find the value of x.

(18. Cor. 2.) dividing every term by x, 3x

.. by transposition, 3x - x = 8 + 10,
or 2x = 18;

10 = 8 + x;

Here 12 is the least common multiple of 2, 3, and 4; (18. Cor. 1.) multiplying both sides of the equation therefore

by 12, 6x + 4x = 3x + 84;

.. (17) by transposition, 6x + 4x 3x

or 7x = 84;

.. (18. Cor. 2) x =

84,

1

(18. Cor. 1.) multiplying by 20, the least common multiple

of 4 and 5,

5x 25 + 120x = 1136 - 4x;

.. (17) by transposition, 5x + 120x + 4x = 1136 + 25,

(18. Cor. 1.) multiplying by 6, the least common multiple

.. (17) by transposition, 6x 2x + 3x = 57 - 22,
or 7x = 35;

(18. Cor. 1.) multiplying by 10, the least common multiple of 2 and 5,

30x + 4x + 12 = 50 + 55x 185;

.. (17) by transposition, 12 50 + 185 = 552 30x

or 147 = 21x;

(18. Cor. 1.) multiplying every term by 3,

6x 4-6-18 4x + 3x;

and .. (17) by transposition, 6x + 4x - 3x = 18 + 6 + 4,

Since 16 contains 8 and 2, a certain number of times exactly, it will be the least common multiple of 16, 8, and 2; and therefore (18. Cor. 1.) multiplying both sides of the equation by 16,

(18. Cor. 1.) multiplying both sides of the equation by 6,

the product of 2 and 3,

6x + 9x

15 = 72 - 4x + 8;

.. (17) by transposition, 6x + 9x + 4x = 72 + 8 + 15,

Since 12 is a multiple of 3 and 4, it is the least common multiple of 3, 4, and 12; therefore (18. Cor. 1.) multiplying both sides of the equation by 12,

36x-3x + 12 48 20x + 56 1;

.. (17) by transposition, 36x 3x - 20x = 56 + 48 -1-12,

(18. Cor. 1.) multiplying both sides of the equation by ×5×7= 140,

20x 20 + 644 - 28x = 980 140

.. (17) by trans", 20x - 28x + 35x = 980 or 27x = 216;

35x;

(18. Cor. 1.) multiplying both sides by 2×3×5 = 30, 70x + 50 96-24x + 180 = 45x + 135;

.. (17) by transposition, 70x - 24х - 45x = 135 + 96 - 50

180,

(18. Cor. 1.) multiplying by 20, the least common multiple of 2, 4, and 5,

12 + 16 70x + 30 = 5x 80;

.. (17) by transposition, 16 + 30 + 80 = 5x + 70x - 120, or 126 = 63x;

3 × 5 = 15,

51- 9x-20х

(18. Cor. 1.) multiplying both sides of the equation by

.. (17) by trans", 90x - 35x - 20x - 9x = 75+70+10 - 51,

or 26x = 104;

10 = 75 90x + 35x + 70;