XII. SIMPLE EQUATIONS. 163. An Equation is a statement of the equality of two expressions. The First Member of an equation is the expression on the left of the sign of equality, and the Second Member is the expression on the right of that sign. Thus, in the equation 2x-3=3x-5, the first member is 2x-3, and the second is 3x – 5. The sides of an equation are its two members. 164. A Numerical Equation is one in which all the known quantities are represented by numbers; as 2x 3=3x-5. 165. A Literal Equation is one in which some or all the known quantities are represented by letters; as 2x + 3a = bx - 4. 166. An Identical Equation is one whose two members are equal, whatever values are given to the letters involved; as x2 - a2 = (x + a) (x − a). 167. The Degree of an equation, in which there is but one unknown quantity, is denoted by the highest power of the unknown quantity in the equation. Thus, 3 x2 d 2x = 65 is an equation of the second degree, etc. 168. A Simple Equation is an equation of the first degree. 169. The Root of an equation containing but one unknown quantity, is the value of the unknown quantity; or, it is the value which, when put in place of the unknown quantity, makes the equation identical. Thus, the equation 5x-7= 3x + 1, when 4 is put in place of x, becomes 20-7=12+1, which is identical. Hence the root of the equation, or the value of x, is 4. Note. A simple equation has but one root; but it will be seen hereafter that an equation may have two or more roots. 170. The solution of an equation is the process of finding its roots. A root is verified, or the equation satisfied, when, on substituting the value of the root in place of its symbol, the equation becomes identical. 171. The operations required in the solution of an equation are based upon the following general principle, which is derived from the axioms of Art. 42: If the same operations be performed upon equal quantities, the results will be equal. Hence, Both members of an equation may be increased, diminished, multiplied, or divided by the same quantity, without destroying the equality. TRANSPOSITION. 172. Any term may be transposed from one side of an equation to the other by changing its sign. For, consider the equation x + a = b. Subtracting a from both members (Art. 171), we have or, by Art. 26, x + a - a=b-a; x = b -u, where + a has been transposed to the second member by changing its sign. or, Again, consider the equation x - a = b. Adding a to both members (Art. 171), we have x-a + a = b+a; x = b + a. where - a has been transposed to the second member by changing its sign. Note. If the same term appear in both members of an equation affected with the same sign, it may be suppressed. 173. The signs of all the terms of an equation may be changed without destroying the equality. For, consider the equation a-x = b c. Transposing each term (Art. 172), we have which is the same as the original equation with every sign changed. SOLUTION OF SIMPLE, EQUATIONS. 174. 1. Solve the equation 5x-7=3x+1. Transposing the unknown quantities to the first member, and the known quantities to the second, we have 5x-3x=7+1. Uniting the similar terms, 2x=8. Dividing both members by 2 (Art. 171), x= 4, Ans. Note. The result may be verified by substituting the value of x in the given equation, as shown in Art. 169. We have then the following rule for the solution of a simple equation containing but one unknown quantity: Transpose the unknown terms to the first member, and the known terms to the second. Unite the similar terms, and divide both members by the coefficient of the unknown quantity. EXAMPLES. 2. Solve the equation 14 - 5x = 19 + 3x. (2x - 3)2 - x(x + 1) = 3 (x - 2) (x + 7) -5. Performing the operations indicated, we have 4x2 - 12x + 9 - x - x = 3x2 + 15 x – 42 – 5. Solve the following equations: 18.3+2(2x+3) = 2x -3(2x+1). 19. 2x - (4x - 1) = 5 x – (x - 1). 20. 7 (x - 2) - 5 (x + 3) = 3(2x - 5) – 6(4x - 1). 21. 3(3x+5) - 2 (5 x - 3) = 13 - (5x-16). 22. (2x-1) (3x + 2) = (3x - 5) (2x+20). 23. (5-6x) (2x - 1) = (3x + 3) (13 - 4x). 2 24. (x - 3)2 - (5 – x)2 = - 4x. 25. (2x-1)2-3 (x - 2) + 5 (3x - 2) - (5 – 2x)2=0. 26. 2(x-2)2 - 3(x − 1)2+ x = 1. 27. (x - 1)(x - 2) (x + 4) = (x + 2) (x + 3) (x - 4). 28.5(7+3x) - (2x-3) (1-2x) - (2x-3)2 - (5+x) = 0. 2 29. (5 x - 1)2 - (3x + 2)2 - (4x - 3)2 + 4 = 0. 3 30. (2x+1)8+ (2x - 1) = 16 x(x2 - 4) – 228. |