V. EQUATIONS OF THE FIRST DEGREE, AND INEQUALITIES. · 102 Equations of the First Degree. ........102 Transformation of Equations ......... Solution of Equations ............ 106 Problems involving Equations of the First Degree . . Equations containing Two Unknown Quantities ... 123 Elimination . . . . . . . . . . . . . . . 124 Equations containing Three or more Unknown Quan- VI. POWERS . . . . . . . . . . . . . . . . . . 153 Powers of Monomials. ............. VII. EXTRACTION OF Roots, AND RADICALS........ Extraction of Roots ............. Transformation of Radicals .......... Subtraction of Radicals ........... Multiplication of Radicals .......... Division of Radicals ........... VIII. EQUATIONS OF THE SECOND DEGREE . . . . . . . . 206 Equations containing One Unknown Quantity . . . . 206 Incomplete Equations ............ 208 Equations containing Two Unknown Quantities . . . 212 Complete Equations ............. Properties of Equations of the Second Degree . .: 229 Formation of Equations of the Second Degree .... Trinomial Equations of the Second Degree ..... SUGGESTIONS TO TEACHERS. 1. The introduction is designed as a mental exercise. If thoroughly taught, it will train and prepare the mind of the pupil for those higher processes of reasoning which it is the peculiar province of the algebraic analysis to develop. 2. The statement of each question should be made, and every step in the solution gone through with, without the aid of a slate or blackboard, though perhaps in the beginning some aid may be necessary to those unaccustomed to such exercises. 3. Great care must be taken to have every principle on which the statement depends carefully analyzed ; and equal care is necessary to have every step in the solution distinctly explained. 4. The reasoning process is the logical connection of distinct apprehensions, and the deduction of the consequences which follow from such a connection : hence the basis of all reasoning must lie in distinct elementary ideas. 5. Therefore, to teach one thing at a time, to teach that thing well, to explain its connections with other things and the consequences which follow from such connections, would seem to embrace the whole art of instruction. ELEMENTARY ALGEBRA. INTRODUCTION. MENTAL EXERCISES. • LESSON I. 1. John and Charles have the same number of apples. Both together have 12. How many has each ? Let x denote the number which John has. Then, since they have an equal number, x will also denote the number which Charles has; and twice x, or 2x, will denote the number which both have, which is 12. If twice x is equal to 12, x will be equal to 12 divided by 2, which is 6: therefore each has 6 apples. Hence we write, — x = number of apples John has. Then x + x = 2x = 12. Hence < -12 – 6. Let NOTE. — When x is written with the sign + before it, it is read plus X; and the line above is read, x plus x equals 12. When x is written by itself, it is read one x, and is the same as 1x. x or 1x means once X, or one X, 2x “ twice X, or two X, 2. What is x+x equal to? 7. James and John together have 24 peaches, and one • has as many as the other. How many has each ? Let æ denote the number which James has. Then, since they have an equal number, æ will also denote the number which John has; and twice a will denote the number which both have, which is 24. If twice x is equal to 24, x will be equal to 24 divided by 2, which is 12: there. fore each has 12 peaches. Hence we write, Let <= number of peaches James has Then x = 2 x = 24. Hence x = 24 = 12. + A verification is the operation of proving that the number found will satisfy the conditions of the question. Thus, James's app?es. John's apples. 12 + 12 = 24. NOTE. — Let the following questions be analyzed, written, and verified in exactly the same manner as the above. 8. William and John together have 36 pears, and one has as many as the other. How many has each? 9. What number added to itself will make 20? 10. James and John are of the same age, and the sum of their ages is 32. What is the age of each ? 11. Lucy and Ann are twins, and the sum of their ages is 16. What is the age of each ? 12. What number is that which added to itself will make 30 ? |