If this equation is solved, it will be found that x=9. For no other value of x will 3 x - 4 = 23. An equation is a statement of equality which is true for a particular value or values of the letter or letters contained therein. 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 equal. ILLUSTRATION. Suppose and then, A AXIOM 3. If equals be taken from equals, the remainders are equal. AXIOM 4. If equals be multiplied by the same number, the products are equal. ILLUSTRATION. A B E If the triangle ABC equals the triangle DEF, then twice the triangle ABC equals twice the triangle DEF, then three times the triangle ABC equals three times the triangle DEF, then four times the triangle ABC equals four times the triangle DEF, etc. AXIOM 5. If equals be divided by the same number (zero excluded as a divisor), the quotients are equal. If angle A equals angle B, then half of angle A equals half of angle B, one third of angle A equals one third of angle B, etc. 60. Solve: 7x-3=4x+12. Subtract 4 x from each member; then, by Axiom 3, (1) 7x-4x-3=12. Add 3 to each member; then, by Axiom 2, (2) 7x-4x=12+ 3. Uniting like terms, (3) 3x=15, From this solution it appears that a term may be transposed from one member of an equation to the other member by changing its sign. This process is called transposition. Accordingly, the above solution may be shortened as follows: |