occupied with results of this kind, that is, with identical equations. An identical equation is called briefly an identity. 155. An equation of condition is one which is not true whatever numbers the letters represent, but only when the letters represent some particular number or numbers. For example, x+1=7 cannot be true unless = 6. An equation of condition is called briefly an equation. 156. A letter to which a particular value or values must be given in order that the statement contained in an equation may be true, is called an unknown quantity. Such particular value of the unknown quantity is said to satisfy the equation, and is called a root of the equation. To solve an equation is to find the root or roots. 157. An equation involving one unknown quantity is said to be of as many dimensions as the index of the highest power of the unknown quantity. Thus, if a denote the unknown quantity, the equation is said to be of one dimension when a occurs only in the first power; such an equation is also called a simple equation, or an equation of the first degree. If a occurs, and no higher power of x, the equation is said to be of two dimensions; such an equation is also called a quadratic equation, or an equation of the second degree. If a3 occurs, and no higher power of x, the equation is said to be of three dimensions; such an equation is also called a cubic equation, or an equation of the third degree. And so on. It must be observed that these definitions suppose both members of the equation to be integral expressions so far as relates to x. 158. In the present Chapter we shall shew how to solve simple equations. We have first to indicate some operations which may be performed on an equation without destroying the equality which it expresses. 159. If every term on each side of an equation be multiplied by the same number the results are equal. The truth of this statement follows from the obvious principle, that if equals be multiplied by the same number the results are equal; and the use of this statement will be seen immediately. Likewise if every term on each side of an equation be divided by the same number the results are equal. 160. The principal use of Art. 159 is to clear an equation of fractions; this is effected by multiplying every term by the product of all the denominators of the fractions, or, if we please, by the least common multiple of those denominators. Suppose, for example, that Multiply every term by 3 × 4 × 6; thus 4×6×x+3 × 6 × x + 3 × 4 ×x=3×4×6× 9, that is, 24x+18x+12x = 648; divide every term by 6; thus 4x+3x+2x=108. Instead of multiplying every term by 3 × 4 × 6, we may multiply every term by 12, which is the L. C. M. of the denominators 3, 4, and 6; we should then obtain at once Thus 12 is the root of the proposed equation. We may verify this by putting 12 for x in the original equation. The first side becomes 161. Any term may be transposed from one side of an equation to the other side by changing its sign. Suppose, for example, that x÷a=b—y. Add a to each side; then x−a+a=b−y+a, that is x=b−y+a. Subtract b from each side; thus x-b=b+a-y-b=a-y. Here we see that -a has been removed from one side of the equation, and appears as a on the other side; and +b has been removed from one side and appears as the other side. -bon 162. If the sign of every term of an equation be changed the equality still holds. This follows from Art. 161, by transposing every term. Thus suppose, for example, that x-a-b-y. and this result is what we shall obtain if we change the sign of every term in the original equation. 163. We can now give a Rule for the solution of any simple equation with one unknown quantity. Clear the equation of fractions, if necessary; transpose all the terms which involve the unknown quantity to one side of the equation, and the known quantities to the other side; divide both sides by the coefficient, or the sum of the coefficients, of the unknown quantity, and the root required is obtained. 164. We shall now give some examples. Here there are no fractions; by transposing we have 7x-5x=35-25; We may verify this result by putting 5 for x in the original equation; then each side is equal to 60. The student will find it a useful exercise to verify the correctness of his solutions. Thus in the above example, if we put 2 for x in the original equation we shall obtain 16-10-6, that is 0, as it should be. 28 53= 5 by 10; thus that is, transpose, that is, =2; the L. C. M. of the denominators is 10; multiply 5(5x+4)− (7x+5)= 28 × 2 − 5(x−1); 25x+20-7x-5=56-5x+5; 25x-7x+5x=56+5−20+5; The beginner is recommended to put down all the work at full, as in this example, in order to ensure accuracy. Mistakes with respect to the signs are often made in clearing an equation of fractions. In the above equation the fraction has to be multiplied by 10, and it is ad 7x+5 visable to put the result first in the form (7x+5), and afterwards in the form -7x-5, in order to secure attention to the signs. 168. Solve (5x+3)–(16 — 5.x) = 37 – 4.. Multiply by 21; thus 7(5x+3)-3(16-5x) = 21(37 – 4x), |