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Xavier Rabasa 1
Xavier Rabasa 2
ÍNDEX
Introducció
3
Equacions Primer grau Segon grau Biquadrades Irracionals Exponencials Logarítmiques
4 17 27 32 37 46
Sistemes
Lineals de dues incògnites Lineals de tres o més incògnites Exponencials Logarítmics
53 69 87 91
Xavier Rabasa 3
INTRODUCCIÓ Aquest llibre està adreçat als estudiants de Secundària i Batxillerat. És un material de reforç per assolir els procediments de les equacions i sistemes d’equacions. El treball fet amb objectiu de lucre produeix una gran satisfacció però el treball realitzat per aplanar el camí dels que venen pel darrera és molt més gratificant i productiu.
Xavier Rabasa
Xavier Rabasa 4
EQUACIONS DE 1r GRAU NIVELL 1 NIVELL 2 NIVELL 3 NIVELL 1
Resoleu 5( x – 1 ) + 10 ( x + 2 ) = 45 RAONAMENT Desfer parèntesi 5x-5+10x+20=45 Separar la variable 5x+10x=5-20+45 Agrupar termes 15x=30 Aïllar la variable x=30/15=2 EXERCICIS:
Resoleu: 1.1. 2034x2 −=− Sol: x=7
1.2. 6x78x9 +=+ Sol: x=-1
1.3. 5x33x4 +=+ Sol: x=2
1.4 x939x7 +=+ Sol: x=3
1.5 11x28x −=− Sol: x=3
1.6 7x21x −=+ Sol: x=8
1.7 x846x6 +=+ Sol: x=1
1
Ex.1
Xavier Rabasa 5
1.8 x517x99 +=+ Sol: x=2
1.9 x33x2 =+ Sol: x=3
1.10 20x3x225 −=− Sol: x=1
RAONAMENT
20x3x225 +=− → -2 x – 3 x = 20 - 25 → - 5 x = - 5 → 5 x = 5 → x = 1 1.11 3x31x4 +=+ Sol: x=2
1.12 6x103x5 −=− Sol: x=3/5
1.13 31x16x81 +−=+ Sol: x=5/4
1.14 19x1511x5 −=− Sol: x=4/5
1.15 30x1548x12 −−=− Sol: x=2/3
1.16 7x317x2 +=+ Sol: x=10
1.17 2xx510 −=− Sol: x=2
1.18 x4x370 =− Sol: x=10
1.19 x5x348 =− Sol: x=6
1.20 10x330x4 −−=+− Sol: x=40
RAONAMENT
103304 −−=+− xx → - 4 x + 3 x = - 10 - 30 →
Xavier Rabasa 6
- x = - 40 → 40x = 1.21 27x415x10 +=− Sol: x=7
1.22 2x6)2x(3x −=−− Sol: x=1
1.23 8x61x3 −=+ Sol: x=3
1.24 )1x(27x3 +=− Sol: x=9
1.25 x115x347 +=− Sol: x=3
1.26 x123)x42(2 +=+ Sol: x=1/4
1.27 21x7x930 +−=− Sol: x=9/2
1.28 3)3x5(7x5 +−= Sol: x=3/5
1.29 1x210x3 +=− Sol: x=11
1.30 17x3)5x(2 −=− Sol: x=7
RAONAMENT
173)5(2 −=− xx → 2 x – 10 = 3 x - 17 → 2 x – 3 x = - 17 + 10 → - x = - 7 → x = 7 1.31 35x3x225 −=− Sol: x=12
1.32 3x)13x(52 −=−+ Sol: x=15
1.33 3x3x575 +=− Sol: x=9
1.34 )15x2(31x2 −=− Sol: x=11
Xavier Rabasa 7
1.35 20x2x85 +=+ Sol: x=5/2
1.36 )x4()2x(2 −−=− Sol: x =0
1.37 5y3y2 +=− Sol: y =8
1.38 14t)49t3(2 +−=− Sol: t =16
1.39 1z3z62 −=− Sol: z=1/3
1.40 )x410(x220 −−= Sol: x=5
RAONAMENT
)410(220 xx −−= → 20 = 2 x – 10 + 4 x → - 4 x – 2 x = - 10 - 20 → - 6 x = - 30 → 6 x = 30 → x = 5 1.41 )x121(31x60 +=− Sol: x=1/6
1.42 45)2s(10)1s(5 =++− Sol: s=2
1.43 67t)1t2(3t2 +=−+ Sol: t=10
1.44 60)4z2(3z12 =−+ Sol: z=4
1.45 30xh4)h25(h23 2 −+=−− Sol: h=3
1.46 2x)1x(x3 −=+− Sol: x=-1
1.47 1x)]1x3(x2[3 +=+− Sol: x=-1
1.48 10w3)5w(3w +=+− Sol: w=-5
Xavier Rabasa 8
1.49 )19t(315t −=− Sol: t=21
1.50 1x18)x2(3 −=− Sol: x=1/3
RAONAMENT
118)2(3 −=− xx → 6 – 3 x = 18 x - 1 → - 3 x – 18 x = - 1 - 6 → - 21 x = - 7 → 21 x = 7 → x = 7 / 21 = 1/3 1.51 1w4)4w(3 +=+ Sol: w=11
1.52 )1y(3)3y(510 +=−+ Sol: y=4
1.53 9t2)t43(2 −=− Sol: t=3/2
1.54 )4z(4z910 −=− Sol: z=2
1.55 )]2x(5x2[4)2x3(2 −−=+ Sol: x=2
1.56 3)t91(2t15 −+= Sol: t=1/3
1.57 x9)x11(2x4)x12(3 +−=−− Sol: x=1
1.58 )4y2(33y −=+ Sol: y=3
NIVELL 2
Resoleu:
127x-3 + 2x = x +
63+x -
41)+(x 3
RAONAMENT
Ex.2
Xavier Rabasa 9
Eliminar denominador
127x-3 + 2x = x +
63+x -
41)+(x 3 ·12·1212·12·12
)73(2412)3(2)1(9 xxxxx −+=++−+
Desfer parèntesi xxxxx 7324126299 −+=+−−+ Separar la variable 3697241229 ++−=+−+− xxxxx Agrupar termes 02 =x
Aïllar la variable 020==x
EXERCICIS
Resoleu: 2.1 4 + x = 2 +
23x Sol.
x=4
2.2 3
6-x - 2x = 8 - x Sol.
x=12
2.3 3 + 7x =
43x - x Sol.
x=28
2.4 3 + x =
35+x 2 ⎟⎠⎞
⎜⎝⎛
Sol. x=1
RAONAMENT
2
Xavier Rabasa 10
3 + x = 3
5+x 2 ⎟⎠⎞
⎜⎝⎛ → 2(x+5)=3(x+3) → 2x+10=3x+9
2x-3x=9-10 → -x=-1 → x=1 2.5
31 +
32x = 6 -
49x Sol.
x=4
2.6 11 - x = 43x -
65x Sol.
x=12
2.7 1 + 62x = 7 -
53x Sol.
x=30
2.8 6) - (x 95 = 10 - x Sol.
x=15
2.9 92x + 10 = x +
3x Sol.
x=9
RAONAMENT
92x + 10 = x +
3x
→ 3 x + 9 x = 90 + 2 x →
3 x + 9 x - 2 x = 90 → 10 x = 90 → x = 9 2.10
3x - 12 = 1 +
23x Sol.
x=6
2.11 3 - x = 2x +
5x Sol.
x=10
2.12 4
6 - 5x = 7 - 4x Sol. x=2
Xavier Rabasa 11
2.13 4 - 5x = 3
2+x Sol. x=1 2.
RAONAMENT
4 - 5x = 3
2+x → x + 2 = 3 ( 5x – 4 ) → x + 2 = 15 x - 12
→ x – 15 x = - 12 - 2 → - 14 x = - 14 → 14 x = 14 → x = 1 2.14
87 =
20-3x10-2x Sol.
x=12
2.15 21 = x + 63x +
4x Sol.
x=12
2.16 65 -
25x =
613 -
4x Sol.
x=-16/27
2.17 94 = 5x +
4x +
3x Sol.
x=120
RAONAMENT
94 = 5x +
4x +
3x
→ 20 x + 15 x + 12 x = 94 · 60 →
47 x = 47 · 120 → x = 120 2.18 16 +
5x = 10 +
3x Sol.
x=45
2.19 3 - 3+x
10 = 3+x7-x Sol.
x=2
Xavier Rabasa 12
2.20 3 - 2x =
5x + 9 - 3x Sol.
x=5
2.21 30x - 2 -
52x = 5 +
4x Sol.
x=60
RAONAMENT
30x - 2 -
52x = 5 +
4x
→ 15 x + 300 = 24 x – 120 – 2 x →
15 x – 24 x + 2 x = - 120 - 300 → -7 x = - 420 → 7 x = 420 → x = 60 2.22
618 + 2x- = 20)-(x 5 -
85x Sol.
x=24
2.23 2x + x =
51+x + x Sol.
x=2/3
2.24 2x + 4
3-x + 1- = 8
x-7 - 3x Sol. x=-1
2.25 9- = 85x -
42x +
103x - 8 Sol.
x=40
RAONAMENT
9- = 85x -
42x +
103x - 8 → 320 – 12 x + 20 x – 25 x = -
360 → -12 x + 20 x – 25 x = -360 -320 → -17 x = - 680
Xavier Rabasa 13
→ 17 x = 680 → x = 40 2.26
3x + 1 =
6x+3 +
21+x Sol.
x=0
2.27 0 = 10x -
23x + 2 -
53x Sol.
x=1
2.28 4
1-5x = 4
1-3x - 6
3-7x Sol. x=0
2.29 1 - 3
2-4x = 4
1-3x - 6
3 - 4x Sol. x=1
RAONAMENT
1 - 3
2-4x = 4
1-3x - 6
3 - 4x
2 (4 x – 3) – 3 ( 3 x -1) = 4 ( 4 x – 2 ) – 12 8 x - 6 – 9 x + 3 = 16 x – 8 – 12 8 x – 9 x – 16 x = - 8 - 12 - 3 + 6 - 17 x = -17 → 17 x = 17 → x = 1 2.30
127x-3 + 2x = x +
63+x -
41)+(x 3 Sol.
x=0
2.31 3 - 10x =
3x - 2 -
52x Sol.
x=30
2.32 4
2-3x = 2 - 93x -
41+2x Sol.
x=-15/7
Xavier Rabasa 14
2.33 2
2-5x = x41 -
25-x +
3x Sol.
x=-18/23
2.34 2
2)+(x-3-x = 2
2)-(x 3 - 3
3-x Sol. x=27/7
RAONAMENT
22)+(x-3-x =
22)-(x 3 -
33-x
2 ( x – 3 ) - 9 ( x – 2 ) = 3 ( x – 3 ) – 3 ( x + 2 ) 2 x – 6 – 9 x + 18 = 3 x – 9 – 3 x – 6 2 x – 9 x – 3 x + 3 x = - 9 - 6 + 6 – 18 - 7 x = - 27 → 7 x = 27 → x = 27/7 NIVELL 3
Resoleu:
1-x2+2x =
1+x3+x -
1-x2+x
2
RAONAMENT Eliminar denominadors
1-x2+2xx =
1+x3+xx -
1-x2+xx
2)1()1()1( 222 −−−
22)3)(1()2)(1( +=+−−++ xxxxx
Desfer parèntesi 223223 22 +=+−−++ xxxxx Separar la variable 232223 +−−=−− xxx
Ex.3
Xavier Rabasa 15
Agrupar termes - x = -3 Aïllar la variable x = 3 EXERCICIS:
Resoleu: 3.1
6 = 1+x
12_ 53x
Sol. x=5
3.2 1 - 1-x
x = 1+x
3
Sol. x=2
3.3 3 = 5+x4x+3 +
5+x10 Sol.
x=2
3.4 1-x2+2x =
1+x3+x -
1-x2+x
2 Sol.
x=3
3.5 0 = 2+x
5 - 10+x
15 Sol. x=2
RAONAMENT
0 = 2+x
5 - 10+x
15 → 0)10(5)2(15 =+−+ xx →
15 x + 30 – 5 x – 50 = 0 → 15 x – 5 x = 50 – 30 → 10 x = 20 → x = 2
3
Xavier Rabasa 16
3.6 1-x
5 = 4-3x+x
3 - 4+x
3 - 1-x
52
Sol. x=0
3.7 2+x
18 = 4-x6+12x -
2-x15
2 Sol.
x=4
3.8 a-x
1 = a+x
1 + a-x
122 Sol.
x=1/2
3.9 2
x + 1 = 2 - 2ax Sol.
x=5a/(1-a)
3.10
2 =
1x1-x - 2
1-x1+x + 1
+
Sol. x=3
RAONAMENT
2 =
x1-x - 21-x1+x + 1
1+
→ 2))1()1(2)(1())1()1)((1(
=−−+−++−+
xxxxxx →
2))3)(1()2)(1(
=+−
+xx
xx
→ ( x + 1 ) 2 x = 2 ( x – 1 ) ( x + 3 ) → 2 x2 + 2 x = 2 x2 + 4 x - 6 → 2 x – 4 x = - 6 → - 2 x = - 6 → 2 x = 6 → x = 3
Xavier Rabasa 17
EQUACIONS DE SEGON GRAU NIVELL 1 NIVELL 2 NIVELL 1
Resoleu: 09x6x2 =+− RAONAMENT Aplicar la fórmula
⎩⎨⎧
=±
=−±
=−±−
=33
206
236366
242
aacbbx
Resoleu: 0x6x4 2 =− RAONAMENT Treure factor comú x(4x-6)=0
Plantejar les equacions: ⎩⎨⎧
=−=
0640
xx
Trobar solucions ⎪⎩
⎪⎨⎧
==
=
23
46
0
x
x
Resoleu: 017x4 2 =− RAONAMENT Separar la variable 174 2 =x
Ex.1
Ex.2
Ex.1
Xavier Rabasa 18
Aïllar la variable 4
172 =x
Trobar solucions 217
417
±=±=x
EXERCICIS:
Resoleu: 1.1
⎪⎩
⎪⎨
⎧
=+=−
=+−
018x)c0x9x)b
018x9x)a
2
2
2
Sol⎪⎩
⎪⎨
⎧====
cap9x,0x6x,3x
1.2
⎪⎩
⎪⎨
⎧
=+=−
=+−
06x)c0x5x)b
06x5x)a
2
2
2
Sol⎪⎩
⎪⎨
⎧====
cap5x,0x3x,2x
Sol:
1.3
⎪⎩
⎪⎨
⎧
=+=+
=++
015t)c0t8t)b
015t8t)a
2
2
2
Sol⎪⎩
⎪⎨
⎧−==−=−=
cap8t,0t
3t,5t
1.4
⎪⎩
⎪⎨
⎧
=−=−
=−−
027x)c0x6x)b
027x6x)a
2
2
2
Sol⎪⎩
⎪⎨
⎧
±=
===−=
27x
6x,0x9x,3x
1.5
⎪⎩
⎪⎨
⎧
=+=−
=+−
09z)c0z6z)b
09z6z)a
2
2
2
Sol⎪⎩
⎪⎨
⎧====
cap6z,0z3z,3z
x=0
1
Xavier Rabasa 19
1.6
⎪⎩
⎪⎨
⎧
−==+−=+
9t)c0t6t)b
9t6t)a
2
2
2
Sol⎪⎩
⎪⎨
⎧−==−=−=
cap6t,0t
3t,3t
1.7
⎪⎩
⎪⎨
⎧
==+=+
3x)c0x4x)b3x4x)a
2
2
2
Sol⎪⎩
⎪⎨
⎧
±=
−==±−=
3x
4x,0x72x
1.8
⎪⎩
⎪⎨
⎧
=+=−
=+−
014z)c0z9z)b
014z9z)a
2
2
2
Sol⎪⎩
⎪⎨
⎧====
cap9z,0z7z,2z
x=0
1.9
⎪⎩
⎪⎨
⎧
=+=−
=+−
08x)c0x6x)b
08x6x)a
2
2
2
Sol⎪⎩
⎪⎨
⎧====
cap6x,0x2x,4x
1.10
⎪⎩
⎪⎨
⎧
=−=+
=−+
048s2)c0s10s2)b
048s10s2)a
2
2
2
Sol⎪⎩
⎪⎨
⎧
±=±=
−==−==
6224s
5s,0s8s,3s
1.11
⎪⎩
⎪⎨
⎧
==−=−
20t)c0tt)b20tt)a
2
2
2
Sol⎪⎩
⎪⎨
⎧
±=±=
===−=
5220t
1t,0t5t,4t
1.12
⎪⎩
⎪⎨
⎧
==
+=
6x)cx5x)b
6x5x)a
2
2
2
Sol⎪⎩
⎪⎨
⎧
±=
==−==
6x
5x,0x1x,6x
Xavier Rabasa 20
1.13
⎪⎩
⎪⎨
⎧
=+=−
=+−
03y2)c0y5y2)b
03y5y2)a
2
2
2
Sol⎪⎩
⎪⎨
⎧====
cap2/5y,0y2/3y,1y
1.14
⎪⎩
⎪⎨
⎧
=+=+
=++
025x)c0x10x)b
025x10x)a
2
2
2
Sol⎪⎩
⎪⎨
⎧−==−=−=
cap10x,0x
5x,5x
1.15
⎪⎩
⎪⎨
⎧
=+=
=+
09x)cx10x)b
x109x)a
2
2
2
Sol⎪⎩
⎪⎨
⎧====
cap10x,0x9x,1x
1.16
⎪⎩
⎪⎨
⎧
=+=−
=+−
0108t3)c0t39t3)b
0108t39t3)a
2
2
2
Sol⎪⎩
⎪⎨
⎧====
cap13t,0t9t,4t
1.17
⎪⎩
⎪⎨
⎧
=+=−
=+−
09x2)c0x9x2)b
09x9x2)a
2
2
2
Sol⎪⎩
⎪⎨
⎧====
cap2/9x,0x2/3x,3x
x=3
1.18
⎪⎩
⎪⎨
⎧
==+=+
8w3)c0w2w3)b8w2w3)a
2
2
2
Sol⎪⎩
⎪⎨
⎧
±=
−===−=
3/8w
3/2w,0w3/4w,2w
1.19
⎪⎩
⎪⎨
⎧
=+=+
=++
09t4)c0t12t4)b
09t12t4)a
2
2
2
Sol⎪⎩
⎪⎨
⎧−==
−=−=
cap3t,0t
2/3t,2/3t
Xavier Rabasa 21
1.20
⎪⎩
⎪⎨
⎧
=+=
=+
01x5)cx6x5)b
x61x5)a
2
2
2
Sol⎪⎩
⎪⎨
⎧====
cap5/6x,0x5/1x,1x
NIVELL 2
Resoleu: 3-xx-2 +
34+4x =
34+x + 2
RAONAMENT Treure denominador
3-xx-2 x +
34+4xx =
34+x x + 2x )3(3)3(3)3(3)3(3 −−−−
)2(3)44)(3()4)(3()3(6 xxxxxx −++−=+−+−
Operar i simplificar
xxxxxx 36128412186 22 −+−−=−++− 3x2 - 18x + 24 = 0 ; x2 – 6x + 8 = 0
trobar solucions ⎩⎨⎧
=±
=−±
=24
226
232366x
EXERCICIS
Resoleu:
2
Ex.2
Xavier Rabasa 22
2.1 1 = 3
2+3x - 3x - 1
2
Sol⎩⎨⎧
−=−=
1x2x
2.2
2)-(x - x = x + x - 2
)3-(x 2
2
Sol
⎪⎩
⎪⎨⎧
=
=
35x1x
2.3 x3 +
1-x3 = 2 - x3 + 3x +
1-x1 22
Sol
⎪⎩
⎪⎨⎧
=
=
0x35x
2.4 2x = 3
1-x - )3-(x 2
Sol
⎪⎩
⎪⎨⎧
=
=
7x34x
RAONAMENT
2x = 3
1-x - )3-(x 2 → 3( x – 3 )2 - ( x – 1 ) = 6 x →
3 x2 – 18 x + 27 – x + 1 = 6 x → 3 x2 – 25 x + 28 = 0
→ x=⎩⎨⎧
3/47
2.5 3x =
1-x1 -
33-x
Sol
⎪⎩
⎪⎨⎧
=
=
0x85x
2.6 5 + 5x = 2x1 +
x2 - x
Sol
⎪⎩
⎪⎨
⎧
=
−=
21x43x
2.7 3 - x3 - 2x =
x5 - 3x +
x3-x
Sol⎩⎨⎧
=−=1x5x
2.8 5)-(x - 2)-3(x = )1-(x + x8 - 3x
Sol⎩⎨⎧
=−=2x2x
Xavier Rabasa 23
RAONAMENT
2x = 3
1-x - )3-(x 2 → 3( x – 3 )2 - ( x – 1 ) = 6 x →
3 x2 – 18 x + 27 – x + 1 = 6 x → 3 x2 – 25 x + 28 = 0
→ x=⎩⎨⎧
3/47
2.9 )2-(x =
23)-(x x + 2)-(x 3)-(x 2
Sol⎩⎨⎧
==
4x1x
2.10 4 - )2-(x = 2
2)+(x 2)-(x - 3
2+x - x 2)-(x 2
Sol
⎪⎩
⎪⎨⎧
=
−=
4x32x
2.11 )2-(x = 1)-(x x)-(3 +
32-x - )3-(x 22
Sol
⎪⎩
⎪⎨⎧
=
−=
38x1x
2.12 2 = x
x+3 - 1+x1-x
Sol
⎪⎩
⎪⎨⎧
−=
−=
21x3x
RAONAMENT
2 = x
x+3 - 1+x1-x → x ( x - 1) - (x + 1)(3 + x) = 2 x(x+1)
→ x2 – x - x2 - 4 x – 3 = 2 x2 + 2 x → 2x2+7x+3=0
→ x = ⎩⎨⎧−−
32/1
2.13 2 = 1-xx+3 -
1+x1-x
Sol⎩⎨⎧
=−=0x3x
Xavier Rabasa 24
2.14 4 = 2-x
1 + x
Sol⎩⎨⎧
==
3x3x
2.15 32x -
92 = x - x2
Sol
⎪⎩
⎪⎨
⎧
=
−=
32x31x
2.16 35x = 2 +
3x2
Sol⎩⎨⎧
==
3x2x
RAONAMENT
35x = 2 +
3x2
→ x2 + 6 = 5 x → x2 – 5 x + 6 = 0
→ x = ⎩⎨⎧
23
2.17 3 =
x2 + x
Sol⎩⎨⎧
==
2x1x
2.18 x
8-4x = 2 - x
Sol⎩⎨⎧
==
2x4x
2.19 x
9+2x = x3 +
2x
Sol⎩⎨⎧
=−=6x2x
2.20 5 - 1-x
6x = 2 - 2x
Sol
⎪⎩
⎪⎨⎧
=
−=
3x21x
RAONAMENT
5 - 1-x
6x = 2 - 2x → (x - 1)(2x - 2) = 6x – 5 (x - 1) →
Xavier Rabasa 25
2x2 - 4x + 2 = x + 5 → 2x2 - 5x -3 = 0
→ x=⎩⎨⎧− 21
3
2.21
0 = 2x+x - 1)+(x x ⎟⎠⎞
⎜⎝⎛
Sol⎪⎩
⎪⎨⎧
=
=
21x0x
2.22 43x+1 =
x3 - 1 + 3x
Sol
⎪⎩
⎪⎨⎧
−=
=
34x
1x
2.23 3-xx-2 +
34+4x =
34+x + 2
Sol⎩⎨⎧
==
4x2x
2.24 3x6 =
x1 + x
Sol⎩⎨⎧
−==
1x1x
RAONAMENT
3x6 =
x1 + x → x2 + 1 = 2 → x2 = 1
→ x=⎩⎨⎧−+
11
2.25
x3-2x = 2 - x
Sol⎩⎨⎧
==
1x3x
2.26 3x
10+3x = x2 +
3x
Sol⎩⎨⎧
=−=4x1x
2.27 1-x1+2x = 3 + x
Sol⎩⎨⎧
=−=2x2x
Xavier Rabasa 26
2.28 x1 =
2-x1+x + 2
1 + x
3
Sol⎪⎩
⎪⎨
⎧
=
=
32x21x
2.29
x1- =
1+x1-x - 1
1 - x
43-x -
23-x
Sol⎩⎨⎧
−==
1x2x
RAONAMENT
x1- =
1+x1-x - 1
1 - x4
3-x - 2
3-x
→ x1- = x - x
43-x -
23-x
21+ →
xx
x1
21
43
−=−
−
→ xx
x 122
3 −=
−− → x2 – 3 x = -2 x + 2
→ x2 – x – 2 = 0 → x=⎩⎨⎧−12
Xavier Rabasa 27
EQUACIONS REDUÏBLES A SEGON GRAU EXEMPLES
Resoleu: 4x4-17x2+4=0 RAONAMENT Efectuar canvi de variable: 2xt = 04174 2 =+− tt
Resoldre l’equació ⎩⎨⎧
=±
=−±
=24/1
81517
86428917t
Desfer el canvi de variable ⎪⎩
⎪⎨⎧
±==
±==
222
14
12
2
xxxx
Resoleu: x6-9x3+8=0 RAONAMENT Efectuar canvi de variable 3xt = 0892 =+− tt
Resoldre l’equació ⎩⎨⎧
=±
=−±
=18
279
232819t
Desfer el canvi de variable ⎩⎨⎧
=+===+==
111288
33
33
xxxx
EXERCICIS
1
Ex.2
Ex.1
Xavier Rabasa 28
Resoleu: 1.1 04x5x 24 =+−
Sol⎩⎨⎧
±==±==
2x4x1x1x
2
2
1.2 03t2t 24 =−+ Sol
⎩⎨⎧
−=±==
cap3t1t1t
2
2
1.3 08x9x 36 =+− Sol
⎩⎨⎧
======
28x8x11x1x
33
33
1.4 027t26t 36 =−− Sol
⎩⎨⎧
===−=−=−=327t27t11t1t
33
33
1.5 6x4+2x2-8=0 Sol
⎪⎩
⎪⎨⎧
−=
±==
cap34x
1x1x2
2
RAONAMENT 6x4+2x2-8=0 → 3 (x2) 2 + 1 (x2) – 4 = 0
x2 = ⎩⎨⎧−
=+±−
3/41
64811 →
⎩⎨⎧
=±=±=
capx11x
1.6 0x4x 24 =−
Sol⎩⎨⎧
±==±==
0x4x0x0x
2
2
1.7 04x17x4 24 =+− Sol
⎩⎨⎧
±==±==
2/1x4/1x2x4x
2
2
1.8 04y3y9 24 =+− Sol
⎩⎨⎧
±==±==
3/1y9/1y2y4y
2
2
1.9 027t6t 24 =−− Sol
⎩⎨⎧
−=±==
cap3t3t9t
2
2
1.10 08x7x 36 =−+ Sol
⎩⎨⎧
−=−===
2811
3
3
xx
RAONAMENT
Xavier Rabasa 29
08x7x 36 =−+ → (x3) 2 + 7 (x3) – 8 = 0
x3 = ⎩⎨⎧−
=+±−
81
232497 →
⎩⎨⎧
−=−===
2811
3
3
xx
1.11 08x2x 24 =−−
Sol⎩⎨⎧
−=±==
cap2x2x4x
2
2
1.12 027z28z 36 =++ Sol
⎩⎨⎧
−=−=−=−=−=−=
327z27z11z1z
33
33
1.13 012x7x 24 =+− Sol
⎩⎨⎧
±==±==
2x4x3x3x
2
2
1.14 018t9t 24 =+− Sol
⎩⎨⎧
==±==
6t6t3t3t
2
2
m
1.15 06x5x 24 =+− Sol
⎩⎨⎧
±==±==
3x3x2x2x
2
2
RAONAMENT
06x5x 24 =+− → (x2) 2 – 5 (x2) + 6 =0
x2 = ⎩⎨⎧
=−±
23
224255 → x=
⎩⎨⎧
±±
23
1.16 015x8x 24 =++
Sol⎩⎨⎧
−=−=
cap5xcap3x
2
2
1.17 027z6z 24 =−− Sol
⎩⎨⎧
−=±==
cap3z3z9z
2
2
1.18 09t6t 24 =+− Sol
⎩⎨⎧
±==±==
3t3t3t3t
2
2
Xavier Rabasa 30
1.19 9h6h 24 −=+ Sol
⎩⎨⎧
±=−=±=−=
3h3h3h3h
2
2
1.20 x4-9x2+14=0 Sol
⎩⎨⎧
±==±==
7x7x2x2x
2
2
RAONAMENT x4-9x2+14=0 → (x2) 2 – 9 (x2) + 14 = 0
x2 = ⎩⎨⎧
=−±
27
256819 → x=
⎩⎨⎧
±±
27
1.21 08x6x 24 =+−
Sol⎩⎨⎧
±==±==
2x2x2x4x
2
2
1.22 048x10x2 24 =−+ Sol
⎩⎨⎧
−=±==
cap8x3x3x
2
2
1.23 20tt 24 =− Sol
⎩⎨⎧
−=±==
cap4t5t5t
2
2
1.24 6z5z 24 += Sol
⎩⎨⎧
−=±==
cap1z6z6z
2
2
1.25 03x5x2 24 =+− Sol: x=± 1; x=±
23
RAONAMENT 03x5x2 24 =+− → 2 (x2) 2 – 5 (x2) + 3 = 0
x2 = ⎩⎨⎧
=−±
12/3
424255 → x =
⎩⎨⎧
±±
12/3
1.26 025x10x 36 =++
Sol⎩⎨⎧
−=−=−=−=
33
33
5x5x5x5x
Xavier Rabasa 31
1.27 510 t109t =+ Sol
⎩⎨⎧
=====
11t1t9t9t
55
55
1.28 0108x39x3 36 =+− Sol
⎩⎨⎧
====
33
33
9x9x4x4x
1.29 09x9x2 48 =+− Sol
⎩⎨⎧
==±==
44
44
2/3x2/3x3x3x :
1.30 8x2x3 714 =+ Sol: x= 7 2− ; x= 7
34
RAONAMENT 8x2x3 714 =+ → 3 (x7) 2 + 2 (x7) – 8 = 0
x7 = ⎩⎨⎧−
=+±−
23/4
69642 → x =
⎩⎨⎧
−=− 77
7
223/4
Xavier Rabasa 32
EQUACIONS AMB RADICALS NIVELL 1 NIVELL 2 NIVELL 1
Resoleu: 11−x = 1 + x RAONAMENT Separar l’arrel i agrupar 12−x = x Elevar al quadrat ( ) ( )22
12−= xx 144242 +−= xxx Resoldre l’equació resultant
0144252 =+− xx ⎩⎨⎧
=±
=±
=9
162
7252
4925x
Triar les solucions vàlides ⎩⎨⎧
==
noxsix
916
Resoleu: 11−x = 1 + x RAONAMENT Separar l’arrel i agrupar 12−x = x Elevar al quadrat ( ) ( )22
12−= xx 144242 +−= xxx Resoldre l’equació resultant
0144252 =+− xx ⎩⎨⎧
=±
=±
=9
162
7252
4925x
Ex.1.2
Ex.1.1
Xavier Rabasa 33
Triar les solucions vàlides ⎩⎨⎧
==
noxsix
916
EXERCICIS
Resoleu: 1.1 30 = x + x Sol. x=25
1.2 11x = 1 + x −
Sol. x=16
1.3 7 = x - 3x - 7
Sol. x=-3
1.4 x 2 = 3 + x 5
Sol. x=9
1.5 x 2 = 5 - 1 + 6x 3
Sol. x=8 , x=1/2
RAONAMENT x 2 = 5 - 1 + 6x 3 → 5+x 2 = 1 + 6x 3
→ 9(6x+1)=4x2+20x+25 → 4x2-34x+16=0
→ 2x2-17x+8=0 → x=⎩⎨⎧
sisi
2/18
1.6
3x = 1 + x + 1 Sol. x=15
1.7 5 = 7 + 3x + x2
Sol. x=3 , x=-6
1.8 1 + x = x - 3
Sol. x=1
1.9 3 + 3x = 6 + 5 + 2x Sol. x=2
1
Xavier Rabasa 34
1.10 0 = 4 - 2 - 3x
Sol. x=6
1.11 1 - x = 1 + 2x
Sol. x=4
RAONAMENT
1 - x = 1 + 2x → 2x+1=x2-2x+1 → x2-4x=0
→ x=⎩⎨⎧
sino
40
NIVELL 2
Resoleu: 6 = 4 + x + 1 - 2x RAONAMENT Separar una arrel i agrupar 4612 +−=− xx Primera elevació al quadrat ( ) ( )22
4612 +−=− xx ; )4(4123612 +++−=− xxx Separar l’arrel i agrupar
43612412 ++++−=+ xxx ; 41412 +−=+ xx Segona elevació al quadrat 168182)4(144 2 +−=+ xxx ; 168182576144 2 +−=+ xxx ; 011052262 =+− xx
Resoldre equació ⎩⎨⎧
=±
=2215
2216226x
Ex.2.1
Xavier Rabasa 35
Triar solucions vàlides ⎩⎨⎧
=2215
x
EXERCICIS
Resoleu: 2.1 1 - x - 3 = 4 + x
Sol. x=13/9
2.2 1 = 1 + 3x - 5 + 4x
Sol. x=5
2.3 6 = 4 + x + 1 - 2x
Sol. x=5
2.4 x = x 2 - x3
Sol. x=0
2.5 4 + x 5 = 4 + x 2
Sol. x=12
RAONAMENT 4 + x 5 = 4 + x 2 → 4( x + 4 ) = 5x + 4
→ 4x + 16 = 5x + 4 → x = 12 si 2.6 3 + 3x + 1 = 10 + 3x Sol. x=2
2.7
4 + 3x6 =
x3 Sol. x=4
2.8 2x =
x2 + 2
3
Sol. x=4
2.9 3 + 2x 6 = x - 15 9
Sol. x=-1
RAONAMENT
3 + 2x 6 = x - 15 9 → )32(36159 +=− xx →
2
Xavier Rabasa 36
)32(415 +=− xx → 15 – x = 16 (4x2 + 12x + 9) →
64 x2 + 193x + 129 = 0 → x=⎩⎨⎧
− sino
164/129
Xavier Rabasa 37
EQUACIONS EXPONENCIALS
Resoleu les equacions: 1.1 x1x 42 =+ Sol. x=1
1.2
1612x = Sol. x=-4
1.3 2x1x 93 −+ = Sol. x=5
RAONAMENT
2x91x3 −=+ ⇔ 3x+1=32x-4 ⇔ x+1=2x-4 ⇔ x=5 1.4 5 = 25x Sol. x=1/4
1.5
51 = 25x Sol. x=-1/2
1.6 9 = 3 2-x2 Sol. x=4
RAONAMENT
9 = 3 2-x2 ⇔ 3x-2 = 32 ⇔ x – 2 = 2 ⇔ x = 4 1.7 81 = 3 3-2x Sol. x=7/2
1.8
41 = 2 3-x2 Sol. x=1
1.9 31-x 3 = 3 Sol. x=4/3
RAONAMENT
31-x 3 = 3 ⇔ 3x-1 = 31/3 ⇔ x - 1 = 1/3 ⇔ x = 4/3
1
Xavier Rabasa 38
1.10 16 = 2 x1+x Sol. x=1/3
1.11 81 = 3 x1-2x Sol. x= -1/2
RAONAMENT
81 = 3 x1-2x ⇔ 32x-1=34x ⇔ 2x-1=4x ⇔ 2x=-1 ⇔ x=-1/2 1.12 x = 271/3 Sol. x=3
1.13 5 = x1/2 Sol. x=25
1.14 2 = 32x Sol. x=1/5
RAONAMENT
232x = ⇔ 25x=2 ⇔ 5x=1 ⇔ x=1/5 1.15 27 = x3/2 Sol. x=9
1.16 32 = 4x Sol. x=5/2
1.17 8 = x3/2 Sol. x=4
RAONAMENT
8 = x3/2 ⇔ 33 2)( =x ⇔ 2=x ⇔ x=4 1.18 27 = 32x Sol. 3/2
1.19 0,001 = 10x Sol. x=-3
1.20
100 = 101
x
⎟⎠⎞
⎜⎝⎛
Sol. x=-2
RAONAMENT ( )2x 1010 =− ⇔ -x = 2 ⇔ x = -2
Xavier Rabasa 39
1.21 9 = 3 1+xx Sol. x=-2
1.22 27 = 92x Sol. x=3/4
1.23 8 = 2 22x Sol. x=3
RAONAMENT 8 = 2 22x ⇔ 22x = 26 ⇔ 2x=6 ⇔ x=3
1.24 100 = 103x Sol. x=2/3
1.25 0,01 = 10 1-2x Sol. x=-1/2
RAONAMENT
010 = 10 1-2x ' ⇔ 102x-1=10-2 ⇔ 2x-1=-2 ⇔ 2x=-1 ⇔ x=-1/2 1.26 4 = 9 + 3 1xx −
Sol. x=1
1.27 2187 = 3 3+2x
Sol. x=2
1.28 91 = 3 2-x
1+x
Sol. x=1
RAONAMENT
91 = 3 2-x
1+x
⇔ 221
33 −−+
=xx
⇔ 221
−=−+
xx ⇔ x +1 = -2x +
4 ⇔ 3x=3 ⇔ x=1 1.29 3 = 3 3+3x-x2
Sol. x=1, x=2
1.30 10 = 10 1+x1-x2
Sol. x=2
1.31 0 = 3 - 3 1+x1-2x
Sol. x=2
RAONAMENT 0 = 3 - 3 1+x1-2x ⇔ ( ) 03·93 2 =− xx ⇔ 3x ( 3x – 9 ) = 0
Xavier Rabasa 40
⇔ ⎩⎨⎧
===
29303
ximpossible
x
x
1.32 500 = 5 . 3 + 5 3-6x1+2x
Sol. x=1
1.33 12- = 2 - 4 1+x2-x
Sol. x=3
1.34 0 = 77 - 3 . 4 - 3 x2)+(x 2
Sol. x=0
RAONAMENT 0 = 77 - 3 . 4 - 3 x2)+(x 2 ⇔ 81·(3x)2 – 4 · 3x – 77 = 0 ⇔
3x=⎪⎩
⎪⎨⎧−=
±=
+±
1621541
1621584
16224948164 ⇔
⎩⎨⎧
<==
xcapx
x
x
.03013
1.35 0333·23 x2x =−++
Sol. x=1
1.36 222·32 1x1x −=+− −−
Sol. x=0
1.37 12·322 2x1x =+− −+
Sol. x=-2
RAONAMENT 13·222 2x1x =+− −+ ⇔ 2x ( 2 – 1 ) = 1 - 3/4 ⇔ 2x=1/4
⇔ 2x=2-2 ⇔ x=-2 1.38 3xx2 2222 =+− −−
Sol. x=-1
1.39 32x23x 222·2 =+−
Sol. x=1
1.40 31255·5 3x21x =−−
Sol. x=3
RAONAMENT
31255·5 3x21x =−− ⇔ 5
4
3
555
=x
⇔ 53x = 59 ⇔
3x=9 ⇔ x=3
Xavier Rabasa 41
1.41 e = e 1)-(x 22-x
Sol. x=0
1.42 2 = 4 3-2x1+x
Sol. cap.
1.43 8 = 2 3-x1-x
Sol. x=4
RAONAMENT 8 = 2 3-x1-x ⇔ 2x-1 = 23x-9 ⇔ x – 1 = 3x - 9 ⇔ 2x = 8
⇔ x = 4 1.44 702- = 9 - 3 2+x1+2x
Sol. x=1
1.45 625 = 5 2-3x
Sol. x=2
1.46 1 = 5 6-x-x2
Sol. x=-2, x=3
RAONAMENT
1 = 5 6-x-x2 ⇔ x2 – x – 6 = 0 ⇔ ⎩⎨⎧
=−=32
xx
1.47 54- = 3 - 3 2x1-2x
Sol. x=2
1.48 32 = 2 - 4 2+xx
Sol. x=3
1.49 25 = 5 3-x2-x
Sol. x=4
RAONAMENT 25 = 5 3-x2-x ⇔ )3x(22x 55 −− = ⇔ x - 2 = 2x - 6 ⇔ x = 4
1.50 27 = 3 2+x5+2x Sol. x=-1
1.51 8 = 2 3+x5+-x Sol. x=-1
1.52
21 = 2 + 4 1-xx
Sol. x=-1
Xavier Rabasa 42
RAONAMENT
21 = 2 + 4 1-xx ⇔ 2·4x + 2x = 1 ⇔ 2·(2x)2 + 2x – 1 = 0
⇔
⎩⎨⎧
−==−=
12/1212
xno
x
x
1.53 120 = 3 + 3 + 3 + 3 1-xx2-x1+x
Sol. x=3
1.54 511 = 2 + ... + 8 + 4 + 2 + 1 x
Sol. x=8
1.55 3280 = 3 + ... + 27 + 9 + 3 + 1 x
Sol. x=7
RAONAMENT
3280 = 3 + ... + 27 + 9 + 3 + 1 x ⇔ 32801313 1
=−−+x
⇔ 3x+1 = 2(3280)+1 ⇔ 3x+1 = 38 ⇔ x+1 = 8 ⇔ x = 7 1.56 1365 = 4 + ... + 16 + 4 + 1 x
Sol. x=5
1.57 19531 = 5 +.. 25 + 5 + 1 x+
Sol. x=6
1.58 55987 = 6 + ... + 36 + 6 + 1 x
Sol. x=6
RAONAMENT 55987 = 6 + ... + 216 + 36 + 6 + 1 x ⇔
559871616 1
=−−+x
⇔ 6x+1 = 5(55987)+1 ⇔ 6x+1 = 67 ⇔ x+1=7 ⇔ x=6 1.59 19608 = 7 + ... + 49 + 7 + 1 x
Sol. x=5
1.60 273·3 x =
Sol. x=2
1.61 29 = 2 + 2 + 2 + 2 3-x1+x1-xx Sol. x=3
Xavier Rabasa 43
RAONAMENT
29 = 2 + 2 + 2 + 2 3-x1+x1-xx ⇔ 2x(1+1/2+2+1/8) = 29 ⇔ 2x(29/8) = 29 ⇔ 2x = 8 ⇔ 2x = 23 ⇔ x=3 1.62 4053·5 x =
Sol. x=4
1.63 442x
= Sol. x=4
1.64 414 1x2 =+ Sol. x=-1
RAONAMENT
414 12 =+x ⇔ 42x+1=4-1 ⇔ 2x+1=-1 ⇔ x=-1
1.65 81 = 3 3-x
Sol. x=7
1.66 91 = 3x Sol. x=-4
1.67 3 =
31
x
⎟⎠⎞
⎜⎝⎛
Sol. x=-1/2
RAONAMENT
3 = 31
x
⎟⎠⎞
⎜⎝⎛ ⇔ 3-x = 31/2 ⇔ - x = 1/2 ⇔ x = -1/2
1.68 042·32 xx2 =−−
Sol. x=2
1.69 11333 2x1xx =−+ −−
Sol. x=2
1.70 8
6522 xx =+ − Sol. x= 3 ,x= -3
RAONAMENT
86522 =+ −xx ⇔
865
212 =+
x
x ⇔ xx 2·658)2(8 2 =+
Xavier Rabasa 44
⇔ 082·65)2(8 2 =+− xx
⇔16
636516
2564225652 ±=
−±=x ⇔
⎪⎩
⎪⎨⎧
−==
==
3812
382xx
x
x
1.71 7293 2x =+ Sol. 4x =
1.72 162 2x3 =−
Sol. 2x =
1.73 75055 1xx =+ +
Sol. 3x =
RAONAMENT 11000 )x2( =+ ⇔ 2+x=0 ⇔ x = - 2
1.74 1x1x2 82 −+ = Sol. 4x =
1.75 3 = 3 1-x1-x 2
Sol. x=0 , x=1
1.76 24 =
22
5
x
x
1+3x
2 Sol. x=3 ,x=-2
RAONAMENT
24 =
22
5
x
x
1+3x
2 ⇔ 5x2x1x3 22 2 −−+ = ⇔ 3x+1-x2=2x-5 ⇔
x2 – x – 6 = 0 ⇔ ⎩⎨⎧
−==
23
xx
1.77 x1x 42 =+ Sol. x=1
1.78 93 2x =+ Sol. x=0
1.79 1x1x 24 +− =
Sol. x=3
RAONAMENT 2x2x 525 −−+ = ⇔ 52x+4 = 5-x-2 ⇔ 2x + 4 = - x -2 ⇔
3x = - 6 ⇔ x = -2
Xavier Rabasa 45
1.80 15333 1xx1x −=−+ +− Sol. x=2
1.81 43·233 1xx1x =−− −+
Sol. x=1
1.82 0422 x2x3 =−−
Sol. x=1
RAONAMENT 033·123 2x1x2 =+−+ ⇔ 3·(3x)2 - 12·3x + 9 = 0 ⇔
(3x)2 – 4 · 3x + 3 = 0 ⇔ ⎩⎨⎧
=−±
=13
2121643x ⇔
⎩⎨⎧
==
01
xx
1.83 4- = 2 - 2 + 2 1+xx1-x Sol. x=3
1.84 1 =
51 - 5 + 5 1+xx Sol. x=-1
1.85 3 2x 2 = 32
Sol. x=2/15
1.86 271 = 3 . 3 x1+x Sol. x=-2
1.87
3 . 2 = 31 + 3 1-2x1-x Sol x=0
RAONAMENT 33-2x a = a ⇔ a2x-3 = a1/3 ⇔ 2 x - 3 = 1/3 ⇔
2x=10/3 ⇔ x = 5/3
Xavier Rabasa 46
EQUACIONS LOGARÍTMIQUES
Resoleu les següents equacions: 1.1 5 = 32 log x
Sol. x=2
1.2 2 = 36 log x
Sol. x=6
1.3 2 =81 log x Sol. x=9
1.4 2 = 49 log x
Sol. x=7
1.5 21 = 5 logx Sol. x=25
1.6 4- = 161 logx Sol. x=2
RAONAMENT
4- = 161 log x → x -4 = 1 / 16 → x -4 = 2 -4 → x=2
1.7
21- = 5 log x Sol. x=1/25
1.8 25 = 32 log x Sol. x=4
1.9 2- = 01'0 log x
Sol. x=10
1.10 21- = 4 log x Sol. x=1/16
1.11 3 = 216 log x
Sol. x=6
1.12 3 = 64 log x
Sol. x=4
RAONAMENT 3 = 64 xlog → x3=64 → x= 4643 =
1
Xavier Rabasa 47
1.13 2log5logxlog −= Sol. x=5/2
1.14 3ln2xln = Sol. x=9
1.15 3xlog21 =+ 1 Sol. x=10
1.16 9xlog3 3 −= Sol. x=1/27
RAONAMENT
9log3 3 −=x → log3x=-3 → x=3-3 → x=1/27 1.17 x16log2 = Sol. x=4
1.18 x000.10log =
Sol x=4
1.19 x27log3 =
Sol x=3
1.20 0xloga =
Sol x=1
1.21 2xlog9 = Sol x=81
1.22 x40log16 = Sol x=1/2
RAONAMENT x=4log16 → 16x=4 → 42x=4 → 2x=1 → x=1/2
1.23 4- = 0001'0 log x Sol. x=10
1.24 x =
321 log2 Sol x=-5
1.25 31 = 10 log x Sol x=1000
1.26 ( ) x = 3 3 log 23
Sol x=5/2
Xavier Rabasa 48
RAONAMENT
( ) x = 3 3 log 23 ⇒ 3·33 2x = ⇒ 2
12x 33+
= ⇒
25
212x =+=
1.27 x= 1 log Sol x=0
1.28 x = 81 log3 Sol x=2
1.29 ( ) x = 2 log log 8
22 Sol x=3
1.30 x = 5 log5 Sol x=1/2
1.31 x = 625 log5 Sol x=4
RAONAMENT x = 625 5log → 5x=625 → 5x=54 → x=4
1.32 2 log= x log Sol. x=2
1.33 3= x log Sol x=1.000
1.34 5= x log Sol x=100.000
1.35 ( ) x = 32 log 2
2
Sol x=10
RAONAMENT x = 2
232log → x=102 2log → 10 = x → x = 10
1.36 3 log 2= x log Sol x=9
1.37 2 = 10 log - x log Sol x=1.000
1.38 81log = 5)-(2xlog 4 33 Sol x=4
Xavier Rabasa 49
1.39 2 = 2)+x+x( log 22 Sol x=1 x=-2
RAONAMENT
3 = 1-2x5+x3
2
2 ⎟⎠⎞
⎜⎝⎛log → 8loglog 2 =
1-2x5+x3
2
2 ⎟⎠⎞
⎜⎝⎛ →
81253 2
=−+
xx
→ 3x2-16x+13=0 → ⎩⎨⎧
==
3/131
xx
1.40 ( ) 2 = 25+3x log Sol. x=25
1.41
1'0 log = 2-x3x-5 Sol x=3/2
1.42 2=1)+(xlog - 1)-(3x log 33 Sol x=-5/3
1.43 8log = 1)-(xlog 3 22
Sol x=3
1.44 (2x)log=1)-(x log + 3 log Sol x=3
1.45 x log - 10 log = 100
x log 4 Sol x=1.000
RAONAMENT
x - 10 = 100
x 4 logloglog → x
x 10000log100
log =
→
x2=1000000 → ⎩⎨⎧
−==
noxsix
10001000
1.46 9xlog3 = Sol. x=2
1.47 3
81logx −= Sol x=2
1.48 5xlog2 = Sol x=32
Xavier Rabasa 50
1.49 x
93log3 =
Sol x=-3/2
1.50 25logx −= Sol x=1/ 5
1.51 xe1ln
2= Sol x=-2
RAONAMENT
xe
Ln =2
1 → ln e -2 = x → - 2 = x → x = - 2
1.52
x33log9 =
Sol. x=1/4
1.53 x8log21 = Sol. x=-3
1.54 8xlog 4
3 = Sol. x=9
1.55 5log2logxlog =+ Sol. x=5/2
1.56 x2log 3
21 = Sol. x=-3
1.57 21xlog =− Sol. x=1.000
RAONAMENT
21log =−x → log x = 3 → x=103 → x=1000 1.58 3log2x3log4xlog −=−+ Sol. x=12
1.59 3ln)3x4ln()2xln(xln −−=−− Sol. x=3
1.60 x3logxlog)2xlog( 2 −=−− Sol. x=3
RAONAMENT
)3log(log)2log( 2 xxx −=−− → xx
x31log2log
2=
− →
3x ( x – 2 ) = x2 → 3x2 – 6x = x2 → 2 x2 - 6x = 0
Xavier Rabasa 51
→ ⎩⎨⎧
==
sixnox
30
1.61 )1xlog(x2logxlog2 −=− Sol. x=2
1.62 )x2ln()1xln()2xln( 2 −=+−+ Sol. x=0
1.63 2logxlog2log2xlog3 2 −=− Sol. x=2
RAONAMENT
2loglog2log2log3 2 −=− xx → logx3-log22=logx2-log2
→ 2
log2
log2
2
3 xx= → 2x3=4x2 →
⎩⎨⎧
==
sixnox
20
1.64 x =3 log 27 Sol. x=1/3
1.65 x = 32 log 3
2 Sol x=5/3
1.66 21 = 7 logx Sol x=49
1.67 3 =
x81 log3 Sol x=3
1.68 25 = 32 log x Sol x=4
1.69 0 = xlog 1000 Sol x=1
RAONAMENT
0log 1000 = x → x=0
1000 → x=1 1.70 x = 7 log49 Sol. x=1/4
1.71 2- =
4x log2 Sol x=1
1.72 2 = 81log x Sol. x=3
Xavier Rabasa 52
1.73 1- =
9x log3
Sol. x=9
1.74 2 = x
100 log5 Sol. x=4
1.75 2 =9 log x Sol. x=9
RAONAMENT 29log = x → ( ) 9
2=x → x=9
1.76
⎟⎠⎞
⎜⎝⎛
31 ln = 1)-x(ln - 1)-(x ln 2
Sol. x=2
1.77 3)+(xln = 2 ln +
x1+x ln ⎟⎠⎞
⎜⎝⎛
Sol. x=1
1.78 x)-(2log=2)-(xlog + 1)+(x log
Sol. x=-1, x= 2
1.79 2log 2 = 1)-(xlog 2
Sol. x=-1, x=3
1.80 2)-(xlog = 1-xlog - 1)+(x log
Sol. x=5
1.81 1)-(4xlog=2)+(xlog + x log
Sol. x=1
RAONAMENT 1)-(4x = 2)+(x + x logloglog → log x(x+2) = log (4x-1)
→ x2+2x = 4x-1 → x2-2x+1=0 → ⎩⎨⎧
==
sixsix
11
→ x=1
Xavier Rabasa 53
SISTEMES LINEALS AMB DUES INCÒGNITES 1.DISCUSIÓ DE SISTEMES 2.SISTEMES COMPATIBLES DETERMINATS 3.SISTEMES COMPATIBLES INDETERMINATS DISCUSIÓ i RESOLUCIÓ DE SISTEMES En tota equació lineal a x + b y = c els punts ( x , y ) pertanyen a una mateixa recta que expressada en forma explícita té per equació: y = m x + n on m és la pendent i n el terme independent Primer cas: Si dues rectes tenen la mateixa pendent m i el mateix terme independent n aleshores coincideixen i la seva intersecció són tots els punts de la recta. En aquest cas direm que el sistema lineal format per les dues equacions és compatible indeterminat S .C .I amb infinites solucions que en funció d’un paràmetre t són:
⎩⎨⎧
+==
nmtytx
Discutiu i resoleu el sistema: 42 =y2x
2=y+x
+⎩⎨⎧
Solució: Primera recta y = -x + 2 m = -1 n = 2 Segona recta y = -x + 2 m = -1 n = 2 Sistema compatible indeterminat S. C. I.
( infinites solucions ) Solucions: ⎩⎨⎧
−==
tytx
2
Segon cas:
Ex.1
Xavier Rabasa 54
Si dues rectes tenen la mateixa pendent m i diferent terme independent n aleshores les rectes són paral·leles i la seva intersecció no dona cap resultat. En aquest cas el sistema format per les dues equacions és incompatible S. I. amb cap solució.
Discutiu i resoleu el sistema: 82 =y2x
2=y+x
+⎩⎨⎧
Discussió: Primera recta y = -x + 2 m = -1 n = 2 Segona recta y = -x + 4 m = -1 n = 4 Sistema incompatible S. I. ( cap solució ) Tercer cas: Si dues rectes tenen diferent pendent m aleshores les rectes es tallen en un sol punt i el sistema és compatible determinat S. C. D. amb única solució i per determinar aquesta solució s’utilitzen els mètodes de: Substitució Reducció Igualació
Discutiu i resoleu el sistema: 1-=y+x-
3=2y+3x
⎩⎨⎧
Discussió: Primera recta y = -3/2x + 3/2 m = -3/2 Segona recta y = x – 1 m = 1 Sistema compatible determinat S. C. D. (única solució ) RESOLUCIÓ: a) Mètode de substitució Aïllem la variable y de l’equació més fàcil y = x – 1 La substituïm a l’altra equació 3x + 2y = 3 ; 3x + 2( x – 1 ) = 3 Resolem l’equació 3x + 2x -2 = 3 5x = 5 x = 1
Ex.3
Ex.2
Xavier Rabasa 55
Substituïm el valor de x a l’equació més fàcil y= x – 1 y= 1 – 1 y = 0
Solució:⎩⎨⎧
==
01
yx
b) Mètode d’igualació Aïllem la mateixa variable de les dues equacions
⎩⎨⎧
−=+−=1
23
23
xyxy
Igualem segones parts 123
23
−=+− xx
Resolem l’equació -3x+3 = 2x -2 -5x = -5 x = 1 Substituïm el valor de x a l’equació més fàcil y= x – 1 y= 1 – 1 y = 0
Solució:⎩⎨⎧
==
01
yx
c) Mètode de reducció
Multipliquem la segona per 3 333 -=y+x-
3=2y+3x
⎩⎨⎧
Sumem les equacions per eliminar la x, 5 y = 0 Calculem la variable y y = 0 Substituïm el valor de y a una de les equacions y = x – 1 0 = x – 1 x = 1
Xavier Rabasa 56
Solució:⎩⎨⎧
==
01
yx
1.DISCUSIÓ DE SISTEMES
Discutiu els següents sistemes: 1.1
1=y-2x
2=y+x
⎩⎨⎧
Sol. S.C.D
1.2
1-=y+x-
3=2y+3x
⎩⎨⎧
Sol. S.C.D
1.3
3-=y+x-
3=y+2x
⎩⎨⎧
Sol. S.C.D
1.4
2=2y+2x
5=y-x
⎩⎨⎧
Sol. S.C.D
1.5
4=y22x
2=y+x
+⎩⎨⎧
Sol. S.C.I
1.6
⎪⎩
⎪⎨⎧
−=−−
=+
31
3y2x
3y2x3
Sol. S.C.I
RAONAMENT
1
Xavier Rabasa 57
⎪⎩
⎪⎨⎧
−=−−
=+
31
32
323yx
yx →
⎪⎩
⎪⎨
⎧
+−
=
+−
=
21
23
23
23
xy
xy → m1 = m2 n1 ≠ n2
→ S. I. 1.7
6-=y4+x-
12=y82x −
⎩⎨⎧
Sol. S.C.I
1.8
10=2y+2x
5=y-x
−−⎩⎨⎧
Sol. S.C.I
1.9
4=y22x
21=y+x
+⎩⎨⎧
Sol. S. I.
1.10
⎪⎩
⎪⎨⎧
−=−−
=+
31
3y2x
5y2x3
Sol. S .I.
1.11
6-=y4+x-
6=y82x −
⎩⎨⎧
Sol. S. I.
1.12
15=2y+2x
5=y-x
−⎩⎨⎧
Sol. S. I.
RAONAMENT
15=2y+2x
5=y-x
−⎩⎨⎧
→ ⎪⎩
⎪⎨⎧
+=
−=
2155
xy
xy →m1 = m2 n1 ≠ n2
→ S. I.
Xavier Rabasa 58
2.SISTEMES COMPATIBLES DETERMINATS
Resoleu els següents sistemes compatibles i determinats 2.1
1=y-2x
2=y+x
⎩⎨⎧ Sol
⎩⎨⎧
==
1y1x
2.2
1-=y+x-
3=2y+3x
⎩⎨⎧ Sol
⎩⎨⎧
==
0y1x
2.3
3-=y+x-
3=y+2x
⎩⎨⎧ Sol
⎩⎨⎧
−==
1y2x
2.4
2=2y+2x
5=y-x
⎩⎨⎧ Sol
⎩⎨⎧
−==
2y3x
2.5
1-=y-2x
1=y+x
⎩⎨⎧ Sol
⎩⎨⎧
==
1y0x
RAONAMENT
1-=y-2x
1=y+x
⎩⎨⎧
→⎩⎨⎧
⎩⎨⎧
→=
+−=→
−=+−−+−=
01
1)1(21
xxy
xxxy
⎩⎨⎧
==
0x1y
2.6
1-=3y+x-
3=y-x
⎩⎨⎧ Sol
⎩⎨⎧
==
1y4x
2.7
1=5y+2x-
5=3y-4x
⎩⎨⎧ Sol
⎩⎨⎧
==
1y2x
2
Xavier Rabasa 59
2.8
0=2y+3x
1=y+x
⎩⎨⎧ Sol
⎩⎨⎧
=−=3y2x
2.9
2-=2y-2x
3=y-5x
⎩⎨⎧ Sol
⎩⎨⎧
==
2y1x
2.10
8=y+7x
5=2y+3x
⎩⎨⎧ Sol
⎩⎨⎧
==
1y1x
Mètode d’igualació
8=y+7x
5=2y+3x
⎩⎨⎧
⎪⎩
⎪⎨⎧
⎪⎩
⎪⎨⎧
→−=
−=−
→−=
−=
xy
xx
xy
xy
78
78235
78235
⎩⎨⎧
==
1y1x
2.11
23=y-2x
7=y+x
⎩⎨⎧ Sol
⎩⎨⎧
−==
3y10x
2.12
17=2y-7x
3=6y-5x
⎩⎨⎧ Sol
⎩⎨⎧
==
2y3x
2.13
3=y-x
9=y+2x
⎩⎨⎧ Sol
⎩⎨⎧
==
1y4x
2.14
7-=3y-2x
6=y+3x
⎩⎨⎧ Sol
⎩⎨⎧
==
3y1x
2.15
0=y+2x
5-=y-3x
⎩⎨⎧ Sol
⎩⎨⎧
=−=2y1x
Mètode de reducció
Xavier Rabasa 60
0=y+2x
5-=y-3x
⎩⎨⎧ →
⎩⎨⎧
=−−−=−
0361026
yxyx
→ ( sumand) →
-5y=-10 → y=2 → ⎩⎨⎧
=−=21
yx
2.16
7-=5y+3x
1-=3y+5x
⎩⎨⎧ Sol
⎩⎨⎧
−==
2y1x
2.17
21=3y-15x
3=7y-12x
⎩⎨⎧ Sol
⎩⎨⎧
==
3y2x
2.18
6=y-5x
8-=12y+4x
⎩⎨⎧ Sol
⎩⎨⎧
−==
1y1x
2.19
4=3y+5x
4=5y+3x −
⎩⎨⎧ Sol
⎩⎨⎧
−==
2y2x
2.20
9=y+5x
5-=3y-7x
⎩⎨⎧ Sol
⎩⎨⎧
==
1y4x
Mètode de substitució
9=y+5x
5-=3y-7x
⎩⎨⎧ →
⎩⎨⎧
→⎩⎨⎧
=−=
→−=−−
−=159
5)59(3759
xxy
xxxy
⎩⎨⎧
==
14
xy
2.21
5=y - 2x
2y=3)-2(x
⎩⎨⎧ Sol
⎩⎨⎧
−==
1y2x
2.22
3=y + 2x
y=2)+5(x
⎩⎨⎧ Sol
⎩⎨⎧
=−=5y
1x
Xavier Rabasa 61
2.23
2y=1)+2(x
5=y +x 3
⎩⎨⎧ Sol
⎩⎨⎧
==
2y1x
2.24
15=2y) - 3(x
5-=y + 2x
⎩⎨⎧ Sol
⎩⎨⎧
−=−=
3y1x
2.25
y) - 2x ( 2=2
1) -y ( 3=3x
⎩⎨⎧ Sol
⎩⎨⎧
==
3y2x
Mètode d’igualació
y) - 2x ( 2=2
1) -y ( 3=3x
⎩⎨⎧ →
⎩⎨⎧
⎩⎨⎧
→−=+
+=→
−=+=
1211
121
xxxy
xyxy
⎩⎨⎧
==
23
xy
2.26
12=3y) + 2x ( 3
5y-=2) - 3x ( 2
⎩⎨⎧ Sol
⎩⎨⎧
=−=2y1x
2.27
5 - x=3 -y
)y - 4 ( 2=x
⎩⎨⎧ Sol
⎩⎨⎧
==
2y4x
2.28
2x +2y =1 - x
6 - x=3y + x
⎩⎨⎧ Sol
⎩⎨⎧
−==
2y3x
2.29
3 +3y + 2x =5y + x
3y- = ) 1 +2y - x ( 3
⎩⎨⎧ Sol
⎩⎨⎧
==
2y1x
2.30
2y + 3x- =5y + 3x
y)+3-3(x =y - 4x
⎩⎨⎧ Sol
⎩⎨⎧
=−=2y1x
Mètode de reducció
Xavier Rabasa 62
2y + 3x- =5y + 3x
y)+3-3(x =y - 4x
⎩⎨⎧
⎩⎨⎧
⎩⎨⎧
=+=+−
→=+−=−
→021882
0294
yxyx
yxyx
→=→=→ 2189 yy ⎩⎨⎧
−==
12
xy
2.31
3=3y+
2x
8=y+x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
==
6y2x
2.32
2=4y-3x
9=2y+x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
==
4y1x
2.33
2=2y+
3x
3=y+x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
=−=6y
3x
2.34
5=2y+3x
6=3y-x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
==
1y9x
2.35
2=2y-x
2-=y-2x
⎪⎪⎩
⎪⎪⎨
⎧
Sol⎩⎨⎧
==
4y4x
Mètode de substitució
Xavier Rabasa 63
2=2y-x
2-=y-2x
⎪⎪⎩
⎪⎪⎨
⎧
→
⎩⎨⎧
⎩⎨⎧
→=−−
−=→
=−−=−
4)42(242
4242
yyyx
yxyx
⎩⎨⎧
==
44
yx
2.36
1=43y+
32x
0=2y+
3x
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
Sol⎩⎨⎧
−==
4y6x
2.37
1-=32y+
2x
0=2y+3x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
−==
3y2x
2.38
1 - 23y=
2x
6y=3x
⎪⎪⎩
⎪⎪⎨
⎧
Sol⎩⎨⎧
==
2y4x
2.39
1=5y-
32x
1=y-2x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
==
5y3x
Xavier Rabasa 64
2.40
1=43y-2x
1=3
y+x+2
y-x
⎪⎪⎩
⎪⎪⎨
⎧
Sol⎩⎨⎧
==
4y2x
Mètode d’igualació
1=43y-2x
1=3
y+x+2
y-x
⎪⎪⎩
⎪⎪⎨
⎧
→ ⎩⎨⎧
=−=++−
43862233
yxyxyx
→
⎩⎨⎧
=−=−
43865
yxyx
→ ⎪⎩
⎪⎨⎧
−=
−=
34865
xy
xy →
⎪⎩
⎪⎨⎧
−=−
−=
34865
65xx
xy →
⎩⎨⎧
==
24
xy
2.41
1514=
6y-
102x
1=4y+
63x
⎪⎪⎩
⎪⎪⎨
⎧
Sol⎩⎨⎧
−==
2y3x
2.42
3 + 5
7y=32x
3y=x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
==
5y15x
2.43
1-x=6-y
4=72y-3x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
==
7y2x
Xavier Rabasa 65
2.44
1=1 +y
x
2=y
1 + x
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
Sol⎩⎨⎧
==
2y3x
2.45
2x-=15y - 4x
1 + 2x=y) - x ( 3
⎩⎨⎧ Sol
⎩⎨⎧
−=−=
2y5x
Mètode de reducció
2x-=15y - 4x
1 + 2x=y) - x ( 3
⎩⎨⎧ →
⎩⎨⎧
=−=−
05213
yxyx
→
→ ⎩⎨⎧
−=⇒=−−=+−
2052
262y
yxyx
→ ⎩⎨⎧
−=−=+=
2531
yyx
2.46
1=3y + 3
3x
5=y - xy + x
⎪⎪⎩
⎪⎪⎨
⎧
Sol⎩⎨⎧
==
2y3x
2.47
4=3y + 2x
4=x
y - 2x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
=−=2y1x
2.48
2- x=2y - 3x
2=y + x
5x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
==
3y2x
Xavier Rabasa 66
2.49
3=3y + 2x
51 - 2=
y + 2x3x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
−==
1y3x
2.50
29=3y-
23x
2x=5y+x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
==
1y5x
3. SISTEMES COMPATIBLES INDETERMINATS
Resoleu els següents sistemes compatibles indeterminats 3.1
3=3y+
2x
81=y2+x3
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
−==
t39yt2x
3.2
2=4y-3x
8x12=y −
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
−==
8t12ytx
3.3
2=2y
3x
12=y3+x2
−
−−
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
−==
4t2yt3x
3
Xavier Rabasa 67
3.4
5=2y+3x
15=y6x +
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
=−=
tyt65x
3.5
4=y2x
2-=y-2x
+−⎪⎪⎩
⎪⎪⎨
⎧
Sol⎩⎨⎧
+==
2tyt2x
Mètode d’igualació
42 =yx
2-=y-2x
+−⎪⎪⎩
⎪⎪⎨
⎧
→+
=→⎪⎩
⎪⎨
⎧
+=
+=
→2
4
24
24
xyxy
xy
⎪⎩
⎪⎨⎧
+=+
=
=
22
422
ttytx
3.6
1=43y+
32x
24=y18+x16
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
Sol⎪⎩
⎪⎨⎧
−=
=
9t812y
tx
3.7
0=92y+
3x
0=2y+3x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
−==
t3yt2x
Xavier Rabasa 68
3.8
1 - 23y=
2x
2=y3x −−
⎪⎪⎩
⎪⎪⎨
⎧
Sol⎩⎨⎧
=−=
ty2t3x
3.9
61=
6y-
3x
1=y-2x
⎪⎩
⎪⎨
⎧
Sol⎩⎨⎧
−==
1t2ytx
3.10
6x5y
1=3
y+x+2
y-x
−=⎪⎪⎩
⎪⎪⎨
⎧
Sol
⎩⎨⎧
−==
6t5ytx
Mètode d’igualació
65 −=⎪⎪⎩
⎪⎪⎨
⎧
xy
1=3
y+x+2
y-x
⎩⎨⎧
→−=→−=−=
→ 656565
xyxyxy
⎩⎨⎧
−==
65tytx
Xavier Rabasa 69
SISTEMES LINEALS AMB DIVERSES INCÒGNITES
1. EQUIVALÈNCIA DE SISTEMES 2. DISCUSSIÓ DE SISTEMES 3. RESOLUCIÓ DE SISTEMES 4. DISCUSSIÓ I RESOLUCIÓ DE SISTEMES
1.EQUIVALÈNCIA DE SISTEMES
Digues si són o no equivalents els següents sistemes: 1.1
⎩⎨⎧
−=−=+
1yx3yx
0yx21yx3
⎩⎨⎧
=−=−
Sol. Si
1.2
⎪⎩
⎪⎨
⎧
=+−=−=−+
2z2yx2zx2
0zyx
0z2yx3yx2
4zy3x
⎪⎩
⎪⎨
⎧
=−+=−
=+−
Sol Si
1.3
2z2yx1zyx1zy3x
⎪⎩
⎪⎨
⎧
=+−−=−+−−=+−
⎪⎩
⎪⎨
⎧
−=+−=+−=++−
3zx30z3yx22zy3x
Sol No
1.4
⎩⎨⎧
−=+=−
1y2x2yx
0yx4yx3
⎩⎨⎧
=−−=−
Sol Si
RAONAMENT
⎩⎨⎧
−=+=−
1y2x2yx
→ ⎩⎨⎧
−==
11
yx
0yx4yx3
⎩⎨⎧
=−−=−
→ ⎩⎨⎧
−==
11
yx
mateixa solució i aleshores equivalents 2.DISCUSSIÓ DE SISTEMES
2.1
1
Xavier Rabasa 70
Discutiu segons els valors del paràmetre els sistemes següents: 2.1.1.
⎪⎩
⎪⎨
⎧
=++=−+=++
2zy2x0zayx
2zy2ax
Sol.
⎪⎩
⎪⎨
⎧
−≠−==
.D.C.S}2,1{a.I.S2a.I.C.S1a
2.1.2
ay0ayxayax
⎪⎩
⎪⎨
⎧
==+=+
Sol.
⎩⎨⎧
≠=
.I.S0a.I.C.S0a
2.1.3
⎪⎩
⎪⎨
⎧
=+=−+=+−
ααzx30zyx21z2yx
Sol.
⎩⎨⎧
≠=
.D.C.S1a.I.C.S1a
2.1.4
1ax0yax
1ayx
⎪⎩
⎪⎨
⎧
+=−=++=+
Sol.
⎪⎩
⎪⎨
⎧
=−=−≠
.D.C.S2a
.D.C.S1a.I.S}2,1{a
RAONAMENT
⎪⎩
⎪⎨
⎧
+=−=++=+
101
axyax
ayx
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−=
01111
aA ⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
+−
+=
10101
111
aa
aA
010111
≠=−
⇒ r(A) = 2 )2)(1( aaA −+=
Discussió: a ≠ 2 , a ≠ -1 ⇒ r( A)=3 S I (sistema incompatible)
a = -1 ⇒⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=
001011011
A 0=A ⇒ r( A)=2 S CD
a = 2 ⇒⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−=
301012311
A 0=A ⇒ r( A)=2 S CD
Xavier Rabasa 71
2.1.5
⎪⎪⎩
⎪⎪⎨
⎧
=++=+=+−
=++
ααα
α
z2yx1zy1yx
1z2yx
Sol.
⎪⎩
⎪⎨
⎧
−==−≠
.I.S2a.I.C.S1a.D.C.S}2,1{a
2.1.6
0z)2k(y2xkkyx
1kz)2k(y2kx
⎪⎩
⎪⎨
⎧
=+++−=+
−=+−−
Sol.
⎪⎪⎩
⎪⎪⎨
⎧
−≠−===
.D.C.S}2,1,0{k
.D.C.S2k.I.C.S1k
.I.S0k
2.1.7
5kykx2kzy
kz2yx
⎪⎩
⎪⎨
⎧
=+=−
=+−
Sol.
⎪⎩
⎪⎨
⎧
≠==
.D.C.S}1,0{k.I.C.S1k
.I.S0k
2.1.8
3az5y)2a(x30z4y3ax
1z3ayx
⎪⎩
⎪⎨
⎧
−=+++=++
=++ Sol.
⎪⎩
⎪⎨
⎧
≠==
.D.C.S}5,2{a.I.S5a.I.C.S2a
2.1.9
⎪⎩
⎪⎨
⎧
=++=−+=++
2zy2x0zayx
2zy2ax
Sol.
⎪⎩
⎪⎨
⎧
−≠−==
.D.C.S]2,1{a.I.S2aI.C.S1a
2.1.10
ay0ayxayax
⎪⎩
⎪⎨
⎧
==+=+
Sol.
⎩⎨⎧
≠=
.I.S0a.I.C.S0a
2.1.11
⎪⎩
⎪⎨
⎧
=+=−+=+−
aazx30zyx21z2yx
Sol.
⎩⎨⎧
≠=
.D.C.S1a.I.C.S1a
Xavier Rabasa 72
2.1.12
1ax0yax
1ayx
⎪⎩
⎪⎨
⎧
+=−=++=+
Sol.
⎩⎨⎧
−≠−=
.I.S}1,2{aD.C.S}1,2{a
RAONAMENT
⎪⎩
⎪⎨
⎧
+=−=++=+
101
axyax
ayx
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−=
01111
aA ⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
+−
+=
10101
111
aa
aA
010111
≠=−
⇒ r(A) = 2 )2)(1( aaA −+=
DISCUSSIÓ a ≠ 2 , a ≠ -1 ⇒ r( A)=3 S I (sistema incompatible)
a = -1 ⇒⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=
001011011
A 0=A ⇒ r( A)=2 S CD
a = 2 ⇒⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−=
301012311
A 0=A ⇒ r( A)=2 S CD
2.1.13
⎪⎪⎩
⎪⎪⎨
⎧
=++=+=+−
=++
az2yax1azy1yx
1z2yax
Sol.
⎪⎩
⎪⎨
⎧
−≠−==
D.C.S}2,1{a.I.S2a.I.C.S1a
2.1.14
0z)2k(y2xkkyx
1kz)2k(y2kx
⎪⎩
⎪⎨
⎧
=+++−=+
−=+−−
Sol
⎪⎪⎩
⎪⎪⎨
⎧
−≠−===
.D.C.S}2,1,0{k
.D.C.S2k.I.C.S1k
.I.S0k
Xavier Rabasa 73
2.1.15
5kykx2kzy
kz2yx
⎪⎩
⎪⎨
⎧
=+=−
=+−
Sol.
⎪⎩
⎪⎨
⎧
≠==
D.C.S}0,1{k.I.C.S1k
.I.S0k
2.1.16
3az5y)2a(x30z4y3ax
1z3ayx
⎪⎩
⎪⎨
⎧
−=+++=++
=++ Sol.
⎪⎩
⎪⎨
⎧
≠==
D.C.S}5,2{a.I.S5a.I.C.S2a
RAONAMENT
3z5y)2(x3
0z4y3x1z3yx
⎪⎩
⎪⎨
⎧
−=+++=++
=++
ααα
αA=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
+ 52α343α3α1
A =0 ⎩⎨⎧
=α=α
⇒52
Discussió: a ≠ 2 , α ≠ 5 r(A) = 3 = r( A) SCD(compatible determinat) a = 2
A = 0 05331≠ → r(A) = 2 ; 0
153042131
=−
→ r( A) = 2
→ SCI (compatible indeterminat) a=5
A = 0 05331≠ → r(A) =2 ; 0
253045131≠ → r( A)=3
SI (sistema indeterminat) 2.1.17
atzy2x
3z2yax1tzy2x
⎪⎩
⎪⎨
⎧
=−−+−=−+=++−
Sol.
⎩⎨⎧
−≠−=
.I.S1a.C.S1a
Xavier Rabasa 74
2.1.18
⎪⎪⎩
⎪⎪⎨
⎧
−=−−=++=+=−+
2z3yx2azyx
5z2x32zy2x
Sol.
⎩⎨⎧
≠=
.I.S3a.D.C.S3a
2.1.19
1azyx1zayx1zyax
⎪⎩
⎪⎨
⎧
=++=++=++
Sol.
⎪⎩
⎪⎨
⎧
−≠−==
D.C.S}2,1{a.I.S2a.I.C.S1a
2.1.20
0zkyx0zykx
0zyx
⎪⎩
⎪⎨
⎧
=++=++=++
Sol.
⎩⎨⎧
≠=
.D.C.S1k.I.C.S1k
RAONAMENT
2)1(1111111
−= kk
k⎪⎩
⎪⎨
⎧
∞===
==≠
)(11
)0,0,0(31
2solucionsArangrangAk
solucióArangrangAk
2.1.21
⎪⎩
⎪⎨
⎧
=++=++=++
0z2y5x30zy2x0z2kyx
Sol.
⎩⎨⎧
≠=
.D.C.S3k.I.C.S3k
2.1.22
⎪⎩
⎪⎨
⎧
=+−=+−
=−+
0zy2x50kzy4kx
0zyx
Sol.
⎩⎨⎧
≠=
.D.C.S3k.I.C.S3k
2.1.23
3zy8x1zy
2kzyx1z3y2x
⎪⎪⎩
⎪⎪⎨
⎧
=++=−
=+−=++
Sol.
⎩⎨⎧
≠=
.I.S2k.D.C.S2k
Xavier Rabasa 75
2.1.24
2z3y4k2kzy3x
0z2ykx
⎪⎩
⎪⎨
⎧
=−=−+=+−
Sol.
⎪⎩
⎪⎨
⎧
≠==
D.C.S}4/5,1{a.I.S4/5k.I.C.S1k
RAONAMENT
⎪⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
→==
→⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
≠−=−
=
→=
→==
→⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=−
=
→=
→==→≠≠
−−=−−
−
.I.S3Arang2rangA
03240
2/531014/5
2rangA
4/5k
I.C.S2Arang
2ranA0
240231011
2rangA
1k
D.C.S3ArangrangA4/5k,1k
)5k4)(1k(340k31
21k
Calculeu el valor o condició dels paràmetres per tal que els sistemes següents siguin compatibles:
2.2
Xavier Rabasa 76
2.2.1
3bzxazyx
1z2y1zayx
⎪⎪⎩
⎪⎪⎨
⎧
=+−−=−+
=+−=++
Sol. a=2 i b=3
2.2.2
⎪⎩
⎪⎨
⎧
=++=++
=+++
0z2y2x20zmyx
0z2yx)2m(
Sol. m=1 ó m=0
2.2.3
⎪⎩
⎪⎨
⎧
=+−=+−=++
0zyax0azy2x3
0zayx2
Sol. a3 + a – 7 = 0
2.2.4
1axa2zay
0zyax21zyax
⎪⎪⎩
⎪⎪⎨
⎧
==+−
=−+=−+−
Sol. a = -1
RAONAMENT
Xavier Rabasa 77
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−−−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−−−−
→=
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛−−
→=
−−
=−
−−−
+−
−−
−=−
−−−
SCD3
1001211001121111
r3
001110112111
r1a
I.SSI3
1000010001101110
r2
000100110110
r0a
)a1(a4
1a011a211a
1a21a
011111
a
100aa21a0
011a2111a
2.2.5
⎪⎪⎩
⎪⎪⎨
⎧
−=+=+−=++=−+
ky3x42zkyx
1z2yx30zkyx2
Sol. k=1 ó k= -25/6 S CD
2.2.6
kkz2x)1k(1kzyxkkzykx
⎪⎩
⎪⎨
⎧
=−−=−−=+−
Sol. k≠ 0 o k ≠ 1 S CD
2.2.7
⎪⎩
⎪⎨
⎧
=+−=++−=−−
az2y0azyx
2y2x)1a(
Sol. a = 2 SCD
2.2.8
0zbyax0zyx20bzayx
⎪⎩
⎪⎨
⎧
=+−=+−=+−
Sol. a2+2b2-ab-2a-b+1=0
Xavier Rabasa 78
RAONAMENT
0=z+by-ax
0=z+y-2x
0=bz+ay-x
⎪⎩
⎪⎨
⎧
el sistema és compatible indeterminat si: A = 0 A = a2 + 2 b2 – a b – 2 a – b + 1 =0 3.RESOLUCIÓ DE SISTEMES
Resoleu matricialment els següents sistemes compatibles i determinats: 3.1.1
8=3z+y-x
0=z- 2x
2-=z-y+x
⎪⎩
⎪⎨
⎧
Sol.
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−
−=−
31
31
31
61
32
67
61
31
61
1A
x=1 y=-1 z=2 3.1.2
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
2
1-
1
=
z
y
x
1-11
21-1-
12-1
Sol.
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−=−
110
132
31
131
31
1A
x=2 y=1 z=1 3.1.3
3=z+ 3x
0=z-y-2x
1-=3z+y-x
⎪⎩
⎪⎨
⎧
Sol.
3.1
Xavier Rabasa 79
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−
−−
−
=−
131
133
133
137
138
135
134
131
131
1A
x =1 y = 2 z = 0 3.1.4
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
2-
3-
3
=z
1
2
1-
+ y
x
1-2
1-3
21
RAONAMENT
RAONAMENT
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
2-
3-
3
=z
1
2
1-
+ y
x
1-2
1-3
21
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
−
23
3
112213121
zyx
A=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
−
112213121
4=A A -1 = 41
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−
−
751531
311
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
zyx
41
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−−
−
751531
311
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−−
23
3 =
41 =
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
− 440
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−110
Resoleu en cas de compatibilitat els següents sistemes: 3.2.1
1= 2y-2x
1-=z+3y-x
1=z-y+x
⎪⎩
⎪⎨
⎧
Sol. .I.S
3.2
Xavier Rabasa 80
3.2.2
1=z+y-2x
3-=2z-y+3x
1-=z-y+x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
===
2z1y0x
.D.C.S
3.2.3
1= y+x
0=z-y
2=z+y+2x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧ −
tt
t1.I.C.S
3.2.4
3=z-3y+2x
2= 2y+x
1=z-y+x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧−
tt1
t2.I.C.S
RAONAMENT
3=z-3y+2x
2= 2y+x
1=z-y+x
⎪⎩
⎪⎨
⎧
213 FFF +=
⇔ ⎩⎨⎧
=+=−+221
yxzyx
⇔
⎩⎨⎧
=++=+221
yxzyx
⎪⎩
⎪⎨
⎧
=−=
=⇒
zzzy
zx12
⎪⎩
⎪⎨
⎧
=−=
=⇒
tzt1y
t2x
3.2.5
1=11z-4y-x-
3=3z+2y+x
0=z-y+x
⎪⎩
⎪⎨
⎧
Sol. .I.S
3.2.6
4-=z-3y+x
2= y-x
1=z-y-2x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
=−==
2z1y
1x.D.C.S
Xavier Rabasa 81
3.2.7
0=3z-4y-x
0=3z+y-4x
0=z+2y-3x
0=2z+y+x
⎪⎪⎩
⎪⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
−===
tztytx
.I.C.S
3.2.8
5y3x
62y+x2
1yx3
−=−
=
=−
⎪⎩
⎪⎨
⎧
Sol.
⎩⎨⎧
==
2y1x
D.C.S
RAONAMENT
⎩⎨⎧
==
⇒⎩⎨⎧
=+=−
−=−
=
=−
⎪⎩
⎪⎨
⎧⇔
−=
21
62213
53
62
13213
yx
yxyx
yx
2y+x
yx
FFF
3.2.9
yz2x
y2z3+x2
z6y2+x
=+
+=
+=
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
−===
1z1y3x
.D.C.S
3.2.10
3y5z
1yz3+x2
y32x+z
−=+
−=
+=
⎪⎩
⎪⎨
⎧
Sol. .I.S
3.2.11
z2x
z1x+y
y22x
=
+=
−=
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
=−=
=
tzt1y
t2x.I.C.S
3.2.12
1= y-x
2=3z+y+2x
⎩⎨⎧
Sol.
Xavier Rabasa 82
⎪⎩
⎪⎨
⎧
=−=−=
tztyt1x
.I.C.S
Resoleu aplicant la regla de Cramer en els següents sistemes compatibles i determinats: 3.3.1
4=z- 3x
1-=z+y-2x
3= y+x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
−===
1z2y1x
.D.C.S
3.3.2
2=z-y+x
5=2z- 3x
0=z+2y-x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
−===
1z0y1x
.D.C.S
3.3.3
4-=z-3y+x
2= y-x
1=z-y-2x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
=−==
2z1y
1x.D.C.S
3.3.4
2-=z+y-3x
4-=z+y-x
1=z+y+2x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
−===
3z2y1x
.D.C.S
RAONAMENT
113111112
−− = 4
112114111
−−−− = 4
123141112
−− = 8
213411
112
−−−− = -12
x=4/4=1 y=8/4=2 z= -12/4=-3
3.3
Xavier Rabasa 83
4.DISCUSIÓ I RESOLUCIÓ DE SISTEMES
Discutiu i resoleu els sistemes en cas de compatibilitat: 4.1.1
1=z-y+x
1=z+y+x
3=z3-3y+3x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
=−=
=
0zt1y
tx.I.C.S S . C
4.1.2
0=4z+3y-x
3=z+ 2x-
1=z+y-x
⎪⎩
⎪⎨
⎧
Sol. .I.S
4.1.3
2=2z-3y+x
1=z-y+2x
1=z+2y-x
⎪⎩
⎪⎨
⎧
Sol. .I.S
4.1.4
1-=3z-y+3x-
5=z-3y+x-
3=z+y+x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
−===
t1z2ytx
.I.C.S
RAONAMENT d)Discussió:
2Arang
0113
531311
;2rangA0313131
111A
=
→=−−
−=→=
−−−−=
S. C. I . ( 1∞ solucions )
4.1
Xavier Rabasa 84
Resolució:
⎪⎩
⎪⎨
⎧
⎪⎩
⎪⎨
⎧
λ−==λ=
⇒+=−−=+
⇔⎩⎨⎧
=−+−=++
12
533
533
zyx
xzyxzy
zyxzyx
4.1.6
4=zy-3x
3-=z-2y+x-
2=z- 2x
2=2z+y-x
+⎪⎪⎩
⎪⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
=−==
0z1y
1x.D.C.S
4.1.7
1=z+ x
3=z+y+x
0=2z+y-2x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
−===
t1z2ytx
.I.C.S
4.1.8
2-=z- 2x
4=z+2y
1=z+y-3x
⎪⎩
⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
===
2z1y0x
.D.C.S
4.1.9
0=2z+ x-
0=z+3y-x
⎩⎨⎧
Sol.
⎪⎩
⎪⎨
⎧
===
tztyt2x
.I.C.S
RAONAMENT Discussió SCI Resolució
⎪⎩
⎪⎨
⎧
λ=λ=λ=
⇒⎩⎨⎧
−=−−=−
zy
x
zxzyx
2
23
Xavier Rabasa 85
4.1.10
1=3z-6y+x
0=z+2y-x
3=2z+y-2x
1=z-2y+x
⎪⎪⎩
⎪⎪⎨
⎧
Sol. .I.S
4.1.11
4=z+3y
0=z-2y+x-
2= y-3x
1=2z-y+2x
⎪⎪⎩
⎪⎪⎨
⎧
Sol.
⎪⎩
⎪⎨
⎧
===
1z1y1x
.D.C.S
4.1.12
5=2y+x
5=y-2x
2=y-x
⎪⎩
⎪⎨
⎧
Sol.
⎩⎨⎧
==
1y3x
.D.C.S
4.1.13
3=3z-2y-x
3=2z+y+x-
4-=z+3y+2x
⎪⎩
⎪⎨
⎧
Sol. .I.S
4.1.14
0=6y-3x
3=2y+x
1=y-3x
⎪⎩
⎪⎨
⎧
Sol. .I.S
4.1.15
⎪⎩
⎪⎨
⎧
−=−−−−=−+−
−=−+
3t4z2yx2t3zyx3
2zy2x
Sol.
⎪⎪⎩
⎪⎪⎨
⎧
=−=−=−=
ttt1z
ty1tx
I.C.S
RAONAMENT
Xavier Rabasa 86
Discussió
ICSArangrangA ..305211
113121
==→≠−=−−
−−
Resolució:
ttt
zttt
y
ttt
xtzyxtzyx
zyx
−=−
−−−−−
=−=−
−−−
−−
=
−=−
−−−−−
−−
=→⎪⎩
⎪⎨
⎧
−=−−−=+−−=−+
15
34112313
221
52341
1233121
15
21341123122
342233
22
Sol.
⎪⎪⎩
⎪⎪⎨
⎧
=−=−=−=
ttt1z
ty1tx
Xavier Rabasa 87
SISTEMES EXPONENCIALS
Resoleu els sistemes següents: 1.1
⎪⎩
⎪⎨⎧
41 = 2
5 = 2 + 2y-x
yx
Sol.⎩⎨⎧
==
2y0x
1.2
⎩⎨⎧
19 = 2 + 37 = 2 + 3
2yx
yx
Sol.⎩⎨⎧
==
2y1x
1.3
⎩⎨⎧
4 = 24 = 2
y+3x
y-x
Sol.⎩⎨⎧
−==
1y1x
1.4
⎩⎨⎧
9 = 3 + 27 = 3 - 2
y1+x
y1-2x
Sol.⎩⎨⎧
==
0y2x
1.5
⎩⎨⎧
7 = 2 - 2 . 212 = 2 . 3
yx
y+x
Sol. ⎩⎨⎧
==
0y2x
1.6
⎩⎨⎧
1 = 3 . 3 - 210 = 3 . 2 - 2 . 3
yx
1+y1+x
Sol.⎩⎨⎧
−==
1y1x
RAONAMENT
1 = 3 . 3 - 2
10 = 3 . 2 - 2 . 3yx
1+y1+x
⎩⎨⎧ ⇔
⎩⎨⎧
=−=−
13·32·1103·62·6
yx
yx⇔
⎪⎩
⎪⎨⎧
−==
==
1313
122yx
y
x
1.7
⎩⎨⎧
8 = 26 = 2 + 2
y+x
yx
Sol.2y1y1x2x
====
⎩⎨⎧
1
Xavier Rabasa 88
1.8
⎩⎨⎧
27 = 32187 = 3
y-x
y+x
Sol.⎩⎨⎧
==
5y5x
1.9
⎪⎩
⎪⎨⎧
21 = 2
1- =y 2 + x 3
y+x
Sol.⎩⎨⎧
−==
2y1x
1.10
⎪⎩
⎪⎨⎧
2 =y + x 292 = 3 - 3 y-1-x
Sol.⎩⎨⎧
==
2y0x
1.11
⎩⎨⎧
1 = 3 - 211 = 3 + 2
y1+x
2yx
Sol.⎩⎨⎧
==
1y1x
1.12
⎩⎨⎧
4 = 22 = 8 . 2
y-x
yx/2
Sol.⎩⎨⎧
==
0y2x
1.13
⎩⎨⎧
47 = 5 - 314 = 5 + 3
yx
yx
Sol.⎩⎨⎧
==
1y2x
1.14
⎩⎨⎧
27 = 3 . 35 = 3 - 3 . 2
yx
2-yx
Sol.⎩⎨⎧
==
2y1x
RAONAMENT
27 = 3 . 3
5 = 3 - 3 . 2yx
2-yx
⎩⎨⎧
⇔ ⎪⎩
⎪⎨⎧
=
=−+ 333
53·913·2
yx
yx
⇔ y = 3 – x
⇔ 2 · 3x – 31-x = 5 ⇔ 2· (3x)2 – 3 = 5 · 3x ⇔
⎩⎨⎧
−===
capx
x
x
5'03133
; y = 2
Xavier Rabasa 89
1.15
⎩⎨⎧
2 = 2 - 21 =y - x
yx
Sol.⎩⎨⎧
==
1y2x
1.16
⎩⎨⎧
5 = 3 - 21 = 3 - 2
1+y2+x
yx
Sol.⎩⎨⎧
==
0y1x
1.17
⎩⎨⎧
80 = 2 . 2 - 321 = 3 . 2 + 2 . 3
1-x2+y
yx
Sol.⎩⎨⎧
==
2y0x
1.18
⎪⎩
⎪⎨⎧
27 = 321 = 2
x+y
2y-3x
Sol.⎩⎨⎧
==
2y1x
1.19
⎩⎨⎧
81 = 3 . 35 = 25 . 5
22y2x
2+y2xx
Sol.⎩⎨⎧
==
3y1x
1.20
⎩⎨⎧
4- = 3 . 2 - 23- = 2 - 3
2+xy
1+y1+x
Sol.⎩⎨⎧
=−=1y1x
RAONAMENT
⎩⎨⎧
4- = 3 . 2 - 23- = 2 - 3
2+xy
1+y1+x
→⎩⎨⎧
−=+−−=−
42·13·1832·23·3
YX
YX
→ ⎩⎨⎧
==−==12213/13
yx
Y
X
1.21
⎩⎨⎧
9 = 381 = 3
x-y
y+x
Sol.⎩⎨⎧
==
3y1x
1.22
⎩⎨⎧
3 = 336 = 3 + 3
x-y
yx
Sol.⎩⎨⎧
==
3y2x
1.23
⎩⎨⎧
64 = 220 = 2 + 2
x+y
yx
Sol.⎩⎨⎧
====
4y2y2x4x
Xavier Rabasa 90
1.24
⎩⎨⎧
23 = 3 - 27 = 3 + 2
2y1+2x
yx
Sol.⎩⎨⎧
==
1y2x
1.25
⎩⎨⎧
3 = 332 = 2
2y-x
y-2x
Sol.⎩⎨⎧
==
1y3x
1.26
⎩⎨⎧
2 = 2 . 23 = 9 . 3
61+y1-x
8yx
Sol.⎩⎨⎧
==
2y4x
1.27
⎩⎨⎧
108 = 3 - 3 . 515 = 3 - 3 . 2
1+y2+x
1-y1+x
Sol.⎩⎨⎧
==
2y1x
1.28
⎩⎨⎧
4 = 3 . 4 - 2 . 542- = 3 . 2 - 2 . 3
1-y1+x
yx
Sol.⎩⎨⎧
==
3y2x
1.29
⎩⎨⎧
28 = 3 + 532 = 3 . 2 - 5 . 2
1+yx
2+yx
Sol.⎩⎨⎧
==
0y2x
RAONAMENT
⎩⎨⎧
28 = 3 + 532 = 3 . 2 - 5 . 2
1+yx
2+yx
→ ⎩⎨⎧
=+=−283·35·1323·185·2
yx
yx
→⎩⎨⎧
====
0132255
yx
y
x
Xavier Rabasa 91
SISTEMES LOGARÍTMICS
Resoleu els sistemes següents: 1.1
3 =y log + x log
110 =y + x
⎩⎨⎧ Sol.
⎩⎨⎧
==
10y100x
1.2
9 =y 2 - x
1 =y log - x log 33
⎩⎨⎧ Sol.
⎩⎨⎧
==
9y27x
1.3
0 = x log -y log
70 =y + x 3-2⎩
⎨⎧ Sol.
⎩⎨⎧
==
100y10x
1.4
3- = y log - x log
12 =y + x3
22⎩⎨⎧ Sol.
⎩⎨⎧
==
4y8x
1.5
1 =y log - x log
2 = y) - (x log
22
2
⎩⎨⎧ Sol.
⎩⎨⎧
==
4y8x
1.6
0 = x 4 -y
4 =y log + x log
⎩⎨⎧ Sol.
⎩⎨⎧
==
200y50x
RAONAMENT
0 = x 4 -y
4 =y + x loglog
⎩⎨⎧ →
⎩⎨⎧
==
xyxy
410000
→ 4x2=10000
→ 2x=100 → ⎩⎨⎧
==
20050
yx
1.7
⎪⎩
⎪⎨⎧
3 =y log + x log 3
1 = )yx( log
Sol.
⎩⎨⎧
==
1y10x
1
Xavier Rabasa 92
1.8
4 log - 1 =y log + x log
1- =y log - x log
⎪⎩
⎪⎨
⎧ Sol.
⎪⎩
⎪⎨⎧
=
=
5y21x
1.9
⎩⎨⎧
2 = 2 . 216 log = y)-(x log + y)+(x log
8yxSol.
⎩⎨⎧
==
3y5x
1.10
0 =y log - x log
2 =y log + x log
⎩⎨⎧ Sol.
⎩⎨⎧
==
10y10x
1.11
3 =y log - x log
1- =y log + x log
⎩⎨⎧ Sol.
⎪⎩
⎪⎨⎧
=
=
1001y
10x
1.12
3 =y log + x log
50 =y 5 - x
⎩⎨⎧ Sol.
⎩⎨⎧
==
10y100x
RAONAMENT
3 =y + x
50 =y 5 - x
loglog⎩⎨⎧ →
⎩⎨⎧
=+=
1000550
xyyx
→ 5y2+50y-1000=0
→ y2+10y-200=0 → ⎩⎨⎧
−=−===
no50x20ysi100x10y
1.13
0 =y log - x log
2 =y log + x log
⎩⎨⎧ Sol.
⎩⎨⎧
==
10y10x
1.14
2 =y log - x log
30 =y + x
33⎩⎨⎧ Sol.
⎩⎨⎧
==
3y27x
1.15
1 =y log - x log
5 =y log + x log
⎩⎨⎧ Sol.
⎩⎨⎧
==
100y000.1x
Xavier Rabasa 93
1.16
e = e
8 ln =y ln + x ln2y-x⎪⎩
⎪⎨
⎧
Sol.⎩⎨⎧
==
2y4x
e = e
8 =y + x 2y-x
lnlnln
⎪⎩
⎪⎨
⎧ →
⎩⎨⎧
=−=
28
yxxy
→ y2+2y-8=0
→ ⎩⎨⎧
−=−===
noxysixy
2442
1.17
5 =y 5 - x
2 =y log + x log
⎪⎩
⎪⎨
⎧
Sol.⎩⎨⎧
==
4y25x
1.18
1 =y log + xlog
10 log =y log - x log
22
2
22
⎪⎩
⎪⎨
⎧
Sol.⎩⎨⎧
==
1y2x
1.19
2 = y log + 2x log
3 =y log - x log2
22
23
2
⎪⎩
⎪⎨
⎧
Sol.⎩⎨⎧
==
1y2x
1.20
4 =y log + x log
2 =y log - x log 2
32
32
⎪⎩
⎪⎨
⎧
Sol.⎩⎨⎧
==
9y4x
1.21
1 = x log -y log
3- =y - x 2
33⎪⎩
⎪⎨
⎧
Sol.⎩⎨⎧
==
9y3x
Xavier Rabasa 94
1.22
21 =
y logx log
4 =y log + x log 2
⎪⎪⎩
⎪⎪⎨
⎧
Sol.⎩⎨⎧
==
100y10x
RAONAMENT
21 =
y x
4 =y + x 2
loglog
loglog
⎪⎪⎩
⎪⎪⎨
⎧
→ ⎩⎨⎧
==
yxyx
2
2 10000 → x4=10000
→ ⎩⎨⎧
=−===
noyxsiyx
1001010010
