If ab, then am<bm; and if am<bm, then that is, the resulting fraction is less than the first. that is, the resulting fraction is greater than the first. 2x 3. Find the value of x+ 8. Prove that the sum or difference of any two quantities divided by their product, is equal to the sum or difference of their reciprocals. & malif. 9. If two fractions are together equal to 1, prove that their difference is the same as the difference of their squares. 10. If the difference of two fractions is equal to 2, show that p times their sum is equal to q times the difference of their squares. x-y 11. Prove that + =1: (a—b) (ac) (b—a)(b—c) (c—a)(c—b) that when the terms are multiplied respectively by b+c, a+c, and a+b, the sum =0: and that when multiplied respectively by bc, ac, and ab, it is h2. CHAPTER IV. EQUATIONS OF THE FIRST DEGREE. DEFINITIONS AND ELEMENTARY PRINCIPLES. ART. 140. An equation is an algebraic expression, stating the equality between two quantities. Thus x-5=3, is an equation stating that if 5 be subtracted from x, the remainder is 3. ART. 141. Every equation is composed of two parts, separated from each other by the sign of equality. The quantity on the left of the sign of equality, is called the first member or side of the equation. The quantity on the right, is called the second member or side. The members or quantities are composed of one or more terms. ART. 142. There are generally two classes of quantities in an equation, the known and the unknown. The known quantities are represented either by numbers, or the first letters of the alphabet, as a, b, c, &c.; and the unknown quantities by the last letters of the alphabet, as x, y, z, &c. ART. 143. Equations are divided into degrees, called first, second, third, and so on. The degree of an equation depends on the highest power of the unknown quantity which it contains. Thus, an equation which contains no power of the unknown quantity higher than the first, is termed an equation of the first degree, or a simple equation. An equation in which the highest power of the unknown quantity is of the second degree, is called an equation of the second degree, or a quadratic equation. Similarly, we have equations of the third degree, fourth degree, and so on; those of the third degree are generally called cubic equations, and those of the fourth degree, biquadratic equations. Thus, ax-b-c, is an equation of the 1st degree. When any equation contains more than one unknown quantity, its degree is equal to the greatest sum of the exponents of the unknown quantity, in any of its terms. Thus, xy+ax-by=c, is an equation of the 2nd degree. x2y+x2-cx=a, is an equation of the 3rd degree. ART. 144. An equation of any degree is said to be complete, when it contains all the powers of the unknown quantity, from 0 up to the given degree. When one or more terms are wanting, the equation is said to be incomplete. Thus, x2+px+9=0, is a complete equation of the second degree, the term q being equivalent to qr°, since x=1. (Art. 82.) x2+px2+qx+r=0, is a complete equation of the third degree. ax2=q, is an incomplete equation of the second degree. x3+px=q, is an incomplete equation of the third degree. ART. 145. An identical equation, is one in which the two members are identical; or, one in which one of the members is the result of the operations indicated in the other. Thus, ax-b-ax—b, 8x-3x=5x, (x+3)(x—3)=x2-9, are identical equations. Equations are also distinguished as numerical and literal. A numerical equation is one in which all the known quantities are expressed by numbers. Thus 2x2+3x=10x+15, is a numerical equation. A literal equation is one in which the known quantities are represented by letters, or by letters and numbers. Thus, ax+bcx+d, and ax+b=3x-5, are literal equations. ART. 146. Every equation may be regarded as the statement, in algebraic language, of a particular question. Thus, x-5=9, may be regarded as the statement of the following question:-To find a number from which, if 5 be subtracted, the remainder shall be 9. If we add 5 to each member, we shall have To solve an equation, is to find the value of the unknown quantity; or, to find a number or expression, which, being substituted for the unknown quantity, will render the two members identical. REMARK.The solution of equations is the most useful and interesting part of algebra. An equation is said to be verified, when the value of the unknown quantity being substituted for it, the two members are rendered equal to each other. Thus, in the equation x-5=9, if 14, the value of x, be substituted instead of it, we have 14-5=9; or, 9=9. ART. 147. The value of the unknown quantity, in any equation, is called the root of that equation. EQUATIONS OF THE FIRST DEGREE, CONTAINING BUT ONE UNKNOWN QUANTITY. ART. 148. The operations employed to find the value of the unknown quantity in any equation, are founded on this evident principle: If we perform the same operation on two equal quantities, the results will be equal. This principle or axiom may be otherwise stated, as follows: 1. If, to two equal quantities, the same quantity be added, the sums will be equal. 2. If, from two equal quantities, the same quantity be subtracted, the remainders will be equal. 3. If two equal quantities be multiplied by the same quantity, the products will be equal. 4. If two equal quantities be divided by the same quantity, the quotients will be equal. 5. If two equal quantities be raised to the same power, the results will be equal. 6. If the same root of two equal quantities be extracted, the results will be equal. REMARK. An axiom is a self-evident truth. The preceding axioms are the foundation of a large part of the reasoning in mathematics. ART. 149. There are two operations of frequent use in the solution of equations. These are, first, to clear an equation of fractions; and second, to transpose the terms in order to find the value of the unknown quantity. These are named in the order in which they are used in the solution of an equation; we shall, however, first consider the subject of TRANSPOSITION. ART. 150. Suppose we have the equation ax+b=c-dx. Since, by the preceding principle, the equality will not be affected by adding the same quantity to both members; or, by subtracting the same quantity from both members; if we add dx to each side, we have ax+b+dx=c-dx+dx. If we subtract b from each member, we have ax+b—b+dx=c▬dx+dx—b. But +b-b cancel each other, so do -dx+dx; omitting these, we have ax+dx=c—b. But this result is the same as if we had removed the terms +b and dx to the opposite members of the equation, and at the same time changed their signs. Hence, Any quantity may be transposed from one side of an equation to the other, if, at the same time, its sign be changed. This is termed the Rule of Transposition. TO CLEAR AN EQUATION OF FRACTIONS. ART. 151. 1. Let it be required to clear the following equation of fractions. Since the first term is divided by ab, if we multiply it by ab, the divisor will be removed; but if we multiply the first term by ab, we must multiply all the other terms by ab, in order to preserve the equality of the members. Again, since the second term is divided by bc, if we multiply it by bc, the divisor will be removed; but if we multiply the second term by bc, we must multiply all the other terms by bc, in order to preserve the equality of the members. Hence, if we multiply all the terms on both sides by abbc, the equation will be cleared of fractions. Instead, however, of multiplying every term by ab×bc, it is evident, that if each term be multiplied by such a quantity as will contain the denominators without a remainder, that all the denominators will be removed. This quantity is evidently the least common multiple of the denominators, which, in this case, is abc; then, multiplying both sides of the equation by abc, we have cx-ax-abcd. From which we derive the following RULE FOR CLEARING AN EQUATION OF FRACTIONS. -Find the least common multiple of all the denominators, and multiply each term of the equation by it. |