Feb 3. 896. Regin for CHAPTER V. EQUATIONS OF THE FIRST DEGREE. 76. AN EQUATION is the expression of equality between two quantities. Thus, ≈= = b + c, is an equation, expressing the fact that the quantity a is equal to the sum of the quantities b and c. 77. Every equation is composed of two parts, connected by the sign of equality. These parts are called members: the part on the left of the sign of equality, is called the first member; that on the right, the second member. Thus, in the equation, x + y = a xy, is the first member, and a 78. Equations are divided into two classes: those containing but one unknown quantity, and those containing more than one unknown quantity. Each of these classes is subdivided into degrees. In the first class, the degrec is determined by the exponent of the highest power of the unknown quantity, in any term; in the second class, the degree is determined by the highest sum of the exponents of the unknown quantities, in any term. C, c, the second member. Thus, br = c, are equations of the first degree; x2 + 2px = 9, are equations of the second degree; 03 - b + cơ = are equations of the third degree; x2 + pxn−1 + 9x-1=8, 2n-2y2 + аx2-3y + bxy-1d, x2-2y2 bx + cy=d, x2 + axy + y2 = m, 2+2x2y + 3yx + 4y = 5, are equations of the nth degree. We shall first consider equations of the first degree, containing but one unknown quantity. 79. The TRANSFORMATION of an equation, is the operation of changing its form, without destroying the equality of its members. 80. The SOLUTION of an equation, is the operation of finding such a value for the unknown quantity, as will satisfy the equation; that is, such a value as, being substituted for the unknown quantity, will render the two members equal. This is called a ROOT of the equation. 81. An AXIOM is a self-evident proposition. 82. The solution of an equation is effected by successive transformations, which transformations depend upon the following axioms: 1. If equal quantities be added to both members of an equation, the equality 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 quantity, the equality will not be destroyed. 4. If both members of an equation be divided by the same quantity, the equality will not be destroyed. 5. Like powers of the two members of an equation are equal. 6. Like roots of the two members of an equation are equal. 83. Two principal transformations are employed in the solution of equations of the first degree; clearing of fractions, and transposing. The least common multiple of all the denominators is 20. If we multiply both members of the equation by 20 (Axiom 3), each term can be reduced to an entire form, giving, 16411236 16x+12 = 15. RULE. In the same manner, any equation may be transformed. Hence, for clearing of fractions, we have the following Find the least common multiple of the denominators and multiply both members of the equation by it, reduc ing the fractional to entire terms. The reduction will be effected, if we divide the least common multiple by each of the denominators, and then multiply the corresponding numerator, dropping the de nominator. The transformation may be effected by multiplying each numerator into the product of all the denominators except its own, omitting denominators. 3. 1. Clear the equation, of fractions. The least common multiple of the denominators is 12. Multiplying both members by 12 (Axiom 3), and reducing to entire terms, we have, 4. 5. 6. 7. x 3x 7 2 2x 5 2. Clear the equation, of fractions. 3 2' Multiplying each numerator into all the denominators except its own, and, omitting denominators, we have, Clear the following equations of fractions: x -- + 12 4 x 3 3 + 3 4 x 4 = 5 EXAMPLES. 3x 8x 10. 7x 5 2 x 21 42x 20x75. - 28 4 12 4 2 6 5 3 = = 8. Yu Ans. 7x 21 42x+7x=140. Ans. -2x+8−x+2=10. 1216 672, 3 Which reduces to 5 85. TRANSPOSITION is the operation of changing a term from one member to the other, without destroying the equality of the members. Take the equation, 3x + 4 + 6x 4 = 5 ·6x+6x 3x+6x= 3x + 4 = 5 6x. both members of the equation the (Axiom 1), we shall have, RULE. 1. 13x+16=7x+20. 2. 2x 8 10 x. 4. Comparing this with the given equation, we see that 4 has been transposed to the second member, and -6x to the first member, by changing their signs. In like manner, any term may be transposed. Hence, the following rule for transposing. EXAMPLES. Any term may be transposed from one member to the other, by changing its sign. Transpose the unknown terms to the first, and the known terms to the second members, in the following 4. 3x. |