The STATEMENT consists in finding an equation which shall ex press the relation between the known and unknown quantities of the problem. The SOLUTION of the equation consists in finding such a value for the unknown quantity as being substituted for it in the equation will satisfy it; that is, make the first member equal to the second. 82. An equation is said to be verified, when such a value is substituted for the unknown quantity as will prove the two members of the equation to be equal to each other. 83. Equations are divided into classes, with reference to the highest exponent with which the unknown quantity is affected. An equation which contains only the first power of the unknown quantity, is called an equation of the first degree: and generally, the degree of an equation is determined by the greatest of the exponents with which the unknown quantity is affected, without reference to other terms which may contain the unknown quantity raised to a less power. Thus, ax + b = cx + d 4x35x2 5 2x2 is an equation of the 1st degree. If more than one unknown quantity enters into an equation, its degree is determined by the greatest sum of the exponents with which the unknown quantities are affected in any of its terms. Thus, 84. Equations are also distinguished as numerical equations and literal equations. The first are those which contain numbers only, with the exception of the unknown quantity, which is always denoted by a letter. Thus, 4x 3 = 2x + 5, 3x2 x = 8, are numerical equations. They are the algebraical translation of problems, in which the known quantities are particular numbers. A literal equation is one in which a part, or all of the known quantities, are represented by letters. Thus, bx2 + ax 3x 5, and cx + dx2 = c are literal equations. 85. It frequently occurs in Algebra, that the algebraic sign + or which is written, is not the true sign of the term before which it is placed. Thus, if it were required to subtract — b from a, we should write Here the true sign of the second term of the binomial is plus, although its algebraic sign, which is written in the first member of the equation, is This minus sign, operating upon the sign of b, which is also negative, produces a plus sign for b in the result. The sign which results, after combining the algebraic sign with the sign of the quantity, is called the essential sign of the term, and is often different from the algebraic sign. By considering the nature of an equation, we perceive that it must possess the three following properties: 1st. The two members are composed of quantities of the same kind. 2d. The two members are equal to each other. 3d. The essential sign of the two members must be the same. 86. An axiom is a self-evident proposition. We may here state the following: 1. If equal quantities be added to both members of an equation, the equality of the members will not be destroyed. 2 If equal quantities be subtracted from both members of an equation, the equality will not be destroyed. 3. If both members of an equation be multiplied by the same number, the equality will not be destroyed. 4. If both members of an equation be divided by the same number, the equality will not be destroyed. Solution of Equations of the First Degree. 87. The transformation of an equation is any operation by which we change the form of the equation without affecting the equality of its members First Transformation. 88. When some of the terms of an equation are fractional, to reduce the equation to one in which the terms shall be entire. First, reduce all the fractions to the same denominator, by the known rule; the equation then becomes If now, both members of this equation be multiplied by 72, the equality of the members will be preserved, and the common denominator will disappear; and we shall have 48x or dividing by 6, 8x 54x12x 792; 9x+2x=132. 89. The last equation could have been found in another manner by employing the least common multiple of the denominators. The common multiple of two or more numbers is any number which each will divide without a remainder; and the least common multiple, is the least number which can be so divided. The least common multiple can generally be found by inspection. Thus, 24 is the least common multiple of 4, 6, and 8; and 12 is the least common multiple of 3, 4, and 6. We see that 12 is the least common multiple of the denominators, and if we multiply each term of the equation by 12, dividing at the same time by the denominators, we obtain 8x 9x+2x= the same equation as before found. 132, 90. Hence, to transform an equation involving fractional terms to one involving only entire terms, we have the following RULE. Form the least common multiple of all the denominators, and then multiply every term of the equation by it, reducing at the same time the fractional to entire terms. We see, at once, that the least common multiple is 20, by which each term of the equation is to be multiplied. that is, we reduce the fractional to entire terms, by multiplying the numerator by the quotient of the common multiple divided by the denominator, and omitting the denominators. Hence, the transformed equation is to one involving only entire terms. Ans. a4bx-2a2bc2x + 4a4b2 = 4b3c2x - 5a6+2a2b2c2 3a3b3. Second Transformation. 91. When the two members of an equation are entire polynomials to transpose certain terms from one member to the other. Take for example the equation If, in the first place we subtract 2x from both members, the equality will not be destroyed, and we have or, by reducing the terms in the second member, 5x68 + 2x. 5x62x=8+2x-2n; } 5x62x = 8. Whence we see that the term 2x, which was additive in the second member, becomes subtractive in the first. In the second place, if we add 6 to both members, the equality will still exist, and we have or, 5x6 2x+68+6; since 6 and 6 destroy each other 5x -2x=8+6. Hence, the term which was subtractive in the first member, passes into the second member with the sign plus. For a second example, take the equation Hence, we have the following principle: Any term of an equation may be transposed from one member to the other by changing its sign. 92. We will now apply the preceding principles to the resolu tion of equations. 1. Take the equation 4x-3=2x+5. By transposing the terms and by reducing dividing by 2 3 and 2x, it becomes Now, if 4 be substituted in the place of x in the given equation, it becomes Hence, 4 is the true value of x; for, being substituted for x in the given equation, that equation is verified. 2. For a second example, take the equation |