by the plus sign; subtract the numerator if the fraction is preceded by the minus sign. Write the result for the numerator of the required fraction, and for its denominator write the denominator of the given fraction. CHAPTER IV THE EQUATION. PROBLEMS 54. The following is an equation: 3x-4=23. If this equation is solved, it will be found that x= For no other value of x will 3 x In the equation 4 = 23. x2+14= 9x, = 9. For the present, equations in which x has two values will not be considered. An equation involving one letter then holds only for a particular value, or values, of this letter. 55. x+7=7+x is not an equation, for in this statement x may have any value and the equality holds good. An expression of this character is called an identity. 56. The letter whose value is sought in an equation is called the unknown quantity in the equation. 57. The process of finding the value of the unknown quantity in an equation is called solving the equation. 58. A value of the unknown quantity in an equation is called a root of the equation, and is said to satisfy the equation. 59. In solving equations, use is made of certain simple statements assumed to be true. These statements are called axioms. An axiom is a statement whose truth is taken for granted. AXIOM 1. Quantities which are equal to the same quantity are equal to each other. ILLUSTRATION. FIG. 1. FIG. 2. FIG. 3. Suppose Fig. 1 equals Fig. 3 in area, and suppose Fig. 2 equals Fig. 3 in area, then Fig. 1 equals Fig. 2 in area. AXIOM 2. If equals be added to equals, the sums are AXIOM 3. If equals be taken from equals, the remainders are equal. |