Page 14 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I reAl numBers

y + z
Proof: = m ⇒ y + z = m(ay + bz) ⇒ (1 − am)y =(bm − 1)z
ay + bz

(i). Similarly we can obtain (1 − am)z =(bm − 1)x

(ii). (1 − am)x =(bm − 1)y , (iii). (1 − am)x =(bm − 1)y z
(1 − am)x
y =(bm − 1)y
(i) × (ii) × (iii) ⇒ (1 − am) xyz =(bm − 1) xyz , which together with xyz =0 leads to
3
3
2
3
3
(1 − am) =(bm − 1) ⇒ 1 − am = bm − 1 ⇒ m = when a + b =0.
a + b
1.29 Given a + b =2, find the value of a +6ab + b .
3
3
2
2
3
2
2
2
2
3
Solution: a +6ab+b =(a+b)(a −ab+b )+6ab = 2(a −ab+b )+6ab =2a +4ab+2b =
3
2
3
2
2
2
2
2
a +6ab+ 2 b =(a+b)(a −ab+b )+6ab = 2(a −ab+b )+6ab =2a +4ab+2b =
2
2(a + b) =2 × 2 =8
2
2
2(a + b) =2 × 2 =8
2
2
2
1.30 Given x + xy =3 (i), xy + y = −2 (ii), find the value of 2x − xy − 3y .
2
Solution: (i) × 2 − (ii) × 3 ⇒ 2x − xy − 3y = 12.
2
2
ab 1 bc 1 ca 1
1.31 If the real numbers a, b, c satisfy = , = , = , find the value
a + b 3 b + c 4 c + a 5
abc
of .
ab + bc + ca
ab 1 a + b 1 1 1 1
Solution: = ⇒ =3 ⇒ + =3 (i). Similarly, we can obtain + =4
a + b 3 ab a b b c
1 1 1 1 1 abc 1 1
(ii), + =5 (iii). (i)+(ii)+(iii) ⇒ + + =6 ⇒ = = .
1
c a a b c ab + bc + ca 1 + + 1 6
a b c
1.32 Given a + b + c + d =4abcd , show a = b = c = d .
4
4
4
4
4 4 4 4 4 2 2 4 4 2 2 4 2 2
Proof: a + b + c + d − 4abcd =0 ⇒ (a − 2a b + b )+(c − 2c d + d ) + (2a b −
2 2
2 2
4
4
4
4
4
2 2
4
4
4
a + b + c
2 2+ d − 4abcd =0 ⇒ (a − 2a b + b )+(c − 2c d + d ) +
2 2
2
2
2 2
2
2
2
2
2(2a b −
4abcd + 2c d )= 0 ⇒ (a − b ) + (c − d ) + 2(ab − cd) =0 ⇒ a = b ,c = d , ab =
4
4
4
4
2 2
4
2 2
4
2 2
4
4
a + b + c + d − 4abcd =0 ⇒ (a − 2a b + b )+(c − 2c d + d ) + (2a b −
2
2
2
2
2 2
2
2 2
2 2
2
2
4abcd + 2c d )= 0 ⇒ (a − b ) + (c − d ) + 2(ab − cd) =0 ⇒ a = b ,c = d , ab =
cd ⇒ a = b = c = d
2
2
2 2
2
2 2
2
2
2
2 2
2
4abcd + 2c d )= 0 ⇒ (a − b ) + (c − d ) + 2(ab − cd) =0 ⇒ a = b ,c = d , ab =
cd ⇒ a = b = c = d
cd ⇒ a = b = c = d.
1.33 Consider two real numbers x, y, find the minimum value of 5x − 6xy +2y +2x − 2y +3
2
2
and the associated values of x, y .
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