PREFACE. THIS Key is intended partly for the convenience of Teachers, and partly for the use of those who do not enjoy the advantages of an instructor. It has therefore been deemed requisite to give every solution at full length, and in several instances to introduce explanatory Notes, and even general Theorems, when they flowed naturally from the subject under consideration. With such a guide, united to moderate application, it is hoped that even the solitary student will be able to make his way through the FIRST PART of the SYSTEM of PRACTICAL MATHEMATICS with great benefit and satisfaction. PREFACE. THIS Key is intended partly for the convenience of Teachers, and partly for the use of those who do not enjoy the advantages of an instructor. It has therefore been deemed requisite to give every solution at full length, and in several instances to introduce explanatory Notes, and even general Theorems, when they flowed naturally from the subject under consideration. With such a guide, united to moderate application, it is hoped that even the solitary student will be able to make his way through the FIRST PART of the SYSTEM of PRACTICAL MATHEMATICS with great benefit and satisfaction. KEY. ALGEBRA. ART. 20. 1. 8x3+(4-2)15-4x3x2= 24+30-24-30. 2. 8(12+15)-2(4x 15-3)= 3. (8-1)(4-1)(3-1)(2+3)=7x3x2x5=210. 5. 64(4×15-3x15)-0(2x8x2+4)= 6. 83 (2×4+15+4)3 +413 (15+3—2)*= 2× (27)+2(16)=6+8=14. 7. Let x=8, y=4, then x3-512, and y3=64, and x2+xy+y2=64+32+16=112. 8. Let a=7, x=3, then (a+x)=7+3=10, and (a-x)=7-3-4, ·· (a+x)(a—x)=10×4=40, also a2x2-49-9-40. 9. Let a 4, b=3, and c=2, then a+b-c-4+3-2-5, and (a-b+c)=-4-3+2=3, :. (a+b—c)(a—b+c)=5×3=15, and a2-b2-c2+2bc-16—9—4+12=15. 10. Let x=4, y=3, then x4-y+-256-81=175, and (x-y) (x+x2y+xy2+y)=64+48+36+ 27-175. In the last four Examples, equal values will always be obtained by taking the same values on each side, whatever these values may be. B |