IMAGE EVALUATION
TEST TARGET (MT-S)
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1.0
I.I
1.25
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■^ 1^ III 2.2
^. lia IIIIIM
1.8
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1
2
3
1
2
3
4
5
6
1
fM.Tr.
wmm
mmm
w
liiHiiilii
w
mmfmif^m
^
\{
It kt
rif — all <' — ,V^' — 20H<' ;
31§fi - 12^6 - \^Y - n^d - 33^ ;
2^l^a - ^/> - A^c - A^{u - 1 3f e ;
lOfa + 2|A _ l|Jc - 1 iV? - 7^ ;
^a + 8|6 + 42o + 0.^(/ — 4^6^ ; 0.
18. i. ; a ; ^(Ht/,"). ii. 2G«" ; a 4- a?.
EXERCISE XVII [6]. ^Page 30.)
1. 6, 18.
4. 4, 3, 10.
7. 120, 137, 163.
10. 1840.
13. 34.
16. 7 months.
19. $100.
3. 30, 40.
5. 8, 40, 13.
8. 130.
11. 7, 43.
14. .$138|, $236^.
17. 450, 180, 140.
20. 13{33it--j5r(30
3. 35, 65.
6. 500.
9. 30 minutes.
13. $33, $36, $44.
15. £1300.
18. 300.
33a;) } = 44a; , a; = i\.
EXERCISE XVIII. (Page 41.)
1. a^ + 6'^ _ c^ - c/'-i ; a'-V ^ d" + d\ 3. a" - 36' + c\
3. 2m — ?i + 6. 4. — 2a; — 3y — 2z. 5. l^a; — 4|y + \\z.
EXERCISE XIX. (Page 43.)
1. — 3a + ^x + 36. 3. a + 6 + c. 3. 3a6 + 46'.
4. — 3a? — 2/ + 40. 5. 5 — 4a;. 6. 3a — 36 — 3c + Ad.
7. —4a. 8. —a;— 102/ + 30. 9. — 3a; + 3i/.
10. 3a; — Qy — my + 4a6 — 5. 11. 3a — 56 — c.
13. 0. 13. -\y. 14. ^a-2h. 15. \x. 16. 9.
EXERCISE XX. (Page 44.)
1. i. a; — (a + 6) ; a; — (a + 36 — 2y).
ii. X — (2m -2n); x — (36 — 3c — 5d).
iii. X — (3m + 3a — 26) ; a; — (6 — a — c — w + w).
iv. a; — (a + 6 — c — 12); a; — |(a + 6) + (p + g) + (m — n)\.
ELEMENTS OF ALOEBKA.
X.
several re-
-e
iniites.
I;i6, 144.
r
!6« + c\
i - 3c + 4d.
• — c.
16. 9.
(m — w)}.
2. i. (2a — 46 — 3c) a; — (6a + 3c) // + (46 — ac) z.
ii. (a — 6 + c) a? — (a 4- 6 — r) /y — (a — 6 — c) ^.
iii. (12a — 15c) x — (12a + 46 + 6c) ij — (126 + 8c) z.
8. i. 2 + (7 - 2c) X + (lia - 3) a.'-" + (9a - 7) x\
ii. (2c — a") it^ + (a — 36) x* + {1 — m) a;' + (4c — 3a6) x.
iii. (1 - a) it-* 4- (1 - 6 + c) aj^" + (6 ~ 1) a;^ + (a - 7) a; + 2.
4. i. - (3c' - 5a) a; - (a6c - 7) ai^* - (a6 - 7) a;".
ii. 1 - (a — 1) a; - (1 - 6) x' _ (a - c + 1) a;" - (a-6 - 1) x*
iii. - (a - 36') a;* - (1 - c) a;' — (1 + 5c') aj' - (6 + c) a;.
5. i. (a — c + 1) a;' — (a + 26 + 1) a;' + (6 + c) a; + 3.
ii. (5a + 4c) x^ + (7c - 66 + 3a) a;' + (2a - 76) x.
iii. (a — 6 + c) a;' — 2 (a 4- 6 + c) a? + a6 — 6c — ca.
6. i. 6: 6.
ii. -17; _9.
iii. — 1 ; — 56.
7. (a + 6 + c) (x-\-ij + z).
8. —^a — rx — (2 — 6) a;' + (4a — jo — 1) x^ + 2x*
9. (6^ + 1) i»' — (^ + 2^) X* — (2z + ^.
EXERCISE XXI [6J. (Page 48.)
1. 36: -48: 5: 9: -168
180.
2. i. m^xyz ; a6ca!' ; — 24a'6^ ii. — 36a®m* ; — aHi^e^x^y^z^.
iii. — 14a'6'.z;' ; — 18a;^^'2r* ; — 5a;'yV.
8. i. 40; -63; -2; —37. ii. 130; —880: 0. iii. \\ 29.
EXERCISE XXII [6]. (Page 49.)
1. a^¥c — ab* + dbc^ ; — fa;' + ^xy + V'^.
2. Mhxy + Mcxz + 15aa: ; 9a;V'^' — \2xHfz'^ + \^x'^yz\
3. — 15a;*2^ — 10a;*y/' + 35a;V' — 5^V' ; 3a* + 2a'6 — a''6'.
4. ^xYz — SxYz + Sxhfz* — 12x* y*z' ; ^a'^x — -^^abx - ^cx.
5. — 2a'a;' + la*x* + a^x* ; — x^'y" + ^^x'lf.
6. ^x*y''z'' - f aj'y^^' + IvYz"" — ^""fz^ ; |<**'^'' - W^^ + «'^*-
2 (a + 6)* + 2 (a + 6)"
3 (a - 6)=* - 2 (a - hy
8. (m" — ny + (w' - nf ; 3 (a + 6)«+i + 2 (a 4- 6)"+*.
9. {a 4- 6)'»+' 4- (a 4- 6)'"+' ; (a - 6)"+' - (a - 6)"+».
^
-'.
HINTS AND ANSWERS.
fl
i
B
1.
3.
5.
8.
10.
13.
15.
16.
17.
18.
20.
21.
22.
23.
24.
26.
28.
30.
31.
34.
EXERCISE XXIII. (Page 50.)
ex"" — Vdxy + ^y'. 2. ISai" — ^x'y — ^bx -i- hy.
x^ — 9a^x. 4. - 106' - 15a6^ + Uab' + 21a'-^6
a" + b\ 6, a' - b\ 7. a" - 6«.
^f _ 5if + 3^2 + 6?/ — 4. 9. a« + -^/a'6 + ia6 — 2ab'' - l6^
(a'^ - b"") x''+\ 11. a;* — a\ 12. 1 - x\
y" + 2y* - 72/' - 16. 14. ^x* + y\
am + {an — bm) x + {ap — 6w) re'' — bpx^.
a — (a^ — b)x + cx^ — (ac — b'^^ x"^ a- bcx*.
x^ — Sx' — Sic" + Gx" + 4x* — n
«3 + 63 _ c" + 3a6c. 19. 0;=' + y' + 3a?2/ — 1
18a.'« +27a;' + 7x'' + dx^ - 2x* + 65.«'' + lloic' +
(x + yy -(z + a)\
16a'' + 24ab + 96' - 4c' - 4c(^ - c^'.
16a' - 24c?6 + 96' - 4c' + 4cd - d\
n>* J^ Oi-3 1 ^2 ^,4 I Oi.3 i/2 OR
12a; + 6.
-1.
+ 49;i; + 6.
x* + 2x^ + ic' — 2/' + 2y' — 2/'.
81ir*
2ma\
9^4 S^/v.3
25. a" 4- 86^ - 27c' + 18a6c.
27. x' + 2a;'2/' + y\
|aif^ + ^a'a?' — %a\ 29. a;*^ — a«.
^2^m+2 ^ tj55.^n+2 _^ Q^2,^~+3 _^ ft2^n+3 _ J^H _ ^^2_
^2m _,_ j^.2m^m _ ^m^ _ ^^Sm^ 32^ ,^8 _ ^8^8^ 33^ ^^^ _ \^
x^ + x*a* + «'. 35. X* — y* — 42/' - 62/' — 4^/ — 1.
EXERCISE XXIV [a]. (Page 53.)
2. a;" — 3.r* + 3ir' — 1.
1. a;' — 5a;* + 10a;' - 10a;' -!- 5a; — 1.
3. 2a;' — 18a;* + 39a;' — 25a;' + a; + 1.
4. 4a;« - 5a;' + 8a;* - 10a;' - 8a;' — 5a; — 4.
5. 21a;« + Ux' — 49a;« — 8a;' — 10a;* + 41a;' — a;' — 14a; + 2.
6. Sa;" + 7a;' — 12a;* + 2a^' — 3a;' + 13a; — 6.
7. 4a;'' — 8a;' + 4a;* — 12a;' + 12a;' — 6a; + 9.
8. x^ - 3a;« + 6a;* - 7a;' + 3. 9. a;" - 57a;* -r «oo
10. 18a;« + 21a;' + 8a;° + a;' + 63a;' + 96a;' + 43a; + 6.
11. a;" — 3a;*a' 4- 3a;'a* — a\ 12. 1 — x\
13. 4 - 12a + 5a' + 14a' - 11a* — 4a* + 4a«.
14. 1 + a; + a;' + x* + x'\
— 57a;* + 266a!' - 1.
ia
50.)
)bx -\- by.
+ Uah' + 21a%
7. a" — 6".
lb — 2ab^ - ^b\
13. l — x'.
+ 6.
-1.
+ 49.1; + 6.
' - 27c' + ISabc.
33. x'^~l.
- 62/« - 4y - 1.
53.)
x* + 3;r'' — 1.
14ic + 2.
66a;'^ — 1.
6.
ELEMENTS OF ALGEBRA. i
15. ahx' + {al + 6A;) iK* + {am + 6Z + cA) a;" + {an + 6m + c?) x'
{bn + cwi) a; + cw.
3 0. 3. t/*-72/^ + 10. 5. 729a;'' - 117649.
6. 2a' - 2ap - 2a^n + p' + 2a7ip - 2an + np -^ 2an\
7. 0, put a = & + c.
EXERCISE XXV. (Page 56.)
1. 3a;; 7a!; - 3a;^ - 5a;^ 2. - aV ; a*; - 7a'6V.
3. 3a''; -2a!; - 2a!'^^ 4. Za'b'c; Ixy; -\x.
5. - a'c ; - a^^ ; ax\ 6. - 2a'»-'6"-^ ; - a ; ma ; - 2a5'''+''.
7. 3a'»-"p"-» ; 4a^ {x - y)"^' ', - {a + 6)""'".
8. - 4?/ia!* -^ 5a'' ; -3a6-i-4c; 12a -r-c.
9. a'd" -^ 6 ; — a"^' ; af-* -^ y""*. 10. maf -t- wjT ; a''6''c'-r■a;^
EXERCISE XXVI. (Page 57.)
1. a!-2y; -a!" + 2/'; «'& - «•
2. 1 - 3aa; — 4a V ; - 1 -t- a! + 2a6a!.
3. -ti + ft + c; -a + 64-6^ ia!y-i.
4. - 3?na;™-" + 2a7?i=' — ^a*mxP-^.
5. (a + 6); 4(a-&)^ a-« + a»-\
6. _ ^xY + K2/ - 2y ; 3a:'' - 1|2/ + 4.
7. _ a" - 0! ; a''" - a'-a;" + a!^". 8. 6a''a;2/ - 5ay' + ^a'^xy - ^ay\
9. -ix'^y-^ K - K?/" + K-
2 3 4 - » 3
10.
11.
12.
_f_ _ — + ^ ; 4a!« - a;' 4- iT^-
5c'' 5a'' 5a'' ' 235"
""4^ "^4 42/' a!" a!" a?
.|. + 3 + ^,;(«-6)"-.
3^'
2a'' 2a
13. |(a + &)='-(« + &)+i- 14. a— -(a-
15. 2 (a; + 2/)"^' {x - yT'' - (^ + ^)"-' (^ " 2/)'"'-
16. (a + 6)"^^ (a - 6)"-" -{a^ 6)-" (a - 6)'»-^
&)
,ni— n
8
HINTS AKL ANSWERS.
EXERCISE XXVII. (Page 60.)
il
1.
5.
9.
13.
17.
20.
23.
26.
28.
30.
33.
35.
37.
1.
3.
4.
5.
2. a — 6.
6. 3ir-7.
10. x — l.
14. 9ic' -f iy".
18. 03^ 4- 3a; + 1.
21. %ah.
3. 3a; — 2.
7. 3ic + 2.
11. 3a; + 4.
15. 8a; + 3^.
19. a' + ^-l.
22. a" — 36a' + 2b''a.
4. a — 24.
8. 4a; + 3. .
12. 5a; -1.
16. x' + Ux.
24. a;'' — 5a; + 6. 25. a;'— 2.x' + 2 ; —100a;.
27. x^ + xif + lax.
29. 'W — rib' + dc\
31. a — b — c. 32. 5a' + 3a;'.
34. a;' — wta; + ??i' — w ; (nf — ni'^) x.
36. a;"* — 2af ?/" + y
2n
6.
3.
5.
7.
9.
10.
11.
12.
13.
14.
15.
16.
a; + 7.
3a; + 1.
3a; — 2y.
x^ — 2/'.
4a;' — X.
a'a;' + aa; + 1.
a;' + "Ixu + 2if
24a;' - 2aa; — 35a'.
- 5a' + \bd - ^cf.
OX X -\- li,
2fq + ^p(f + 2(f.
^ + 2/ ; iT^^ + 2xif.
ax" — 6a;' — a'x + a6a; + a^ — a'6. (Read a^6' for a'6' in text.)
EXERCISE XXVIII. (Page 61.)
a^ + 6^ + C — a6 + 6c 4- ca. 2. a;' — (a + 6) a; + a6.
2/* — (m — 1) 2/^ — (w — ?^ — 1) 2/' — (wi — 1) y + 1.
jp2 + g2 ^. ^.2 ^ j^g _|_ g^, _ j,p
1 — a; 4- 2?/ + a;' + 2a;?/ 4- 42/', 1 4- ^k — 22/ 4- ic' 4- 2a;2/ 4- 42/'.
a;' 4- 2/' + ^"^ 4- 1. 7. 3a;' — a; — 2, rem 2a; + l. 8. 2a;'— a;4-l.
EXERCISE XXIX. (Page 64.)
2a;' 4- 3a;' 4- 4a; + 7. 2. a;' — 2x 4- 4.
5a;' — 10a; 4- 2, 3a;' — 10a; 4- 1. 4. x^ — 3a;' 4- Sa- - 1.
4a;=' - 3a;' + 2a; 4- 2. 6. 5a;' — V2x 4- 12, 12a; -72.
5a;' + 10.f + 5, -5a;'-10a;4-27. 8. 10a;^ lOa;* - 100.
1 - 2a^ + 3a;' - 4a;=' 4- 5a;* ; a* 4- 2a' + 3a' 4- 4a + 5.
a;' 4- 2xy 4- 32/' ; m' — 2m 4- 3.
a;* + 2a;' 4- 3.2;' 4- 2a; 4- 1 ; a* - 2a'6 4- 3a'6' - 2a6' 4 6*.
3a;* — 2a!' _ 2a; + 3.
a;* — 3a;' — 4a; 4- 15, 54a;' — 56,2; 4- 27.
oc^ _ 3^-4 _ 2a'' 4- 2a;' + 3.i" - 1 , 5,r.
2x^ — .r' - 2a' 4- 4, 24.c- - 12a! + lo. '
X* 4- 4a;' 4- 6a;' 4- 9,r — 4, rem. 5. 17. 2a;' — 4a; 4- 3.
ELEMENTS OF ALGEBKA.
1)
4. a — 24.
8. 4x + S.
13. 5a; -1.
16. x" + Ux.
ha"" + 262^.
1*^^ + 2; -lOOiC.
ax.
r a'6Mn text.)
I.)
X + ab.
1.
h 2xy + 4y\
8. 2*-'^— ^+1.
5^-1.
13, 12iC-73.
-100.
5.
>' "+ 6*.
18. .-•' — 3re" + 3a;* — 2a;' + 1 ; a;* — a;' + a-' — x + 1.
19, a;' + (« + 6) a; + «6 ; a;'' — aa; + 6^ — ""wi.^ 20. '2if — tuj — h i'.
31. 5a;' — 34a;' + 99a;' — 400a; + 1601, rem. - 6400.
33. a?'" — X* — x^ + a;' -f 2a; + 1, rem. 101.
33. 2a;* — a;' + 3a;' — 3a; + 1, rem. 10.
34. ^x^ + ix* — x\+ ^x" + t, rem. — 3a.-' + 21a'- - 3.i; + 14 ; take
factor 3 out of divisor and divide resulting- ([uotieut by 3.
EXERCISE XXX.
1. 8. 3. 1. 3. 4w'-18. 4.
7. 3. 8. 4. 9. 13. 10.
11. {n — h + a?)-r-{a — m + 3a6 + c).
(Page 69.)
3. 5. 2.
6. 1.
(a' + ab + b')
■T- 3 (6/ + 3).
13.
13.
15.
16.
17.
19
30.
73
31.
^
33.
3;
24.
(be — ab) -T- {a -f c + 6' 4- (fl^ — cf].
(a" + 6' + ab) -r- (a -f 6). 14. 6c -^ (30 — 11a + 36 + 3c).
xz=^{a ■\-b)\ write P for a; — a, and Q for x — b, and equa-
tion becomes P^-t-Q^= {P— (a — b)\-~\Q + {a — b)\, and
on clearing of fractions P^ — Q'^ will prove to be a factor;
.-. P' -Q'' = 0, P + Q = 0, etc. Or, multiply out.
(c' — a&) -j- (a + 6 — 3c) ;
equation is (x -{■ a) — (x + b) = {2x + a + c) -r- (3a; + b -\- c);
complete the divisions, square and transpose ;
.-. {c ^b)-i-{x + b) = (a — b) -T- (2x + b + c), etc.
18. 9. 19. 9.
remove brackets and combine numerical quantities.
33. 4.
('(luanon is ^\x + y^a; + fc - Ux = A + ^ 4- A 4- 1. ; or
Ifla- — ffa- = 8051 -j- 66 x 91 ;
^■. e. 8051a; ^ 77 x 334 = 8051 -r
(2rfb'' - rm) -^ (2a — 2b + 3).
v + 3.
5.
7.
9.
340.
na^ nb
Price =3 |(23a' ^2U)y^ 80a; (x
- n^ = A + f +
3
IT
+ i:
-66
X 91, etc.
35.
8.
>CI.
(Page 72.)
^
miles. 4.
8
men.
EXERCISE XXXI.
8. 12 miles. 3.
-a. 6. mn (a — b) -\- (mn — m — n).
2). 8. 50tral.
One-tliird. 10. 188 oz.; ^(.^c 4- 32) = ^J^ (a- — 56), where
a; = \vt. of lump.
10
HINTS AND ANSWERS.
11. 30 eggs. 12. 4. 13. 40. 14. $78|.
15. 16300, 23000. 16. 63. 17. 30 gal. . 18. 1080 -r- 351 miles.
19. B in ac -f- (a — b) days, Amac-T-{c — a + b).
30. 33 gals., 95. 31. $3. 23. 37,38,39. 33. 6,9,18. 34. 335.
35. Let X = increase of rate, then c-T-a + x = c-i-a — b,
x^a^b -T- (c — ab).
36. 7, 8, 9. 37. 2pqr -i- {j)q + qr + rp).
28. {ma — 6) -f- (m — 1), m (b — a)—- {m — 1).
39. 433. 30. n{m~p)-r-p.
31. 30|, 24f , 11|, 44| ; if a; = 1st part, a; + 4 = 2(1, i^ (a;+2) = 3d,
2 (if + 2) = 4th, and their sum is 100.
EXERCISE XXXII [6]. (Page 78.)
1. 169a'' — 53a + 4a'» ; 335a;'' — 15a^ + ^a"" ;
UlxY + 12Qx''y + 9a!^ ; lUa%* — lUa^h'c + 30«*6V.
3. |a;* + i^V' + -^jiV* ; 18-ri-ga*6^ - 2a'b' + i^Sa'b* ;
389irV"2'' — 2xYz'' + ifir^'y"^*.
3. 1,024,144; 1,096,004; 13331; 5635; 3401.
4. 35(i"''6" + 60a»*'*6'^ + SOa'^^ft'" ; a^^^ - 3a" '6^" + b'" ;
5939a'" + 13553a"6«« + 77446'^".
5. mx" - 4 ; }aj= - ^ ; 4^* - ^l^i/.
6. 49a;'* — 256a;Y ; ^.^'^ - ^-^xY' ; i^"* - y""-
7. 999,856 ; 9879 ; 4875 ; 2499.
8. a;"* — V ; 25a"''6'** - 36a''=&"* ; 5929a"^'' - 77446''''.
9. 4a;^ 10. a" ^rb"" Jr Mb — a^ ; x"" — 2x1/ + y"" — 2\
11. 4a^ - 6^ + 66c - 90" ; y^ - 4x'' + 12xz - 92\
13. {w + yy -(x + zf ; (* + tf - (u + r)\
13. (a + cf)' - (26 - dcy ; (3^ + .?)'' -(x- 2k)\
14. 3m' + Qms — 8jf + 4pk — 12ps— 2mk.
EXERCISE XXXIII [a]. (Page 80.)
6. 1 + 3a; + 2x^ + 2a;=' + a;* ; 1 — 2a; -f 3a;^ - 2a;' + x* ;
1 + 4a; + 6a;' + 4a;' + a;* ; 1 — 4a; + 6a;' — 4a;' + x*.
7. 16 + a?' + 4y' + 8a; - 16y/ - 4a'/y; 35 + ?/'+92''-10^-305'4-6y^;
1 - 3a; - a;' + 2x'' + x* ; a;* + y* + ^* + 3a;'y' + 3.yV' + 32rV.
ELEMENTS OF ALGEBRA.
11
14.
I78i
1080 -T
).
- 251 miles. >
h 9, 18
. 24. 235. ^
= c~-
a — b, 1
E 78.)
+ nGa*b-o\
^b'"-
r44b"\
— z'
80.)
X*
-\-x\
S. 1 + 2a;' + 6:1;^ + a;* + 6a;^ + Oa;" ;
1 — 2.^' + 60;^ + a;* — 6a;' + 9a;« ;
4 _ 4?/ + 9?/' - 4?/ + 4?/* ; 4a;* + i/ + 1 + 4a;''2/ - 2^/ - 4a;^
9 1 _ 2a; + 5a;' - 4a;« + 4a;^ 1+ 2a; - 5a;' - 6a;^ + 9a;* ;
4a* - 7a' + 4 - 4ci^ + 4a ; 1 + 2a' + 2a),+ 2a'^ + c^.
,0. 1 + a;' + 6'2/' + 2a; + 26?/ + 26a;2/ ; ^'^
1 + a'a;' + 6't/' + 2aa; + 262/ + 2a&a;i/ ;
1 + a/x^ + 6'/ — 2aa; - 26?/ + 2a6a;?/ ;
1 - 2aa;' + 26a;« + a'a;* - 2a6a;' + 6'a;«.
A 1 + 2a; + 3a;' + 43;" + 3a;* + 2a;'^ + a;« ;
1 _ 6a; + 15a;' - 20a;« + 15a;* - 6a;^ + a;« ;
1 — 2a; - a;' + 3a5* + 2a;'* + x\
12 1 - 4aa; -h lOa'a;' - 12a^a;'' + 9a*aJ* ;
^6 __ 6a;'' + 13a;* ~ 14a;' + 10a;' - 4a; + 1 ;
x^ — 4a;' + 10a;* - 4a;« - 7a;' + 24a; + 16.
13. 4a' 4- 6' + 4c' — 4a6 + 8ac — 46c ;
a^ + lb"" + K - a6 + ac -■|6c ;
^^2 _|. ^^2 + c^ _ ^a6 - 6c + ac ;
^a' + 6' + ic' - a6 - f 6c + \aG.
[6.]
1 (2a; + 2/)'-^-160. 2. (a + 6-c)'-100. 3. (a; + 2i/ + 3i^^+40)'
4. 6 (a6 + 6c - ca)\ 6. 4 (t^' + a;' + 2/^ + z\ 7. 8a;'2/.
8. «« + 2a'' + 3a* + 2a' + 1.
10. 2a'6' + 26'c' + 2c'tt' _ a* - 6* - c* ; see Ex. 3, p. 128.
EXERCISE XXXIV [6]. (Page 84.)
4. (3a; + 4?/)' - 25^' ; (2a + 4c)' - 96'.
5. \{x^ + 2a;' + 4) - 3a;} x K^' + 2a;' + 4) - 5a;}
= «;'' + 4a;' - 4a;* — 8a;' + 31a!' - 32a; + 16 ;
\{x-^z^w)^-y\\{x-vz\ w) + ^y}
= (x + z + wy + 4y{x + z + uf) + dy\
6. x' + 9a;' + 26a; + 24 ; a;' + 14a!' -r 55a! + 42 ;
x^ + 9itf + 23a! + 15.
7. x^ - 9a;' + 26a; - 24 ; a'' - 14.1;' + 55a; - 42 ;
.^3._9^j.-2 ^ 23a!-15.
8. a;'-' + 3ii;' - 10a; - 24 ; x^ - t2.t'^ + 29.'« + 42 ;
^■.' 4-0;'- 17.1; + 15.
la
HINTS AND ANSWEKS.
9. 8a;' + ISa;" + 22a; + 6 ; 8a;'' - 12a;'^ + 22a; - 6 ;
8a;' — Ax"" — lOo; + 6.
10. X* -{-x^(y + z + w + k) -{■ x^ iwy -{- wz + wh ■\- yz -\- yk + zJc)
+ X (yztv + yzJc + zwk + ykw) + yzwk ;
X* — {a + h -^-c + d)x'^ + {ah -\- ac + ad ■\- he -{- hd -\- cd) x'
— {ahc + ahd + acd + hcd) x + ahcd.
11. 10^ + Qw^r + 'Swr'' + r' ; w* + iiv^r + ^iv'^r'^ 4- 4^^r' + r* ;
8m;» + l^ijfr + Qwr^ + r' ; ?^;* + 8?/'V + 24?//'/'' + 32w;r'4-16;'';
12. A;"* + 15A;*6' + 90^V + 270^;'^*' + 405A-.S-* + 2436-'^ ;
a" — 12a'6 + 60a*6'^ - leOa'fe' + 240a'6* — 192a6' + 646" ;
8a' — %ahv + faw'' - ^iv^ ; |a' + ^a'w + 6a«^;'' + 8t<;' ;
27a'' — 9a* + a-- ^V-
13. 1320aW; - 22680a*6' ; - 2aV.
14. 1485a''6'^' + 55a6" + h^" ; 2145a;Y* + 66a;,y''* + 2/"" ;
- eeSSa" 4- 121a - 1.
15. 54a'^6'; 540a'6='; 1680a*. 16. 1.21662924 ; 1.7101875.
EXERCISE XXXV [a]. (Page 87.)
1. «* + 2a;' - 85a;' - 86a; + 1680.
3. Write A; for a; + a, rn for x -V h, .-. product = k* + k'^m'^ + w*
= 3a;* + 6a:' (a + h) + x" {la"" + 4a6 + 76')
+ X (4a' + 2a''6 + 2a6' + 46') + (a* + a'6' + 6*).
3. a'^6'^ + c'rf' - o'g' — ¥d\ 4. a' + 6' + C - 3a6c.
5. x^ —px* + qx^ — qx^ +px—l,
6. a' (a;' — l) - «« ^^.s ^^''-2) -{-a {ix'' + 3a; + 2) - 3 (a; + 1).
7. w'' — z\ 8. 8a;'. 9. 24a;y^. 10. zw + xy.
11. 6a;y^. 12. 4a;y. 13. See p. 85, H, (3).
\
m^
EXERCISE XXXVI [6]. (Page 88.)
7. a + 6 - c ; a; - 2y — 3z. 8. a^ — 2a6 + 6'' ; a' + a6 + 6^
9. 1 + 2a; + 3a;'^ ; Ba^ + 2a -f 3. 12. a; + 4 ; 2a; - 36.
13. a + 86 ; 2a — 76. 14. 1 + a\ 15. a;'' - 2a; + 1.
5
a
3a_
• "5 3a' 26
2;
8m
dw
[c]
+ 2.
ELEMENTS OF ALGEBRA.
13
J;
' + y2 -\-yk + zk)
■be + bd + cd) x'^
+ 4ivr'^ + r* ;
r^ — Awr^ + r*.
is";
192«6' + 646" ;
Qaw'' + Sw^ ;
+ y''i
; 1.7101875.
GE 87.)
= k* + k'^nf + lit
I. ^x^^-^x-X'^x" ^x-^'^x'^^x + l.
4y
X
+ 6*).
— Mho.
-2)-3(a;+l)
v + xy.
(3).
lGE 88.)
^ a' + a6 -1- ft'*
2a; — 36.
-2x + \,
3. 2a^ + 3y-5^; 2x^-x + \. 4. -^ - 4 + ^ ; o)- 2
1
o.?-i.^;|.2.
5 ?^_1 + ?^. 6. 4a-36; ;^-3a;.
^- 6 5 3« ^
7. l-2;i! + 3x^-, -^'-l. 8. a + 26-c'. „. ^
10. Cube both sides by formula G (2), p. 85.
EXERCISE XXXVII.
1 a'6 + 6^a; a(« + 6)^ + 6(6 + a)^;
^5 (6 _ c) + he (c - a) + ca (« - 6) ; a%o + 6^ca + c^a6 ;
a (6 4- c) + 6 (c + a) + c (a + 6).
3 («_6)(6-c) + (6-c)(c-a) + (c-«)(«-6);
^^ (6 - c) + 6^^ (c - a) + c« (a - 6) ;
«(6_n)''4-6(c-a)^ + c(a-6)^ _
ix-d)(b-e)'^-ix-h){e-ay^{x-c){a-h)..
3. a' + 6^ + c« + cZ^
a'^ (6c + hd + cri) + 6^ {ac + «f/ + ccZ)
+ c'^(«6 + «^ + &c)+^^(«6 + 6c + «c);
«. (6 + c + cZ) + 6^ (c + c? + a) + cnrf + « + &) + ^' (a+6+c) ;
a + 6 + a + c + a + rZ + 6 + c + 6 + 6Z + o + r7;
a6 + ac ■\- ad -\- he -r hd + cd ;
lalhf \\a - c)3 + (^ - ^« + (6 - cf + (b-d? + (c-cZ)'.
4 («_6)X6-e)^ + («-c)^ (e_.?)H(a-6)^ ^^-^^^^^^-^ f-'?;.'
13. a,6, -c;a, -6,c;a, -6, -c. 14. a, 6, c ; a, 6
15 a,6,c;a,6; a, -6. 16. «^, 6, c ; 6^^ and 6y, a; and y.
17. a and 6 ; a, 6, c. 18. a, 6, c. 19. «,&;«'^-
90 ^ ft r 31. a%. 22. a^ a='6, a'h\ abc .
23 aj" a^V a^^- ^^' '^'^- ^^- "^^ ' ""
26*. ^;^ ^^'.. 2^- ^^ -'y ^ -^ ^^^' ^^^^ ^ "^' f ^; ^T;
28. x\ x-y, xyz. 29. a*, a% a^b\ a^he ; ^^ x% x^yz, x^z, x z .
30. a\ a% abc ; x\ x'y, x'y\ xUf-
31. a' + 6' + c' + cZ' - 3 {ahe + «6d + 6cf/ + cda).
^•KW,
w
^
14
32.
HINTS AND ANSWERS.
a + b — c •, a — h + n; — a -f b -{■ c \ a — b — c-
33. if{(a-by + (b + cy + ic + af\;
^{(a + by + (b + cy + (c-ay\;
^{(a + br + (b-cy + (c + ar\;
^\(a + br + (b-cy + (c-\-ar\;
the three expressions are derived from the first by, respect-
ively, substituting — c for c, —b for b, — a for a ; observe,
also, that (—a^cy = + ia + c)'.
(Page 96.)
3. 2 {xy + 'i/z + zx).
EXERCISE XXXVIII.
1. 3 {a" + 6' + c'') - 2 {ab + &c.). 2. 0.
4. 6 (./' + 6' + c'') - 2 {ab + 6c + ca).
5. 2 (a;'* + y"^ -{■ z"^ — yx — yz — zx).
6. 14 {a" ■\-b'' + C-) - 14 {ab + 6c + ca).
7. 4 {a" + 6^ + c' + rf'). 8. 4 (a'V + 6^'^ + c^^"^).
9. 2 (a' + 6» + c^*) + 6 {a'b + etc.) - 13a6c.
10. a'' + 6' + c' + d\ 11. 3 (a'' + 6" + c- + (/-) + 3 {ah + etc.).
12. 0. 13. 6a6c. 14. ahc{a -\-b + c).
15. 4 {x* + 2/* + 0*) + 24 (a^6^ + 6^6-- + d'a"").
16. Note.— The first term in each of the binomial factors should
have index 2 ; i. e., a"^ for a, etc. Multiply out, or use identity,
x^ + y^ + z^ — Sxyz = {x -i- y + z) {x^-\-y'^-\-z-—xy—yz—zx).
17. Multiply out and subs, for 5.
18. rs = {a-^-by — (e — d)"^^ the other pairs by symmetry ; result
is 4 {ab + ac + ad + be + bd + cd).
20. Type terms are a^ 2a^(6 4-c), a^b"^^ and both expressions
reduce to same form. Or, use identity, Ex. 7, p. 105, put-
ting a — 6 for a, 6 — c for 6, and .-. a — c for a -f b.
EXERCISE XL.
1. {a-b){x-{-2y).
3. {a ■\- x){a — b).
5. {m -!- n) {x^ — a).
7. {a + 6) (3.« -h y).
9. (,^ _ 6) (,; + y).
11. (3a — 6) (ic — ?/),
(Page 98.)
2.
{a
-f b) {2x -
-m
4.
{('
- (Z) {ab -
-c).
6.
{a
-f6)(a-
■0).
8.
(a
- ?>rr) (1 -
-x).
10.
in
+ ;r) (a -1- b).
12.
(7
- X) {a -
- 6c).
ELEMENTS OP ALCfEBRA.
15
c-
^rst by, respect-
'iov a\ observe,
: 96.)
' {^y + y2 ■\- zx).
f-'z').
+ 3 (a6 + etc.).
factors should
or use identity,
'—Xy—ljZ—ZX).
nmetry ; result
'th expressions
7, p. 105, put-
2x - dy).
lb — c).
2 — c).
(1 - X).
% + h).
1 — he).
■I
13. (r-s)(3i) + (/).
15. (3.1; - a) (3.*; + Z/).
17. (^- -1) (36^-1).
19. (a-l)(«' + l).
21. ix" + a') {a - 3ft).
23.
25.
37.
39.
31.
33.
35.
37.
{x" - 1) (3«*^ - 1).
ia'x' - c) {a'^x"- - b).
(^a _ a") {x"" + ax + a""),
{a + b) (ax + b!/ + c).
(a -l){a + 6).
(1 _6)(a — & + c).
(1 _ x") {1 +X'' +P + q)-
{a ■\-b — c){(l-e +./).
14. (l-«)(l-6).
10. (a'-'i){(i + 1).
18. {xy — z) {a + be).
30. {x+f){2a + b).
33. ix-y){x-S).
34. (6-l)(c-l).
36. (3?>^-l)(l-3«').
38. {a-b){x-y + z).
30. (aof - b) (6af + «)•
33. (3 + a;") (3 - y").
34. (« — a;) (3it>9 - 36/).
36. (3i)» - 3^") (r" - 35").
38. (1 +jO + g)(l-« +6).
EXERCISE XLI. (Page 100.)
m \ 2
\b (^1
13. (a; + 2/ + ^)'; (P-^i + ^')'.
14. (a-36 + 3c)^ (l-'^ + yy-
15. (3a + 36 + cy ; (3a^ - 3a + 4)^ 16. (3aa; + 36^ + cz) .
17. (3a^ - 36 + 4cy ; {a' -b'- g^.
18. ±4a;y; ± a;y ; a; Y ; -10a;?/; ± 4a;Y.
19. ±6a2/; ± 10a^& ; ±13.-^;^; ±3a"6« ; a\
30. a-, 0- i; i; 4; 6^ 31. i6-, ±460;^/- ±3; ±3; x\
33. 7; i; 6^ -^ 4a^ ; 35-4; 49-4.
23. 81 -^ 16 ; x' + 4:; x' + ld: - c + ^6^ •
EXERCISE XLII [6]. (Page 103.)
Note.— The two factors in each case are expressed with the
double sign ± .
1. a + 6±c; 2(x + y)±z; a^±(y + ^)\ 3 ± (a + 6).
2. p + 2q±r ; ix ± {a + 36) ; 2m ± (p - g) ; 2x (- 4?/).
3';i±(6_c); a + 6 + c±^; (8 + a;) (10 -(T) ; 6-c ±(a-a;).
16
HINTS AND ANSWERS.
4. 3 |2 (a' - be) ± (b' - ac) \; a -- nb ± 1 -, 1 ± (x - y + z) ; V
(a* + b*) {a' + b") (a + b)(a-b); (a- 3c) {a + 4b + 3c). >
5. (_ a + 6 — 4c) (3a — 56 + 4c) ;
(1 _ a + ?,) (1 _ a - 6) (1 + 2a - a^ + 6") ;
(12a; - 1) (2^ + 7).
6. ix-z±y){x + z±y); 4(x + z)(y + ti)',
{x±{y + z)\{x±{y-z)\.
7. {x-z)±(y-'u); a±{x-y); x ± (y -{■ z).
8. x±(y-z)', x±(y + z)', x + z±y', x:' ± {x - 1).
9. {x + a) ± (y + z); ia — c)±(b — (T);
(a' + 6«) {a* + b*) {a' + b') (a + b) {a -b);
(a'' + 6« + 5) (a'' + 2a + 3).
10. a - 6 ± (a; -^ y) ; a' + a ± (?'" - 6).
11. (x -i- b) (a ± X) ; {a- d ± {b - c)}; ab ±c(a — b).
12. \c±(a-b)\\a + b±c\- x'' -\- y' ± (z' + 1) ; a - d ± (b - c).
13. 2« ± (6 - 3c) ; 6 ± {2a - 3c) ; 2a ± (6 + 3c).
14. 3c ± (2a - 6) ; (a + c) ± (6 + rZ) ; (a + (Z) ± (b + c).
15. (6 + c) ± (a + cT) ; (a + r?) ± (26 — 3c).
16. 3c + rf ± (« — 26) ; (a - 3c) ± (26 - c?).
17. {a + d±{b-c)\ {b + c±{a-d)\.
18. (a;^ + 1 -f- ^'^) (a; + 1 -^ y) (a: - 1 -4- 2/) ; i»* ± tV. etc.;
x" (x* - 25) - ^ (a;* - 25) = {x* - 25) (x^ - ^), etc. ;
{X* - 16) (a;^' + 1), etc.
EXERCISE
1. 3a;^ + 2/' ± xy.
3. 3a'' + 6' ± 5a6.
5. aj" + 1 ± a;.
7. a;' + 25 ± 5x.
9. a;' — ?/- ± 3a;y.
11. a'' — 7f±2ay.
13. 9a'' + 6' ± 3a6.
15. ^p" -^q" ±pq.
17. 2a;'' — 1 + 2a;.
19. oc' + 2a''y'' ± 2a.:cy.
XLIII. (Page 105.)
2. 4a'-6'±3a6.
4.
6.
8.
10.
12.
14.
16.
IP.
20.
Vrnf 4- 4«^ ± 7/nw.
a;' + 4 ± 2a;.
^+ I ± t«.
X' + -g^- ."X ■ja;.
m^. — M* ± 4»i??.
4a^ + 6^^ ± 6a6.
9.r"' — y^± 4xy.
ia;^ + ;y' ± xy.
2a'±y'±^ay.
ELEMENTS OF ALOEBUA.
17
(x-y ^z);
(« + 46 + 3c).
).
[X - 1).
a — d ± {h — c).
{h + c).
'^, etc.;
I), etc.;
.05.)
3a6.
± Imn.
imn.
ixy.
vy.
lay-
21. a:* + 2/* ± a^Y, <^tc. ; x* + } ± ^'^
23. a'x* + 1 ± «^"^ ; '^* + ''^y' ± 2^^-
23. (a + 6)^ + c'±3c(a + 6); 1 + 2a;^ ± 2a^.
24 4a!^ + 2(2/-2r)^±r,^(//-^)' 1 + •''>^* ± ^^'•
25. l + 2a*±3a^; a^ + 96^ ± »«&•
(a + by + {a- by ± (a^ - &")•
28. c' + 2(a4-6)='±2c(«^ + 6); 1 -t- a^ + 1 - ^'.^ ± ^
3 ^ a^ + 1 -T- 6'^ ± 3 -r a6.
-r-a6;
EXERCISE XLIV [u]. (Page 108.)
17. C?)!" + 21) (w^* + 19).
19. (af + 7) (af + 12).
18. {a'x + 29) {a^x + I).
20. (05 + 17) (^ + 33).
21. (af + 12)(a5" + 4).
23. («•+ 27) («-''+ 13).
25. (a'aj + 81)^
28. (a;-19)^
31. {x'-S)(x'-2n).
33. {x-Vdyy.
35. (a -276) (a -26).
37. (26-a6)(5-a6).
39. a;\+l± V-«- n,
41. 3a; (a? - 2) (a; - 8)
22. (a; + 33) (a; + 27).
24. {a + lSby.
26. (a; -4)^
29. (a? -20)^
27. (a; -15)'
30. (a; -50)'^
■U.U'
43. a;':l-60±17a5.
45. (a + 6-4) (a + 6- 3)
82. {m - 17/1) (m - 5??).
34. (a;-^ - S//'^) (a;' - 42/'0.
36. (4 -a;) (3 -a;).
38. (a - 25) (a - 15).
±:£ 40. (a;'' - 27) (a;^ - 8).
^--^ 42. a(a; — 5)(a;-6).
^:r/ 44. (a;» - 7) (af' - 37)
46. (13-aa;)(ll-«i^).
47. il-SxY)0--^^^y
Y). 48. (a -276)'
xUa-^tnbx){a-bbx). 50. (m-19)
49.
51. ip-mf
52.. U^-2/)
[6.] (Page 109.)
-v)"-33i {{x-yr-'^^'
1. (a» + 1) (a' - 2).
3. (a; - 3) (a; + 2).
5. (a; 4- 12) (aJ - 7).
2. (a + 3)(«-2).
4. {X — 16) (a; + 3).
6. (^ 4- 12) (*/ - 5).
18
HINTS AND ANSWERS.
■J
i
\
., ./'•
7. {a + 30) {a - 7).
9. {x+ 13) (a; -11).
11. (//' - 10^/^) (//' + 5a0-
13. S (az'' - U) (az'' + \).
15. {aha + n){(ihc-2).
17. Or' - 48) (x' + 8).
19. (x + y-m{x + y + \S).
21. (aj'-"* + 4) Of'^" - 3).
23. (13 - ah) (5 + «6).
25. 3y (rt + 146ic) (a — 26ii!).
27. (3a; + 7) (3.t; + 5).
29. {7a--Sbf.
31. a;* (82/» - 10^)'. 33. {a' -
34. 3 (a;" + 2/') (Ba;' - 4,^') : w
35. (Saf* — 36") (8.t'» + ?/').
37. (f« + 76) (|rt - 86).
8. (a + 25b) (a- 136).
10. (u! - 10) {X + 2).
13. (ah - 4) {^;6 + 1).
14. (a* - 30) (a* + 5).
16. {a'b' - 30) («"6'' + 3).
18. (of — 16) (O!" + 3).
20. {a-30(6+6')H« + 13(64-c)
23. (30 + rt)(19-a).
24. (13-7/0(17 4- w). I
30. (3a; + 7) {2x -{- 5).
28. {2x''y - 7-2-") (3a;''?/ + O^-^).
30. X (36 - y) (36 - 5y).
- 406^) (a' + 56^). 33. (llo;'^
here x = a — b and y = ''•
36. (^a;' + 7) (ia;' - 6).
- e-)(:-«)-
/
39. (^. + l^)(^-2^)- ^^- (•">.*'' + ''il)('^>^''- '^D-
EXERCISE
1. (3a; + 1) (3a; + 3).
3. (3a.' + 3) (5a; + 4).
5. (3a; + 5) (3a; + 4).
7. (4a + 9)(« + l).
9. (a; + 5) (4a; + 3).
11. (4a; - 3) (3a; + 3).
13. (4a; + 7) (3a; - 5).
15. (3a; + 3) (3a; — 1).
17. (3a; + 4) (5a; - 3).
19. (5a; + 3,v) (3a; - 52/).
21. (dm + 20) (2wi - 19).
23. (3a; + 7^/) (4a; — 5y).
25. (5a;'' - 1) (4.^•^ + 1).
27. (13a; - 7) (3a; + 3).
29. (8a + 6) (3a — 46). -
XLV. (Page 112.)
2. (4.a; + 1) (x + 3).
4. (3a; + 3) (3a; + 1).
6. (3a; + 7y) (4a; + 3^/).
8. (1 + m) (7 + dm).
10. (a; + 7) (3a; + 2).
12. (4a; + 3) (3a; - 2).
14. (4a! - 7) (3a! + 5).
16. (5a; - 1) (2.1! - 3).
18. (a; - 7) (7a; - 1).
20. (a'-- - 19) (a'^ + 17).
23. (2a + 20) (3a— 19).
24. (3 - 12a;) (5 + 11a;).
26. (15a _ 1) (a + 15).
28. (6-»/)(3-52/).
30. (8 - 97/) (3 + 8y).
Vly)'
1*
S
I I
i;ij:.mi:nts ok alcjkhua.
10
(a -12b).
r + 2).
'lb + 1).
(n* + 5).
) (a'b-' + 3). ,
(*•" + n). I
l+f)} \a + \2{b+c)\.
19 -a). /
|(17 + ^/0. /
'^ ^- 5).
i'^b ~ ny).
33. {Ux'-V^yf.
»>(1 y=:c.
h'^^ - 6).
(•^o;'^ - 31).
112.)
» + 3).
5« + 1).
'M + Zy).
+ 3;/0.
+ 2).
» + 5).
»--3).
-1).
'^ + 17).
ia-19).
> + llic).
+ 15).
%).
81.
183.
5.
i7.
J9.
41.
43.
45.
47.
1.
3.
5.
i.
9.
11.
13.
15.
17.
19.
21.
23.
1.
3.
5.
7.
9.
11.
13.
15.
(2H.T^ - 25) (^' + 5).
4(7.c — 5^)(2.f — //).
(8« - 56) (7« - 'Uj).
(8?/ + 52-) (Oy - 82-).
(50^''' -H 4Z;'0 (r/' - 56").
{Ux + 12//)(3aJ— 4y).
(39.<;— 2G)(.r + I).
(l-13.r-)(l + 11 ir').
(3.f='-21)(4.i;' + 11).
{x
EXERCISE
a){x'' — 2x — 1).
(my — n) (ay^ + by - c).
x"^ —X — 1
32. 4 ('14.?'+ ryy){x- y).
34. 4(H.r— 5//) (.r + //).
3(J. 2(28.y + 1) (/y- 10).
38. ({)// + 3r0 (4// - 5a).
40. (50^ — 56) {(( — 46).
42. (3.r + v/)(13,c- 11//).
44. {VZx -\' Vdy) {nx — Hy).
40. (a" — 136") (//" + 116").
48. (\7x-\}(x+ 17).
XLVI. (Page 114.)
2. (x — a) (X- — />^ + (j).
4. (26 — c) (ic" — 26a' + 6).
(7ix — a) (x"^
i)-
6. (6a; — a) {{m + 1)6V + (m + 1)(». + ^)(ibx + (y^ + !)«=[
multiply out, take v^i-torms for one group, etc.
7. (y-b)Oj-rry
8. (x — 6) {X — a) (x + 26).
(x+p + q) (x + q—p) (x—2q). 10. (x — a) (x + 6) (a; + 3).
(x+b) \x(x—l)—a(x-\-l)\. 12. (2x - a) (2x'' + 4x — S).
(pi/-^q)iy''-y + i)-
14. (mx — n) {pxr + qx — r).
{mx — n) {ax^ — cx — b). 10. (/;.*; — q) {^x"^ — cx — b).
(x^—px + q){ax''-\-bx—a). 18. (2a; + 3c) (a;' + 0^.^ — 26).
(2a; + 3c) (x" — 2ax + 36). 20. (ap — bq) (2p^ + ?ypq + q").
(ap — bq) (np' —pq — 2q'^). 22. {ax + b) {ex"" + rZa; + c).
(aa; + 6) (2ca;' - dx — 3c). 24. (3y — «6) (3v/ — 6c) (3i/ + 5).
EXERCISE XLVII.
(Oa; - y + 1) (a; - Oy - 1).
(4a + 56 + 4) (3a — 46 — 5)
(3a; 4- 2/ + 3) (.7; - 3^ + 9).
(3a -6- 7) (4a -36 + 8).
(a + 37/)(a-4y-5).
(Page 118.)
2. (3a; + 2y + 1) (2a; — 3y — 1).
4. (a; — y + 3^) (a? + 2y — 2-).
0. (2a — 56 + Oc) (3a + 46— 8c).
8. (7a;-.v-l)(a;-2/ + 3).
1 0. {2x — 5/y — Iz) (2x + 3,y + 3^).
(3a; + y — 4^) (3a; — 3//— 2^). 12. (3a; — 2// + 3^) (2.a;— 3// + 2^).
(5a; — 3y + 2^) (a; — // — 2-) . 14. (a — 26 + 3c) ( 14a — 6 — c).
(2a - 6 - 3c) (4« - 36 - c). 10. (1 - 3a; + 4y) (1 + 7a; - 5y).
30
HINTS AND ANSWERS.
EXERCISE XLVIII. (Page 122.)
7. 8. 8. —8a\ 9. -205. 10. 1.
11. a^+pa^^qa + r. 12. —36. 13. 1555.
14. X + 2, X- 3. 15. (X + 1) (Hx + 2) {2x — 1).
16. Last term should be 52^1 18. (x — 2) {x — 5) (x + 7).
19. -535. .20. -800. 21. 101. 22. 115.
23. — a^ —pa^ — qa — r.
24. dbc — Aah (a + 6).
29. -1. 32. 2. 34. 2 (a + 6)1 35. 0. 39. 0.
40. 0. 41. yes ; put 1 for a; + //. 42. 0.
44. a^+pa + q, a'' +p'a + q'.
EXERCISE XLIX. (Page 126.)
5. (p - ly -(p-l)(q + l) + (q + If ; a» - 6".
6. x'° ~ x'a' + a'" ; 1 - (a-b) + (a -by - {a-b)\
7. ie' — 1 + 1 -=- a?'' ; x' — 2x'' + 'Sx* — 2a;' + 1.
8. x'' + y''±xy; (a- + 4b') (a + 26) (a - 26) ; 2x {x" + 12^^^).
9. {a + b){a' + ¥±ab)-
x" — 2y^ and x^ + 2a;y + 4a; V' + SxY + Uy\
10. (a' + be) (a* + we - Aa'bc) ; {x + 1)^ {x - 1)« ;
{x+\){x-\){x' + \)(x' + \).
11. «^-(26^)='=^a-26^H«' + 2«6^ + 46'»}; («-6)«6.
12. Expression=(4a'-96'^)(a:^-8a='),ete. ; la - ^\la' -^ i + _L\ .
1 5. Expression = {x"" — y"") {x — yf = 128a='6=' {a- + 6'-').
16. a? — 1, factor dividend.
17. (a* - aV + x") (a« + a'x' + x^).
19. Factor and divide hy a -\- \ -^ a, .-. «'■ — i + i -j. ^^^ = o
.•.a' + -^-, = l, .-. «=+^^ + 2 = 3, etc.
20. Expression = ( 1 — ,'c) (1 — ,»;«).
21. Divisor — (x — \) (x' ~x + 1) and ^nveu expression vanishes
for each of these factors.
22. i^r - y) (x:" + y') (x' + y*).
ELEMENTS OF ALGEBRA.
U
tE 122.)
1.
1555.
') {2x - 1).
-•5)(.T + 7).
115.
EXERCISE
(a + b).
39. 0.
42. 0.
126.)
■b\
i — b)'\
ix(x^ + 12y2).
X — 6) a6.
1 ^a' = 0,
ression vaiiislie.s
L [a]. (Page 129.)
2. (a-6)(6-c)(a-c).
4. (a-6)(6-c)(c-a).
6. 3 (a' - 6^) (6^ - f^ (0' - «')■
1. 3(aJ + ?/)(2/ + ^)(^ + ^)-
3. (a_?>)(6-c)(a-c).
8. _(a-6)(6-c)(c-«)(fl'. + & + C).
«). (a_6)(6-e)(^^-«)(«^ + ^ + ^)-
10. (a. + 2/)(y + .)(^ + ^). 11- 3(««-6)(6^-c)(c'-a).
13 («. + 6 + c)(a^ + &-^+c^). 14. (x + y + ^y. 15. (« + 6 + c)-'.
16. 6a6c. Insert in text - (a + &)^ and read - before (c + «)^
17. (a — 6) (6 — c) (c - ^f) (^'^ + ^'^ + c*^)-
[6.] (Page 130.)
By symmetry ; or formula (H) (4), p. 85.
By symmetry ; or transpose dabc, then « is a factor, etc.
5(^ + ^)(^ + 0)(0 + a-)(x^+/+^^ + a.-2/ + F + ^^)-
(a + b-c){a' + ¥ + a' - ah + hc + ca) ;
(^a + b + c)ia' + b"- + c' + ab-bc + ca) ;
(^-a-b + c){a'' + b' + c'-ab-^bc-\-ca).
12f ; 2d term should be 13a;'.
Use synthetic division ; 4m — 12 = 0, m - 3.
Given expression = (a; - 3) {x + 2) (a. - 5), which is true for
all values of x, :. coefficient of like powers of x are equal ;
1 e.,a=-6, 6 = 1, c = 30.
5:=_6,c = 5,a = 12. 11. l-r-(« + 6 + c).
2 ; dividend is 2 (divisor). 13. 16a&c (6 - c) (« - c) {a - h).
-abc{b-G){c-d)(a-b).
- abc (a + 6 + c) (6 - c) (c - a) (a - 6) .
2.
3.
4.
5.
6.
7.
8.
9.
12.
15.
16.
EXERCISE LII [a]. (Page 133.)
1. ax-, x + 2; 2{x-y)\ 2. 2(.^-«); ah(x^a)(x^h).
r + 3; .t: + 9. 4. .^-2; x + 2.
5 ,, + ?>; .'1- + 1; .'-3. 6. .-^ + 5: ^ + 4; (^ + 1)'.
7 ^ + 3;^--ll^ «• 3(a;+t/); (a; + y)^^'' + 4.
3
PX'V
n
HINTS AND ANSWERS.
Q. X + 2 , X -{■ a. 10. X — a ; X — y..
11. a; + 3^; a + 3. 12. a — 1 ; r» — a — 4.
13. a + b^ c; a + b + c. 14. a + b + x + y\ x -\- a.
15. a; — «; 8(a; — 3?/); a; + «. 16. a? + « ; a; — 5 ; ((t— a:V''
17. ic'' + icy 4- 2/' ; ir'^ + «' ; a;^ - y'.
18. 3 (^ - y) (a! + 2^) ; 3 (a + &) {a^ + 6").
19. 5(i?-^)(p + g); a3 + y.
20. X + y.
22. 2a + 5 ; « + 5.
24. a^ +ab + b^; a -\- b.
26. x"" (Sx + 2).
28. 3iC + 4a ; unity.
30. a^ + V
32. 2a + 36 — c.
34. mx -\- m — X.
86. a; + 2a6 ; omit a in ax^.
88. 3 (2a - 7).
89. (2aa; — y) ; last term of 2nd expression should be 3y'
40. (x-iy.
21. 3a; + 1 ; SiC — 1.
23. a; + 3 ; (a; — 1)'.
25. 2a; + 1 ; x^ -\- y.
27. a;'' + 2y^ — 2a;2/.
29. 4; 1 +-••
a
31. 2a; — 1.
33. 5 (a; + 2y).
35. ap — bq.
37. (a — b) (X + a).
EXERCISE
1. 2 (a; + 1)'. 2. x
4. 7a' + 3a - 1. 5. y'
7. x"" -2x- 3. 8. a;'
10. 3a;' — 2xy + y\ 11. a;
13. a; (2a;' + 2a;j^ — 2/').
15. (a; -l)(x + 1) or a;' —
LI 1 1 [a]. (Page 141.)
1.
3. a' - 8
5. 12a; + 5.
cT' — 3.'^ 4- 2.
2
-5.
" 2x' - 3a; - 1
' + 82/-
2.
6. 2/' -32/ -5.
-3.
9. 5a;' -1.
-1.
12. ix + 2)".
14.
X —
2.
1.
[h.]
2.
x'-
- 13a; + 5.
4.
(X-
- 3).
6.
2x'
— 4a;' + .t; — 1.
ELEMENTS OF ALGEBRA.
23
-y.
'' + y, X + a,
-5; ((t~a:)5.
\x — 1.
'2xy,
+ a).
uld be dy^.
s 141.)
2x' — ^x -
-1
y"" - 32/ -
5.
hx^ - 1.
(a; + 2)".
5.
%i. ic'^ + 3a; + 5.
a^ — a" — a — 1.
\\. 2x'' -dx — 1.
Sx — 5a.
Is. x-'-lx-d.
8. (a; + 1) (X'' + 1).
10. 2y'' — 7.
12. 03" + x"" — 6x + 3.
14. 20?" (2a; + 9).
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
2.2a^
A2a^¥
y.xY
EXERCISE LIV [a]. (Page 143.)
2a;' -6^0; ah-cx^y, a-12a¥ ; Az-lxhjz.
4?/ -a;*; y-10x*y'; Sc-la'b'c'.
2'da'b^; 7a'-36^2; ab-c.
2y'Sx'y-2; Sy'-lx*y; 2'2xY-2xy; ax'-Qxij. .
2.3a6c-2; S-ix'^y-Sy, p-Qpq^-p-, ax^-by\
1 . a'&^c* ; xy' • irya'x'y ; n'' • dm'p* ; i?"g".
3a; (a - a;) ; Sa'^b (a + &) ; a (a - &') ; ahc {a' - c%
Aa'x (a + X) ; 21 {x + y) (a + b) ; a{p + q) {p - q\
a (a+&) (b+G) ; x (a;+l) ; x' (a.--3) ; (aJ-1) (.x' + l) ; «=• {a-bf
ah {x + a) (^ + &) ; «& («' - «') I (a; - 1) (^ + !)•
(a: - 2) (OJ - 1) ; 21 {x - 2) (a; + 2) ; x (a;+l) (a-+2) ; {x + yY
{X + 1) (a; + 2) {X + 3) (a; + 4) ; (a + b) {a - b).
X (3a3 - 2) (2a; - 5) (a; + 7) ; x'y {a' - 6^).
^3 (-^2 _ ^2) . a;3 (a? _ «) (a; - 6) (aj - c).
6(a;-2/)^ 6 (a; + 2/)^ (a; + 6) (a;'' - «')•
(2a: - 5) (9x' - 1) ; a' (a + Sf ; (a - 6)^ (« + bf.
{a + by {a - xy ; {a + bf (a' + b').
(^ _ xy {b - yf ; 6 {a + 1)' (« - !)'•
- (1 - 2a;) (1 + 2a;) ; (a;^ - y'Y ; a'*?'^^- 1) (& + 1).
- 6a"6" (6^ - 1) ; x'-y' ; (a;^ - 1) (^' + 1) or a;* - 1.
[&.] (Page 144.)
f X — 1.
21. a6(4a''-l); 6a; (3a; - 1) ; a; (a; - 2) (a; + 2).
22. (x + 1) (a; - l)'^ ; {x - 1) (a; - 2) (a; - 4) ;
{X ~ 1) (a; + 2) (a; - 3).
23. (a; + 3) (a; + 4) (a; + 5) ; (x - 1)' (a; + 2).
24. ia-r)(a- 2) (a + 2) ; {x' - fY or {x - yf {x + yY
24
HINTS AND ANSWERS.
25. (X + 2){x- 4) {X - 10) (X + 12) ; (a; + 3) (« - 3) (x - 12).
26. (x - 2) (a; - 4) (x - 7) ; {x + 1) {x + 3) (.i^ - 4).
27. (a; + a) {x' - ¥) ; (1 - a;) (1 + xy.
28. (a; — a) (ic + a) (a? — 6) ; (a + 6 — c) (a + 6 + c)'.
29. (X - 2)' (X + 2y ; (x + S) {x - 3) {2x - 1).
30. (2/ + 2)iy- 3) (3// + 1) ; (2a; + 3) (2x - 3) (3a; - 2).
81. (2a; + Sy) (2a; - 3y) (3a; - 2y); 3 (a; - 1) (3a7 - 1) (3a; + 2).
32. 20aV (4a - 1) (pa + 1) (3a + 1) ; (4a - 1) (4a + 1) (5a + 1)
33. (a; + 2) (x - 2) (Sx - 7) ; x"" - y''^.
34. {X — 2) {x - 3) (a; - 4) (-i; — 5) ; 2a?/' (4a;— 1) (a;+3) (3a*-2j
35. (,r ^y){x- y) [x" + ?/") ; 12 (a;' - l)^
86. (a; — a)' {x + a)" or (a;' — a^f ; — 12a; {x — 1) (a; + l)^
37. 6c {a' - ¥) ; a;'^ (a; - l)'^ (a; + 1)'.
38. (x + 2/) (x'' + xy + y"") {oc — y) or {x^y) (x^—y') ; 24 (1— a;^
39. a*x* — h*y* ; a" — b\
40. 12a'' (a"" — y'), (a;« + a').
41. 1 —x'-, (1 - 2a;)^ (1 + 2a: + 4a;') ; (x + yy (a;=' + '
(x + a) (a; + c)
(X - c)^
a;
-2c' (x + a)(x-b){x + b)'
xj- b
X ■]- c
(x + 2)', i-l
so'
I
',
H'h
L-^"
35J
HINTS A XI) AXSWKUS.
[&.]
1.
8.
5.
6.
(X
X + a a
• I
x — a' b
x-'ix-l)^
a' {x + 1305)
~W(x — 'ia)
2. 1
a
. (a^'-l)(.r+ l)
' (^ - 7) (i» - 5)"^
— 6 2r///
rt + 6' 15ct/"
\1H ,
{mxyT ; Tj + -, - 1
a'
put a; for a in first term of numerator and a
for a" in second terra
Kfift
7.
a + b
b — c — a
1.
625^
8
i?* + i>^' + q
EXERCISE LXI [a]. (Page 176.)
. 4a — 3 8a 4- 1 a; — 3 a^ — be a''
1 • - • • • . . • rth
" ' 28 ' 6 ' b' ' b*'
2.
16 ' 28 ' 6 ' b""
2(a — b) ^ 16 + x Qx — 2 ^ 2a; — 5 12a; — 8
3 ' 16-a;' 12a; + 5' x - 10 ' 12.i; + 9"
X
, o^ 2 46 + 9a a;^ -^
*• &'' 12a + 86' ^'M^' ^°
%1
ELEMENTS OF ALGEBRA.
33
5.
1).
■')'
6.
7.
8.
rator and a
9.
'>■
10.
11.
a
ax—1 1
2ax 2x — 35
a — ax+1' a' d^ -f- x^ '
a;» - 2ic' — 3 1
X'
— - - ; " ; a + X.
x^ (3 - X) (3 + av a;
a* JfX^ — a" 12
X
a^ + x^ — x^' (x — 1) (a; — 2) '
(a — 6)"
_2ah^_ 4abia -' +_h')_ (a' + b') (a* + Uffh'' + b*)
a' 4- 6" ' a* + 6a''6'' + 6* ' 2ab (3a* + aoa'^6'^ + ;5>/) '
6 — g + 4 6a <'"
6 — aTs' r^;*;'
g' + fe" , ab + 6c + (to
b ^ a * a^ +~a6^+~6o^ '
[6.]
4ab (a + b)
4 (ax'' — 4)
(a-b) (a» + 2a'-'6 + ab'' - a'^ZT^'^) ' x' {dx - 2) (3a; + 3) '
1 + X
1 + X'
2.
1
2
a — S
(^' - 2/' - '£'') (^' + ;y' - 2'') ' 63 ' (a _ 1) (a + 3)'
b^ — ab
a' + ab ^
• x'+l' x* + dx'' + l'
6. 0; ^-^1+-^
' 8^(y + 6)
8.
5. 1;
1
a'^ft (6 - a)
1 + 2iz;
(a + a;) (a" — iB') ' x
2 + 3a;
9 -i^^ . 1
1. 1 ; a + b.
4 ^-y . ^JT^.
x-\-y' 4 4- 4a;"
2. 16a; + 11//.
5. 1.
3. T^x + ll.y.
o.
00
34 hijn'TS and ansvveks,
7. (^.). . 8. ^. 9. -^*,.
10. 2. 11. — ^or— /y. 12. c,
16. Take the fractions in pairs, thus :
/.j_ + _j_\ + /^^ n = .£ I
\s — a s — hi \s — c sf {s — a) {s — b) s {s —c) '
by substituting for 25, etc.
17. a. 20. Multiply given relations out and transpose,
a + & 4- 6' 4- ri = dbc + abd + bod + acd = abed { 4- etc. l etc.
\« 7
EXERCISE LXII. (Page 185.)
21. 1. 1 ■\- X + x" + x^ + X* +
^. 1 4- 3ic 4- Ca;' 4- 27a;=' 4- 81iB* 4-
3. \ — X + x^ — x^ ■\- X* —
4. l — ^x + ^x"" — 27^5" 4- 81a;* — '
- a" a'b a'b^ a'b' a'b*
5. — ■{ — 2 H ;- H 4- — r +
X X^ X^ X* x"
^ X^ , X^ X* x''
a a^ a^ a*
7. a + abx + ab^x^ + ab^x^ + ab*x* +
^ , 2x Sx^ 4x^ 6x*
a a^ a^ a*
9. 1 i- x — x^ — x* + x'^ + ,, .
10. 1 4- ttic 4- a V + a^x"" 4- a*x* 4-
27 ^^ + y' ■_ (^ + yf - ^^y i^ + y) ^ ^^^
• x'-y'^ (X - ijf + dxy {X - y)
Substitute from given conditions.
EXERCISE LXII I [a]. (Page 191.)
1. x = 7. 2. x= 1^. 3. x = 2.
4- ^ = 3. 5. X = 4. 6. x~ f I, read x for 2x,
7. x = ^. S. x = 5. 9. X = 6.
«(l + 6) da-b
10. ^=-^^^-^- 11. ^ = -^ • 12. .. = 1. •
ELKMEKTS OP ALGEBRA.
35
18. x =
16. x =
_ a''{b- a)
b{a + hj'
d{a + c)
6
14. x =
3a -6
4
15. X =
2ab
17. x = b.
19. X
22. X
25. x =
28. x =
31. iB =
33. x=z
35. ic =
38. x =
3
4*
^ 1.
ab
20. a- = 3.
23. x= — ^,
a'
a + b — c
7
TS"'
26. a; =
29. x =
bn + d 7 n+a mn + cmn
b + d + am ■i-'cn
acn—abn— abm — bcm
nb — no — ma — mc
7. 36. a; = 3.
f 39. a; = 11.
b — a
3
a + 6
18. a; = 9.
21. a; = 60.
24. a; = - f
27. a; = — ^.
30. a; = 3a.
32. aj= 14.
34. a;=3a; right mem. sh. be 2^.
37. a; = 3.
40. a; =-6. / b
1. X
4. a;
ifi
3.
[6.J
2. a; = — 107.
b{a—b+c)
7. a; = or -^ 44,
5. a? =
8. a; = - ^T
a
3. X
6. a?
_ a+6+c+c2
w — «i
9. a? = —
10. a;
13. X
16. a;
18. X
20. a;
: 15.
01
^~2~*
1 = and 4aj^ + 5a; + 3 = 0.
11. a; = 3.
14. x = — 6.
17. aj= ± 3 or oo
= 2.
21. x = a.
12. X
15. a?
19. X
22. a;
a&
m
c
H-
7.
a + 6.
76c
96 + 4c''
1. aj = 13 ; second numerator should be 3.
3. (x _ 3) (2a; - 5) = ; x= 3.
4. a; = 41, ;,_>, 5. x = —i
2. ic = — 9.
36
HINTS AXD ANSWERS.
ah — he — ea
6. a; = —
it! =
a(a — h)(a~ c)
e {a + b) {a + c)
8. 05 = 0. 9. a; = 5.
10. x{h — a) = 0; whence x = 0, unless h — a = 0, in that case
X may have any finite value. ,^
11. x = ^^/r^orO. 12. x = c. V^'L^^-N-*^^-^*^
13. x(2x + o) = ; .-. X = or — 2|.
14. a; = a + & + c. 15. a; = a'' + 6" + cr.
16. a; = a*. (First numerator on right hand should be x — 1.)
17. Take in pairs the fractions with like numerators ;
_ np (c — a) + inp (a — b) + m7i j h — c)
~~ m{a — c)+n(b — a)+p ( c — 6)
(_ 7a; + 49) \ - — ^ - —~- — -- i = ;
^ ^lx' + x — 2 x'+x—\2)
.-. X = 7 or 00 .
19. Complete the divisions, cancel and transpose ;
^11
■i%
18.
or
a? — 4
1
1
X
4 x—2 X — T) a; — 4'
whence (x — 8) (x — 4) = ; .*. a; = 8.
The value 4 is not admissible.
20 X = ^^^ ^ ^ ~^^ ^^ ~^ ^ + ap(q- ii) (m — n)
6 {q —p) {m —p) + a{q — n) {m — n)
21. a; = |m (6 — c) — ?i (a + «) f -;- (m — w)
22. (a'^x + b^x - a^b - ah'' - ft'^c + a'^c) x
i i ^ Uo
{{x — a) {x — b) {x — a — c)(x — b + c) )
.: X = {a"" (b-c) + b^ {c + a)\ -r- (a" + 6'').
23. (a — b){ 1 = 0, a; = ■
\n — o p — qf p — q
24. ^ '- 1 4- anal. + anal. = 0, whence
b + c — a
{x — a — b — v)\ h anal. + anal. >■ = ;
{b + G — a S
x = a + b + c.
ELEMENTS OF ALGEBRA.
3t
hat case
25.
a — x
a^ — he a + 6 + c
4- &c. + &c. = 0,
or
a6 + 6c + ca — (a + 6 + c) a? ,
r—^ — Y-— 7 h anal. + anal.
03 = (a6 + 6c + ca) -T- (a + 6 + c).
= 0;
1. lO^'y dozen.
4. $300.
EXERCISE LXIV [a].
PROBLEMS.
2. $36000.
ma — 126
5.
(Page 198.)
3. 12 years.
6. "*"
12 — m "' 6 + c
7. 754. 8. $3.75. 9. 142857.
10. $8000. 11. I9O3-W bushels. 12. 90 and 91.
13. Equation reduces to (4 — • 4) re + 40 = ; a; = 00 ; i. e., condi-
tions of problems are inconsistent. In fact, area will
always he 45 ft. less, under the given conditions; for
using 45 for 85, the resulting equation is an identity.
14. $1857.35|f| and $142.64|ff.
15. 857142 ; x representing number, equation is
TT
{X — 2) + 200,000 = ^x.
16. $7.60.
1.
4.
17. 550 ; read 6 in first line ; 4 times and 6 cts. in second line.
18. 13|- feet and 16^ feet.
19. A, $2800 ; B, $3900 ; C, $5138 ; D, $2196 ; E, $2966.
30. -fH. y
( [6.]
960 gallons. 2. 420 acres. 3. $1280, 7|^.
, , g(100 4-^?)~100p ,.
Gain or loss io = ^^ , according as
^ < 100 + n
5. B makes 1740 yds. in 4 m. 34 sec; C makes 1700 yds. in
4 m. 32 sec. Let x = time in min. from starting at which
A overtakes B, then -^-1760 = 20 + t^- 1740, a;=l|^min.,
distance 775^^ yds. from start. Similarly A is found to
pass C in 3f^ m. ; distance 1456^^ yds. from start.
38
HINTS AND ANSWERS.
6. lOi's^r miles. 7. 5 gal. 8. $7400. 9. 57 miles.
10. mnpqr -^ {mnpq — mpq — npq — mnq — mnp).
11. 484. 12. 1, 2, and 3.
13. 12000 sq. yds.; 45 cts. 14. 189.
■it ^- 4r ^1, 2a? X , px — x
15. aj = distance; then -^ = h^—i
6 op op
16. Let 2aJ and x be digits ; (2001a;)' - (10020?)" = 2999997a;»
= W X 749,999^^. 17. 180,000.
p{ll7n — 21n)
18. (^1)""'
19.
20 {7n — 71)
20. Regular rate 40 miles, diminished rate 38,^ miles ; 100 miles.
ap — a7i
21. 221 : 273
187 : 231.
23.
7n — 7i
23. f 24. 14172.
EXERCISE LXVII.
1. a; = 3 ; y = 2. 2. x =
d. x=:l ; y — S. 4. x =
5. X = 4: ; y = — ii. 6. x =
7. x = l; y = 2. 8, a; =
(Page 212.)
6; y=-4.
5 ; y = 4.
-2; y = B.
-20|; y=-lH.
9.
a; = 10;
y = Q.
•
10. a^ = m; 2/ = iHI-
EXERCISE
LXVII I. (Page 213.)
1.
a; = 2;
y = S.
3. ^ = H; y = -¥.
3.
a; = 3;
y = -2.
4. a; = 5 ; y = — 5.
5.
a; = 4;
2/ = 4.
6. a;^i?^; 2/ = -i|.
7.
a; = 3:
y = 2.
8. a; = 3; 2/ = 1-
9.
a; = 4;
2^ = -3.
10. a; = 3i; y = ^.
EXERCISE
LXIX. (Page 214.)
1.
X — 5;
y = 7.
2. a; = l ; y= —1.
3.
a; = 2;
y = -3.
4. a!=-2|i; 2^ = S^f .
6.
a; = 5;
y = — 4.
6. a; = f; 2^ = |.
7.
a' = iAV; y = ^^'
8. a; = 5 ; 2/ = ^■
9.
«=iW; y^-
■1t%-
10. x=Q^; y = -d^.
ELEMENTS OF ALGEBRA.
39
iles.
7a;'
00 miles.
4. 14173.
19f
f
EXERCISE LXX. (Page 216.)
a + b a
1. x = —,r-\ y — —
2
2
1 _ 62 a6 - 1
2. 03 = ^; y =
a — b
a — b
mp — nq _np — mq
4. a; = 1 ; y = 1.
_ 6(46'~7a '')
2
y
7. x — y —
9. x — y —
10. a;
5. a; = a.+ 6; y = — 1.
8. aj = ^w + ?i ; y = m — n.
a + c
_ a {bc—^ac—c^—a'^ + 3a6) ^
2(2a6 + 6c— ac— c")
; y
a {Za^ ■\-ab-\- 4ac + ac'^—bc)
2"(2a6 + 6c— ac-rc')
EXERCISE LXXI [a]. (Page 218.)
1. a; = 6; y = 12.
3. x = 7; 2/ = 10.
5. a; = 6 ; 2^ = 12.
7. a; = 8 ; y = —^.
9. x = y = ^.
11. a; = 8^; y = -i.
^ „ 2mw (n" — m')
13. a; = —J — T^-5-^ ^ •
n* + 6?7i.W + w
4, y
14. a; =
a"
y =
a — b''^ b — a
15. a; = 4 ; y = l.
17. a; = — 2 ; y = 4.
2. a? = 4 ; y = 3.
4. a;=13T8j,; y=-4i|.
6. a; = 4/^; y = — 12.
8. aj = 2 ; y=7.
10. a; = 4 ; y = 5.
12. x = y =
a'^ + b^
n* — m*
n* + Qni^n^ + m*
16. a; = 6 ; y = — 2.
18. a; = 3; y = — 4.
[6.]
1. x = y = ^\.
2. a; = 14 ; y = — 14.
3. a;=:4^; 2/=ll^.
4. a; = 21; y = 20.
5. a; = 3; y^l.
40
HINTS AND ANSWERS.
X=:
a6~l
(l-a)a-- 6)
; y
a — h
ah H- 4iy — ?-:
7. X — —T — I — ^7- ; y =
ab — b + 2a' ^
8. x—±^\ y=±3.
10. a; = 9; 2^ — ^'
12. a; = 8; 2/ = 2.
... 6 + c' -a — ./ V
4{bc — ac!) ■ ^
14. ^=-3f; y = -6f
16. aj = 6f; y = S.
b'c - be'
{l-a)(l-b)
a' + 2a + ac— ab + b — c
ab — b + 2a
9. x — a; y = b.
a
11. x= r ; y = .
a — b'^ a + 6
15. x — y = d^.
17. a; =
a6' - a'6
'h '
y
ac' — a'c
obT^a/b'
1. a; = y = 00 , and the equations are inconsistent ; thus, put
a
— =: — z=k, and .-. a = A;a', 6 = kb', and substituting these
a
values of a and b m ax ^by=. c, we get a'x + b'y z=~ ,
n
which is inconsistent with the second given equation.
2. x = y = %, i. e. the equations are not independent ; thus,
put —= — =- = m. Then a = ma', b = mh', c = mc\ and
a' b' &
substituting in (1), we get ma'x + mb'y = mc', which is a
multiple of the second given equation.
EXERCISE
LXXII. (Page 222.)
1.
x — S;y = Q.
2. 0! = ^; y = l.
3.
a5 = i; i/ = l.
4. a? = — 2 ; y = f .
5.
^ = -1; y = h
6. x=-^- y = l.
7.
x = y = a + b.
6(a»-6c)
cia'^—bc)
a—c
9.
«' = -tV; y = Th-
10. .i; = y = -- ^ . _
ELEMENTS OP ALGEBRA.
41
EXERCISE LXXIII [a]. (Page 233.)
1. x = y =z — 4,,
3. X=\^^, y=-l^; ^=-H
5. x=:lS', y= — 122; z=-'.9.
7. a; = 4; 2/ = 5; ^ = 6.
9. x=l; y = 2; B — d.
11. x=:S; y = 2i; = - 3. 12. a; = 3.
IS. x = e; y=8; z = 10. U. x = 6; y= -2; z
15. a? = 3; y = 6; 2- = 8. 16. a." =-5; y = 9; ^
17. x=-4\, y=-il ^=-s^.
2. a.«=l|; y/ = 7i; ^:
4. a; = 3 ; 2/ = — 3 ; 5'
6. a; = 4; .y = 3; ^ :=z
5. X--= -1; y/=_2;
10. = - Ifi .
= -2|.
2- — 5.
= -3.
= -8.
■be)
1. a; = |^; y = ^; z = — :^, Divide through by xyz in each
equation.
2. iB = 5; y = li; 2- = ^-
3. ^ = 1]^; y = -|; ^ = 2|.
^ 1 1
a;
a- " 6'
a6 4- c<2 — 6c
2a
1
= -•
c
2/ and ^ symmetrically.
6. 2' = 3«6c -^ (c — «)(& + c), aj and 2^ symmetrically.
7. ic = a — 6; y =:h — c\ z = c — a,
S. x = 2a -^ (6 4- c —a) ; y and symmetrically.
9. x = \ -rr {a — h) {a — c)] y and 2' symmetrically.
10.
12.
13.
14.
15.
^ = li
11. 05 = ^; 2^ = ^; ^=1-.
rfl
a^
a? = «'' — 6" ; y z=ilf — c^-^ z = c
^ = ^ (a + & + y symmetry.
6. a? = ^ (a + cZ) ; y, z, u, and d by symmetry.
7. a; = 30 ; ^ = 20 ; = 42 ; ?* = 72 ; (y should by in second
equation.)
8. x = a + b + c, y, z, &q., by symmetry.
9. x=:^{a + b + G + d + e — 4/) ; y, z, «&c., by symmetry.
10. Divide each side of every equation by xyz ; x = l-hb — o;
y and z by symmetry.
EXERCISE LXXV [a]. (Page 229.)
1. 03 = 714285; ?/ = 142857. 2. a; = 40; y = 65.
8. Willie 4; Charlie 8. 4. a? = 1.234; y = 5.678.
5. a; = 147; 2/ = 63. 6. 76.
7. 13 : 17. 8. 73.
9. 480 gallons; 400 gallons; 560 gallons.
10. $10260; $7560. 11. |.
12. 10a; + y = 6 (a; + y) ; .-. 4a; = 5?/; .-. 10y+a;=9a;=5a;+52/, etc.
13. 98 or 89. 14. A, 200 lbs. ; B, 250 lbs. ; C, 350 lbs.
ll
1'
21
2J
ELEMENTS OF ALGEBRA.
43
15. 82 apples; gave away 2. 16. $5000; .$3000; $4000.
17. 40; 88; 104. 18. 486.
a(b — c)^ h{c — a) ^^ .. 2h 26
19
20. x =
y
31. I, h
b~a b — a 1 — a' " 1+a
22. A, 105; B, 52^; C, 210 minutes; A, B, and C in 30 minutes.
1. First, 220 gallons ; Second, 100 gallons.
2. 3674. 3. A, $40; B, $24; C, $16.
4. d(m ■{■ n) -T- 2mn ; d{n — m) -r- 2mn.
5. X + y : x — y : xy :: 5 : 1 : 18; x = 9', y = Q.
_l(m — l){q — mr) — (1 — mn) (mp — Iq)
~ mH {\ — n)-\- mnl (1 — m) — in (1 — mn)
7. 130. 8. 315 miles. 9. 9; 8^\ miles per hour.
10. a; = (p + 1) w ; y = (pq — l)n;z—(q-\-l)n.
11. x = l^; 2/ = 2f; ;? = - 12.
^^ cj(« + &) c(6_-a), 13. _^!:!!L_ hours.
2ab 2ab qr—ps
14. A, $2.60; B, $1.26f ; C, 61^ cts. 15. ^m*. 3^ miles.
16. 3000 ft. from first station; x = distance from first station;
y = A's rate per second ; ^r = B's rate ;
,, 4000 4000 >n ^ ^ on
then — = 40 ; = 30 ;
y z y z
substitute in this the value of , viz. ^^,
y z
equation and x = 3000.
17. Gold coin, $2 ; silver, $1.
18. 11|^ miles; 7 miles; and 5| miles.
20. x = time for A, y for B, f or C ;
from first
19. $5200; $2480.
^, 11 m 111 ml w + 1
then -+-= — , or -++-=— + -= ;
y z X X y z x X x
... ^ + 1 =, ^y + yz + z ^ similarly for n + 1 and^. + 1.
yz
44
HINTS AND ANSWERS.
21. x = rate of locomotive, y = rate of coach, z = distance;
then Qx + Sy = z=:mx+1 f) + 7^ (y + 1|)
= T^V (^ - H) + ^tV 'y - H) ; ^' = 38^, y = 7,z = 387.
^/iler should be befo7'e in first line of equation.
32. Let m, n be the required dividends;
m n
then given fraction =.:
+
33
Multiply out and equate coefficients, and we get
6w + dn = 37, 7m + 4n = 34; m = 2, n=z 3.
be — ad ab — be ^. 9 6 3
cf—de'' cf—de
35. 16, 30, 43.
34.
8aj-7' 5a? -4' 3a;- l'
EXERCISE LXXVI. (Page 339.)
6. -I, -10; ±(a-6); ± (1 - a) ; ± a + |.
7. ±(3«-26); ±— t-^; 3, f.
8. 36, -3a; ±133; ±375. 10. ±3; ±3^; ± 3^.
13. ±5; ±|a; ±(a: 13. ±^{mn)\ f|, |; ».
16. ± /^(ac) -r- y^(6f^) ; ± ^(flft); ± 1.
17. ± 1 ; 3a, 36. 18. ± y'im" + w«) ; ± ,
35. See Ex. 8; n, -, —n± ^{n'' — 1).
It/
48
HINTS AND ANSWERS.
EXERCISE LXXXI [a]. (Page 264.)
1. 11 or - 24. 2. 26 and 19.
4. ^(^5-l)a, i(3-^5)a.
6. 50 coffee, 60 raisins.
8. 100. 9. 12.
11. A, 11 miles; B, 10.
3. ^ or - 6f
5. 16^.
7. $240.
13. 25 cts.
16. 7.
18. $90 or $10.
21. $3.
23. 2.414 inches.
14. 4c?. a dozen.
10. ^11 vases.
12. 3, 4, and 5
15. 3 and 18.
17. A, 72 miles ; B, 54 miles.
20. A, $1800; B, $1600.
22. ^ { y'(2/i'^ -d') +d\,:^{ y/i^h'' - d') - d\.
19. '-.
5
3. 63.
5. ^6"ti^ /
/St
[6.] (Page 266.)
1. 3 hours and 5 hours. 2. 36 and 30.
4. _i(a + 6) ±y/^^ + ^V(« + *)=)•
6. 4200 ; read 780 in question. 7. 14 acres at $75.
8. 10 seconds. 9. 5f miles. 10. 5 miles an hour.
11. 15 miles.
12. If X be cost and s selling price, then x = s -{- -— ; on solving
it is seen that 4s cannot be greater than 100 ;
see Art. 175 (i).
13. $333^, $666|.
14. 72, 12, 8 ; Let x"^ = number remaining in smaller bag after
handful is taken ; then x^ is left in Imager bag, and x^ =
number in handful, and a;* is number in larger after
second lot is taken out ; then x^ + .-r'* = |- {x^ + x^)^ and
a; = 2, etc.
15. If X represents per cent, then 620 = 82a; + (3790 + 82;f) — ^•
X =3 22.
x = 5.
16. Let 2x = distance, then ■ H
X — b X —
6 _3^.
7 ' x~-^'
ELEMENTS OF ALGEBRA.
49
17. Let X = rate backwards, Ax = rate forwards, tlien •
d\.
i + ^ , i.e. j\x = J
+ , ^^ e. yVa; = -— ^ — - ^ 5:.. • i mile an hour.
X " 4a; + 3 it' — i
18. 90. 19. 4900 ; x^ being number of lines, equation is
\%\ {X - 10) p = 2 {a;" - ^^ {x - 10) f , or 601a;^ - 20a;. 3254
= — 100.49-43; a; = 70.
20. A, half -past 4 o'clock ; B, 5 o'clock.
EXERCISE LXXXII \a\ (Page 273.)
1.
3.
X
X
\,V = \, -If
5. X —
±6, y = a± h.
3, ■
4, -10^,2/ = 5, -13H-
± 20, 2/ = ± 16.
30, 10, y = 10, 30.
7. X
9. a;
11. a;:-3, -3|, 2/ = 4,
Q19
13. a; = l, -f, 2/ = 2, 2f
15. a; = 7, - 4|, y = 3, - 2|.
17. a; = ±7, ±5, 2/=±ll, ±9.
19. a; = 2, 5, y = 6, 3.
21. a; = 7, 1, 2/ = 3, 9.
23. a; = 3, y = l.
3. a; = ± 6, y = a^^h.
4. a; = 3, - 2^, 2/ = 3, - 8.
6. a; = ± 7, 2/ = ± 3.
8. a; = ±15, 2/= ±3.
10. a; = l, — 5, 2/= — li
12. a; = 2, - i, 2/ = 3, f .
14.
16.
18.
20.
22.
24.
'i.
a; = 10, 115, 2/ = 6, -69.
x = \, -If, 2^ = -4, HH-
a; = 2, 3, 2/ = 5, 4.
a; = 5, I, 2/ = 3, - 1^.
a;= ± ^13, 2/= ± 'v/13.
i« = 3, f , 2/ = - 'J',-
1
1. a;
3. a;
5. X
7. a;:
9. X
11. a;
12. a; =
13. X
15. a; =
16. X —
17. a; =
[&■]
= ± 5, 2/ = ± 1. 2. a; = ± 11, 2/ = ± 2.
= 0, ±2, 2/ = ± V-B^, ±1. 4. a; = 0, ± 3, 2/ = ± B, ± 9,
= ± 3, ± i[, 2/ = ± 5, ± -V. 6. a; = 0, ± 1, 2/ = V?' ± 1-
= ± 2i, 2/ = ±i. 8. a; = ± 3 ./f , 2/ = ± i Vt
= ±2^, ±1, 2/=±l, T3. 10. a; = ±7, ±3, 2/ =±3, ±6,
= ±1, ±H, 2/=±5, =Flf.
= ± « -^ V(^ + 6), 2/ = ± & ^ V(^ + ^)-
= ± 6, 2/ = ± 3. 14. a; = ± 3, ± 8, 2/ = ± 5.
± 2, ± v'f , 2/ = ± ^, T 3 ^|.
± 4, ± 3 -v/3, y = ± 5, ± ^3.
7, 4, 2/ = 4, 7. 18. a; = 7, - 5, y = 5, - 7.
50
19.
21.
23.
25.
27.
29.
1.
3.
5.
7.
8.
9.
10.
12.
14.
15.
16.
18.
19.
20.
21.
22.
23.
24.
25.
27.
29.
31.
HINTS AND ANSWERS.
/
x= ±5, ±S, y
x = 4, j/ = S.
a; = 5, 4, y = 4, 5.
a; = 4, y = 3.
X = 13, 9, y = 9, 13.
a? = 3, 1, y = 1, 3.
± 2, 7. 20. x= ± 5, ± 4, 2^ = ± 3.
22. X = 14, 19, y = 19, 14.
24. a; = 4, 2, y = 2, 4.
26. a; = 3, 2, y = 2, 3.
28. a; = 7, 4, y = 4, 7.
30. a; = 3, 2, y = 2, 3.
EXERCISE LXXXIII [a]. (Page 277.)
a5 = 2, 1, 2/ = -l, — 2. 2. a: = 4, — 3, y= — 3, — 10.
^= ±1, Ti, y = |, 0. 4. x=±l,y=±7.
x=±5,y=±2. 6. a;= ±1, y= ±^.
« = i(a±26), 2/ = |(a=F26).
a; = ± (a + &), y= ±(a — b).
x= ±
a=» + 6'
, 2a6
y = ± 1-
a —
a; = 6^, y = If
± 9, ± 5, 2/ = .-t 5, ± 9.
«' = i>y = f 11-
x = 0, a, y = a, 0. 13. a;
a? = ± 7, ±3, y=±3, ±7.
^=±4, ±3, y=±3, ±4.
;r = 2, -1, y = l, — 2. 17. ir = ll, y = 9.
a;=-2, -3, 3±i^ ^56, y=-S, -2,STi ^5Q.
^ = 5, -1, ^(±^41 + 5), y = l, -5, i(± v'41-5).
Treat x + y rs the unknown ; x = }{a± ^(a^ — 48) f ,
y = i{aT ^{a^ - 48) f , where a = ^ (- 3 ± ^853).
^ = i(9±V-47, ^ = ^(9^^-47).
a! = 5, -2, -^(l±y^41), y = 2, -5.
^ = 17, - 6, ± V(118) - 4, y = 3, - 8f 2 ± i ^^118.
a;=5, P, 7, 8, y = S, 7,6, 5.
a; = ±13v/^, y=±7^/^j. 36. a; = 4, 2, y = 2, 4.
ic = l, 10, y=10, 1. 28. a? = 3, 2, 2/ = 2, 3.
a; = 8, 4, y = 4, 8. 30. a; = 1, 1^, y = 2, -^i,.
(a'c - a&y + {ah' - a'h) {b'c - be') = 0.
3.
5.
6.
7.
8.
9.
10.
12.
14.
15.
16.
17,
18.
19.
20.
ELEMENTS OF ALGEBRA.
51
(Page 281.)
x= ±4, ±9, y=±Q, etc.
£» = !, 2/ = 2, 2- = 3.
EXERCISE LXXXIV.
1. a; = 8, 2, 2^ = 4, = 2, 8. 2.
3. a; = 7, y = Q, z = 5. 4.
5. ic = f y = *, ^ = 4.
6. a; = ± t V'S, ?/ = ± .^3, = ± 2 ^3.
7. a; = 4, — 7, ?/ = 3, — 8, ^ = G, 28, - 2^ in text.
8. a? =1,9, y=±4, = 2, -G.
9. a; = 2, - 14, v/ = 3, - 15, z = \, - 16.
10. a;=l, y=z—2, z = 4. 11. a? = 4, y = — 5, 2r = 7.
12. a; = 2,. 7, y = 3, = 7,2. 13. x = y = z= ±1 -^ ^2.
14. a; = 2a6c -j- (ac + be — ab), y, z symmetrical.
15. a; = ± a'^ -T- '\/(a'' + 6' -I- c"), ?/, z by symmetry.
16. a; = ± ^{ (a + 6 — c) (a + c — 6) -4- 2 (6 + c — a)\,
y, z by symmetry.
17. a; = a6c -r- («6 + «c — 6c), ?/, by symmetry.
18. X — ^ \{e + a — b) {a -\-b — c) -^ {b -^ G — a)\.
19. Add 71^ to each equatior and factor ;
x= — n ±{a + n){t + 'n)-^{b-{- n), y, z by symmetry.
20. a; = ± a /y/Kft + c — a) -^ [(a + 6 — c) (a — 6 + c)][,
y, by symmetry.
21. a; = (6c + ca + a6) -^ a, ; y, zhy symmetry.
22. a; = l, 2, 4, 2/ = 2, 4,1, = 4,1,2.
33. x=i ±{—m -if-n +p)-i- /y/2 (??i + ti + ^) ; y, zhy symmetry.
24. x= ± (— be + ca + ab) -^ y'(2^^^) I Vi ^^Y symmetry.
25. a; = 0, or ± 1 -T- (c — a) ; y, zhy symmetry.
26. x = -y/(6V -f- a) ; y, z by symmetry.
EXERCISE LXXXV [a]. . (Page 285.)
1. Ml + V^), H3 -f V^)- 2. i ± ^ ^2193, -^±1 V2193.
3. 36, 16, or - 36, - 16. 4. ± ^(pq), ± ^{p -r- q).
5. 20. 6. ± i (P + g) VC^ -^M), ± i (P - 9) V(^ -^i'?)-
7. 7, 21, 35. 8. 343, 64.
11. 34, 17, 51, or - 204, 612, - 306. 13. ^M.
9. 4,
10. 36.
5^
hf ,'^^H
13
pj'^H
^''^H
14
i-*<'i ' .i^^^H
|f'?H
16
18
fI
20
I'l
22
l"l
24
4.
5.
7.
9.
10.
11.
13.
14.
15.
16.
17.
HINTS AND ANSWERS.
'V^llO ± ^ y'6 ^ y'llO. Add and subtract the equations.
8 ft., 10 ft. 15. 88 yds., 55 yds.
63 ft., 45 ft. 17. 20 m., 30 m.
6^,7^. 19. 102 from length, 114 to width.
18, 9-6. 21. 100 at $75 each.
13, 10. 23. A 40 at $1.20, B 30 at $1.60.
3, 5, 10. 25. 3, 4, 5.
[6.]
Edges (x, y, 0) are 1, 2, 4 ;
x + 7j + z=7, a;'^ + 2/' + ^' = 21, x^ + y^+z' = 73.
Cube first equation by formula H (3), p. 85, and substitute
from third, second, and square of first.
864. 3. 2, 5, 8. See Ex. LXXXV, 16. Add first two
equations and subtract third, then symmetry.
76 ; the one digit remainder is of course 9.
$2145, 2f years. ' 6. 4, 7, 10.
290 yds. 8. 48, 10.
8, 9, 10 ; see Ex. 4, p. 284.
342 ; in last line of problem read 29.
12, 4, 3. 12. 3, 6, 9. See LXXXIV, 19.
^{h±^(c-a')\ :^{a±^(c-b')\ :: ^{aTV(^-^")\
x= ^1(1 + «) (1 + b) -r-{l + c)\, y and z by symmetry.
See 4, p. 284.
28 workmen, each 45 lbs. , or 36 workmen and each 77 lbs. ;
X — number of workmen, y lbs. carried each load, z number
loads in one hour ; then ^xyz is whole weight moved, and
7 (ic + 8) (y - 5) 2r = Myz = 9 (^ - 8) (y + 11)^.
— — ^- ^-^ = 5i_ £ii where p = product of ex-
a - ^{a' -Ap) 6 + ^{h' - 4p)
tremes (or means), and = (a^ -+- b^ — <■) -4- 3 (« + b).
Of the first, 135, 62; of the second, 182, 57, (yds.). 18. 126.
1.
3.
5.
6.
8.
10.
16.
2.
4.
8.
13.
ELEMENTS OF ALGEBRA.
53
EXERCISE LXXXVI [a]. (Page 296.)
1. 05' — 2a7 — 2 = 0. 2. f|.
3. 32i»'' - 1412aj - 23205 = 0. 4. x''-a^b.=zO.
5. p^-q; Pip""- dq) ; (p' - 2q) -^ g^ ; see Art. 178.
6. x" - (4a _ 66) a; + 9a' - 10a6 + 86' = 0. 7. 3 • 14159.
8. Positive for all values of x, expression = {x—2^y. 9. 789|f .
10. p-2q + 3r. 11. See Ex. 1, p. 294.
16. Assume of* = Ax + Bq, then, since a, /? are values of x,
«" — A;r + Bq, and /?" = A^ + Bg, whence A and B.
2. 6' — ac = 0.
4. i. a'a3' - (6' - 2ac) a; + c' = ;
j^. ^V — (p' — 2g') i» + 1 = ;
.?. x^ — (p"" — q) X + q (p"" — 2q) = 0;
4. os' + \p~^(p'-4q)\x-p^/(p'-^4q) = 0.
8. ec' — aa' = 0, 6'c -\-a'b = Q.
[c]
4. 579 and 135 are the roots of the first equation, 579 and — 135
those of the second.
12. 4a6' + a'c — aa'c' = ; let roots of first equation be a, /?, of
second a + in, ^ + m ; form equations from relations of
roots and coefficients and eliminate 7a.
13. (Right side of first equation should be 1.) Substitute for y
m second equation, and apply condition of equal roots to
resulting equation in ic.
^1
EXERCISE LXXXVI I [4 (Page 300.)
1. Min. 4. 2. Min. - V- 3- Max. f
4. Min. tV. 5. Min. -g^. 6. Min. - ||^f .
7. Min. f. 8. Min. 2. 9. Min. -J, Max. J.
10. Max. 36 area, i.e. line is bisected.
'\i%
54
HINTH ANJ> ANHWJiUb.
3. 81. B. Min. i«', i. e. lino in biwsctod.
4. ^a ^^2, the Hidos arc equal. 0. (a 4- ff)'^ -^ 4a/>.
7. il// numbers between ^ and 3. 8. | /y/1 -f |.
11. (6' - 4ac) -^ a' = (/*' - 4mr) + 7W». 13. ^ = 6, or g.
8.
EXERCISE LXXXVIII [a]. (Paoe 305.)
1. 1. 3. 346". 3. 5. 4. 1.
6Vr/^6; y^^ ^6 ^c ; ^«« ^?/ <^c">.
7. i. a'ft-"; a-'6-'c; a^6-» ; 7aW ; a-'-ft-"; a'ft';
« 111 711
5 6__
a-6-v' ^-fftHc**
3^-8 '
1.
[6.]
1. (a' -6")"; {x-\-y)'^', {x-yf.
8. iT^ — Ax^ ; at?> - af 6" + ^a%^ ;
a6^ -\- a^ — fi'h^ — a^b^.
- _c a _6 1 c feW 1
• a'6'' 6»-^^ ^V' 3S"^^"^/J
a^h* b^c' ' ay y+'' ryx
7. ^^+_^^_>^i7; .^.^..^1/^; 31 V^
2. fa^; 1.
X^ffyl
liLKMKf^TS or AI/n-'-HUA.
56
■ 'v^a'^^^ '^//^ ^li' ^a"
0-8; A; A; A; Vi; %-
13. 244140035; 2.
10. 1024a; « -j- 32.
12. (7a; -0^/)-"; (5a- 76)'.
14. 0.
[c] (Page 806.)
arAr. 2. re'-'--'; a^^S ;
1. ^J; aW, ?1^; .A. . .«^.; eW. ^^^-,
3. a'*6-"; ajfr-^ ; Fj . 4. a'i'-""; 6
TO"— nn
a
llmnp
; «*
.mi--mn
mn
_1 ^
5. a~T; a'^i"; ar V ; a """ -6 " . 6. a"6V; a''6"c.
EXERCISE LXXXIX [a]. (Page 309.)
m:»
2. 2""; -t; (a«.6"V'.
8. 3^ + ic* + a;^ + 2a;« + a;§ + a;''' + aj^^ + a; + 1
aj + y.
4. Q^ — \f^
5. ic* + ^x^y^ + y* — ^!/] ^^ — ^^y + y"^
x" + 2/''
6. 4a''-6'
7. 2a;'"' — 42 - 9a;" + 6ar'-"» + liar" ; 4a;'
im
9y'
2p
8. a'
+ a"" — 2 — ai — a"§ + 2ai
l_
9. a;' + 4a: V"^ — Ax^y — IGajyt + 16a;yff — Uy
10. 3 + 2a;~^ + 2a;^ + or" + a;" ; aW — h\
11. a;^ — 2/i ; a; + y + s^^y* + x^y^.
f^ — or^vz
— o'Jiiz 4- ?y*
12. x^ — xhjt + x^y^ — x^y^ + yi ; Sa"' + 7a-' + 6,
13. 56^ + 46^ + 36"^ + 26"i
56
HINTS AND ANSWERS.
14. x~^ + X »y a -I- x~^y~f 4- x sy-' + x »y s + yl -^
oT^ - 2a-''b^ + 4arh^ - 8a-'b + 16arh^ - 326^.
15. x^ — 4x + lOajt _ iQx^ + 19 — lQx~^ + 10a?~l — ix-' + x~^.
16. (at — it'^y^ + y^) (x's + x'syn + y^);
(x^ — 2x^y~^ + 2y~^) (x^ + 2x^y~^ + 2y~^).
17. (x^ — 8) (a;^ + 7) ; (Sx^ — ?/^) (Sx^ + 2.v^).
18. (x —l)(x — xi + l); {Sx^ - 2j/i) {2x^ — 3y^).
/a\i(»»+l)
19. —a~^{l + &-*), b~^ in denominator; 1^1 ;
a'a -f- 6¥ + c^ — a^feT _ ft^ci — c^^t
I 71 1 ' '
30. a6-' + 1 + «-'&. 21. at + 2a^2^^ + 3y^.
[6.] (Page 310.)
1. ; expression = {ab — a-^b~^) {ab + ar^b'^) — ab (ab-^ + ar^b)
+ a-'b-"^ (a-'b + ab-') = (ab-a-'b-') {ab + a-'b-')^ (ab-' + a-'b)
X (ab-a-'b-')={ab - a-'b-') (ab + a-'b-' - ab-' - a-'b)=0.
2. See last question.
3. For last term read a^. Arts, a^ — 2a* + a*.
4. {x^'y*^)^. 5. 2x (3 + x"") -v- (1+ xf.
6. air _ 2a6^ + 3a^6 — 2a^bi + 6^; in divisor read 2afe^.
7. X* + 2^" - 8a;' — Qx-\.
8. ax^ — aa^r = a^x {a^x"^ — 1), and second factor of this is con-
tained in the product of the other two quantities,
.-. L. C. M. = a^x (aV - 1).
9. ab-' + 1 — ^a-'b; in second term in text read b^ for 6~'.
10. tV; in first term read al
TJETi
11.
(x — ai) (x + a)
X — a
in second term of numerator read a.
1.
12. '^(xY) -f- ^(a'b').
ELEMENTS OF ALGEBRA. 57
13. Expression = a^ («« + 6f)s + fta (^^J _|. 6a)a =; etc.
( m — n n— m'\
Denominator = same factor x (if" + or),
15. a^ + a"'-" + a*'-""" + «.=^-="' + a""-*" + a'-'" + a-**.
16. abcde ; divisor = rf'" (say) = rZ'" x d^, apply Law IV.
17. x^ + (1 — m) a^x^ + a.
18. Expr(3Ssion = x^ — y" — (y-" — xr"^) = icy (icy-* — x-^y)
— xrHj-^ {xy-^ — x-^y), etc.
19. a'-^rc + 2 ; second remainder is — 5 (ax^ + 7a^u; + 10)
= — 5 {a^x + 3) (a^ic + o), and H. C. F. is a^ic + 2.
20. (aa;" - 1) {ax" + 1) (aV - 1) = a'a;'" - a'aj" - a'x* + 1.
21. 1 - i^a-^x\
22. asa; {a^x — 1) -;- i^d'^x — 1 ; read x^ in first term of numerator ;
denominator = {M^x + 1) {2a^x — 1) ;
(1 — a^)-^ |a(5a + a^)};
3 1 1
denominator = d^{a + a^) (5a + a^).
' ^-y + ^=-\ jpz:c)'(c^) "^ ((^:::6)^6^^) "^ (a-6) (c-a) f
= 0;
^ . — (a — 6) (6 — c) (c — a) .
Expression = —r^ — r-77 -7 r— = 1.
^ (a — 6) (6 — c) (a — c)
23
24. (a + 6)
2?nn
EXERCISE XC. (Page 313.)
1. ^x'; H^ixy'); ^^{xY)\ 6^(a;Y); ^/{a'b*)-
2. 27T; 512^; H; (#; si
3. i. -1(6")^; iia'h'f; (4-")^; UA)'}^) (a'^-'O^.
^. -|(6-«)-i; |(a-«6-^^)-i; (4")-i; KY)*f"*; (a-*ftV)-i.
58
HINTS AND ANSWERS.
6. v^(a"6); v^(a'»+»); ^| (a» - a^T (« + <») f ;
7. Sy^lO; 5^5; 3^5; 9^6; 18^2; f^l2; 7^.
8. 8^2; 6^48; 2^5; 2^3; 10^3; 2; 2^18;
12; ab^b.
10. 1^150; ^375; a« (aj + 5) ; (^- f y) ^(aj - y).
11. (a; - a) /^K^ + a) («'-«')}; a;i (a; + y) ; 2(a-b)^(ab).
12. 10 ^3, J V3, 1^ ^3, ^ V3, i V3-
13. 4^,3^; 8*, 6^; 10,000^, lOOoi ; 33^,32^; 80^,50^;
as, a^ ; a^, a^ ; a^, a^.
14. 2^3 = 24f; 3^2 = 181; 24t = 576^ ; 18^ = 5832^;
and I ^i = (244^^)1
EXERCISE XCI. (Page 317.)
1. 2^2; 8^5. 2. - 12^ ^3 ; llf <^9.
3. 60y'3; 80^3; 24. 4. 6-5^6; 6 y3 + 3 ^30.
5. —32.
6. iV2 + iV3 + 2V5; i V6 + f ^32 + I <^120.
7. f(V"^ + -v/3); i(7 + 8>y/5).
8. ^(17-3y'5; i(16-13y^2);
^ (7 y^l4 - 13) ; 20" - 1 + 2a y(a» - 1).
9. 288 *^72 ; see 4, p. 316.
lO: x + y + z + 2 ^{xy) - 2 ^{xz) - 2 y'Cy^) ;
13a;'' + 4 + 12a; ^{x" + 1).
11. ^/x — A^a\ ict - {xa)^ — a^', a — ^{ah) + b ;
^(25~6y'2)(3-^2).
ELEMENTS OF ALGEBRA.
59
12. Square and transpose radicals, square again, then
{ax + bi/ + (izf — 4 {ahxy + hcAjz + mtxz)
4- 8 ^{ahcxyz) \ ^{ax) -f- ^Khij) + ^Jiliz)\, etc.
13. Rationalize. 14. 3.1003. 15. 3.160.
16. (/y/S + 1) {4 - ^('10 + 3 ^5)} -I- 4 ; |« + ^{a" - a;')} -r- x.
17. 4aj ^(a;» - 1) ; 1 -v- (1 - a;'). 18. %x^ -f- a^
19. a ; rationalize and substitute.
20. /y/(a — a?) -J- (y'a + ^x) ; factor out y^Ca + x) in denomi-
nator of first fraction and rationalize, resulting numerator
cancels denominator of last fraction, etc. 20. lOf.
EXERCISE XCII \a\ (Page 320.)
1. V3 + V'^. 2. y/\{ ^- \/2. 3. VlO - \/6-
4. 2 4- V2. 5. Vll + V^' 6. VS 4- Vs.
7. V^ + 1- 8. 2 + V^' 9. 2^/3 + 3 Vs.
11. ^7-^/d.
10. X4(^7_V3.
'V/2
12. 3 — V^ ; change 13 to 11.
13. 3 VS — 3. 14. 3 Vll - a/41. 15. V^ - V^-
1. -^3 (1 + ^/2).
4. 5 + V^.
7. v^6 (1 + V2).
[ft.]
2. ^5 (1 + V^). 3. -^2 ( V3 - a/2)-
5. V^ - "^^ 6- Vl7 + \/l9.
8. 'V^2 (V^ - a/2). 9. 3^/5 + 6 a/S.
10. ^ (a/3 - a/I). 11- V^O - A/f 13. \/2 + ^^.
13. v^3 (1 + a/3). 14. a/(«* - «*') - a/(«&').
Vr-Hr + a/H^. 16. 3 A/m - 5 a/^T.
18. ^{x + 2/) + /v/(^ - y)'
15.
2
17. • I-
10. j!^^ ^7^ f '' ^"^^ ^y formula G (2), p. 85.
18. — t'sVt' ~ HM 5 divide by right member, and
/I + rr\ 1 3 /I - a;\^ , .31,^
• (^— --r H 1: 1 = li or v/ + —. = 1, etc.
61
EXERCISE XCIV [a]. (Page 327.)
4« -3ir' + 2.T — 1. 2. 8a;' - 12ii;' + 6a? — 1.
,. 8u. — 12a;V + 6a:.y' — y\ 4. .^•« — 2x''y + 2xif — ?y\
T). 2 — 3.r — a?' 4- 2a;\ Q. x* — 2x^y + 3d;y' — y\
[6.] (Page 328.)
1. x"^ — X ■\-\
3. 1 + 3.X' — a?'.
5. x^ — a;',y 4- ^y'^ + y^
7. 1 +^a;-^a;' + A^''-
2. 2 — 4a7 4- a; .
4. y-'-y + 2
6. a' - ^a-' + ^a-'.
1. If.
O 1 1
5. (a* - -;■#• ■
f:-
X
^^BIPPP
/