EJERCICIO 176

1. x + 6y =
27 ; 7x – 3y = 2. 3x –2 y =– ; 5x +8 y =–60 3. 3x +5y = 7 ; 2x – y = –4
9 2
x = 27 – 7x = 9 + 3x =–2 + 2 5x =–60 –8 5y = 7 – – y = –4 –2x
6y 3y y y 3x
9 + 3y 7 –3 x
x= 2 –2 y 60 y= y = 4 + 2x
+
x 8=–
y x =–
7 3 5
9 + 3y
27 – 6y = 7 – 3x
= 4 + 2x
5
2 –2 y 60 +8y
7 – 3 =– 5 5

7 (27 – 6y ) = 9 + 5(2 –2 y ) = 3(60 7 –3x =5(4 + 2x )

7 –3x = 20 +10x
3y +8y)
–13 =13x
189 – 42 y = 9 + 10 –10 y = 180 + 24 y –1= x
–170 = 34 y
3y 2 (–1)– y = – 4
–5 = y
180 = 45y
–2 – y = –4
4=y 3x –2 ( –5 ) =–2
– y = –2
x + 6(4) = 27 3x +10 =–2
x + 24 = 27 3x =–12
x=3
sol : x = 3 y = 4 x =–4 y=2
sol : x =–4 y =–5 sol : x = –1 y = 2
4. 7 x –4 y =5 ; 9x +8y =13
7x =5 +4 y 9x =13–8y
5+4 y 13 – 8 y 7. 15x –11y =–87
x= 7 x= 9
15x =11y –87
9 (5 + 4 y)= 7 (13 –8y)
11y –87
45 +36y =91–56y x = 15
92 y =46

46 1 –12x –5y =–27

y = 92 = 2
{ 1⎝ 12x = 27 –5y
9 x +8 | | =13 27 –5y
⎝ J2 x = 12
9x +4 =13 12 (11y –87)=15(27 –5y)
9x=9
132 y –1.044 = 405–75y
x =1
207 y =1.449
sol : x =1 1
y= 1.449
2 y = 207 =7
5. 9x +16y = 7 ; 4 y – 3x = 0 12x +5(7 ) =27
16y = 7 –9x 4y= 12x + 35= 27
7 – 9x 12x =–8
3x
y= 3x
16 y= 8 2
4 x =– =–
4 (7 –9x)=16(3x) 12 3
28 –36x = 48x sol: x =– y=7
2
28 =84x
3
28 1
x= =
4 y –1= 0 84 3

4 y =1
1 8. 7x + 9 y = 42
y=
4 9 y = 42 –7x
42 – 7 x
1 =
sol : x = 1 y
9
3 y =4
6. 14x –11y =–29 ; 13y –8x =30 12x +10y =–4
14x =11y –29 –8x =30 –
13y 10y =–4 –12x
11y – 29 x= 4 + 12 x
x = 14 13y –30
8 y =– 10
8(11y –29 ) =14 (13y – 10(42 –7x ) = –9 (4 +12x)

30) 420 –70x =–36 –

108x
88y –232 =182 y –420
38x = –456
188 =94 y
x =–12
2=y
13(2) –8x =30 12 (–12 )+10y =–4

26 –8x =30 –144 +10y =–4

10y =140
–8x =4
1 y =14
x =–
2 2
sol: x =–12
sol: x =– y =2
1
y =14
 EJERCICIO 177

y = 6x +85y = 5+ 24x
–18y = –6x –85–5y = –5– 24x
9. 6x –18y = –85 ;24x –5y = –5
1. x +3y =6 ; 5x –2 y =13

5 6x +85 =18 5+ 24x

6| | –18y = –85
sol : x = 5y =5

30x + 425= 90 + 432x

(
⎝ 6J
{5

x =6 –3y
y =5

5–18y = –85
–18y = – 90

335= 402x 5(6 –3y ) –2 y =13

x = 335 = 5
6

) (
402 6

18
30 –15y –2 y =13
–17 y =–17
y =1
)

x +3(1) =6
x +3 =6
x=3
5

y =1
sol : x = 3

4. x –5y =8 ; – 7x +8y = 25
2. 5x + 7 y = –1 ; –3x + 4 y = – 8y = 25+ 7x
5x = –1–7 y 24 25+ 7x
y= 8
1+
x y= –
7
5 { 25 + 7 x ⎝
x –5 | |
{ 1 + 7 y⎝ =8
3 | | +4y=– ⎝ 8 J
24
⎝ 5 J 8x –5(25+ 7x ) =
3+ 21y
+4y=– 6. 10x +18y =–11 ; 16x –9 y =–5
64
24
8x –125–35x =
64
5
– 27x 10x =–11–18y
3+ 21y + 20y = – 11+18y
=189
120 x = –7
41y = – x =–
–7 –5y =8 { 11 +18 y⎝ 10
123 –5y =15
y = –3
–3x + 4 (–3)= –24 y = –3
–16|
⎝ 10 J
| –9 y =–5

–3x –12 = – sol : x = – 7 y = – –16(11+18 y)–90y =–50

24 3
–3x = –
5. 15x +11y = 32 ; 7 y –9x =8 –176 –288y –90y =–50
12 7 y =8 + 9x –378y =126
x=4
sol : x = 4 y = –3
3. 3x +4 y =8 ; 8x –9 y =–77 8+ 126 1
9y x=
y =– =–
4 y =8 – 7 378 3
{ 8 + 9 x⎝ { 1⎝
3x 15x +11 | |= 10x +18 | – | =–
⎝
8– 11
3
y x= 32
⎝ 7 3
4
{ 8 – 3x⎝
8x –9 | | =– 105x +11(8 + 9x ) = 10x –6 =–11
10x =–5
77 224
105x
224 +88 + 99x =
⎝ 4 J
 1
32 x –9 (8 –3x ) =– 204x =136 x =–
2
136 1
308 2x = =
sol x =– y =–
{ 2⎝ 204 3 1 3
32x –72 + 27x =–308 :
59x =–236 2
x =–4 15| | +11y = 32
⎝3J
3 ( –4 ) + 4 y 10 +11y = 32

=8
–12 + 4 y =8 11y = 22
4 y =20 y=2
y =5 y=2
y =5 sol : 2x =
sol : x =–4
3
 8. 32x – 25y =13 ; 16x +15y =1
16x =1–15y
7. 4x +5y =5 ; – 4x –10y = – 1–15y
7
10y = 7 – 4x x=
16
9. 11x –13y =–163
y =7 –4x { 1 – 15 y⎝ ; –8x + 7 y = 94
| |– =
10 32 25y 13
⎝ 16 J 11x =–163+13y
{ 7 – 4 x⎝
4x +5 |
⎝
| (
2 1–15y –25y ) x =–
163–13y

=5 11
=13 { 163 –13y⎝
10 8 | 11 | +7 y = 94
⎝
8x + 7 – 4x 2 – 30y – 25y =13
=10 2 –55y =13
–55y =11
4x = 3
3
{ 3⎝
x=4 ( )
8 163–13y + 77y =1.034
y=–
4 | | +5y =5 1 1.304 –104 y + 77 y =1.034
{ 1⎝ –27y =–270
5
⎝ 4J
16x +15| – |
⎝ 5J y =10
3+5y =5 =1
5y = 2 16x –3=1 11x –13(10) =–163
2 16x = 4 11x –130 = –163
y=
5 11x =–33
3 2 1 x =–3
x =4
sol : x = y=
4 5 y =– sol : x =–3 y =10
sol : x=
1 1
4
5

EJERCICIO 178

1. 6x – (2) 3x – 4 y = 41 (3)
5y = –9 3. 5. 10x – 3y = 36
4x + 3y = 13 11x + 6y =47 (2)
(– 2x + 5y =–4 (–5)
3)
12x –10y =–18 9x –12 y =123 10x – 3y = 36
–12x – 9 y = –39 22x +12 y =94 –10x –25y = 20
–19 y =–57 31x = 217 –28y =56
y=3 x=7 y =–2
4x +3(3) =13 3(7) –4y = 41 10x –3 ( –2 ) =36
4x + 9 =13 21–4y =41
10x =30
4x = 4 –4 y = 20 x=3
x =1 y =–5 sol : x =3 y =–2
sol : x =1 y =3 sol: x =7 y =–5
2. 7x –15y =1 6. 11x – 9 y = 2 (5)
2. ( ) 4. 9x +11y =–14 (4) 13x – 15y =– (–3)
2
– x – 6y =8 7 6x –5y =–34 – ( 55x –45y =10
7x –15y = 1
36x +44 y =–56 6 ) – 39 x +45 y = 6
– 7 x – 42 y =56
– 36 x + 30 y = 204 16x =16
–57 y =57 74 y =148 x =1
y = –1 7x +15=1
7x –15(–1)=1 7x = –14
 y=2 y=2
11(1
6x –5 (2) =–34 11–9y =2
6x –10 =–34
) –9
–9 y = –9
6x =–24 y =1
x = –2 x =–4 sol: x =1 y =1
sol : x = –2 y = –1 sol : x =–4 y =2
 7. 18x + 5y = –11
( ) 9. 12x –14 y = 20 (7 11. 36x –11y = –14 (2)
12x + 11y = 31 2
– ( –14x +12 y = –19 ) 24x –17 y = 10 (–3)
3 ) (6
)
84x –98y =140
36x +10y = –22 72x –22 y = –28
–84x +72 y =–114
–36x –33y = –93 – 72 x +51y = –30
–26y = 26
– 23y = – 29 y = –58
y =–1
115 ( )
12x –14 –1 = 20 y =–2
y =5 12x +14 = 20 36x –11(–2)=–14
18x +5(5) = –11 12x = 6 36x + 22 = –14
1
18x + 25 = –11 x= 36x =–36
18x = – 36 x = –1
x=–2
sol : 2
x=– y =5 y =–1 sol: x = – y =–2 ( )
sol: 1
x=
1
2
8. 9x + 7 y =–4 (–11) 2
12. 12x –17 y =104 5
11x – 13y = –48 (9 10. 15x – y = 40 (8 15x +19 y =– (–4)
) ) 31

–99x – 77 y = 44 19x +8y = 236

60x –85y =520
99x –117 y = –432 120x –8y = 320
–60x –76y =124
–194 y = –388 19x +8y = 236
–161y = 644
y=2 139x =556 y =–4
9x + 7 (2) = – 4 x=4
12 x –17 (–4 ) =104
9x +14 = –4 15(4) – y = 40
12 x + 68 =104
9x = –18 60 – y = 40
12x = 36
x = –2 – y =–20 x=3
sol : x = –2 y = y = 20 sol : x =3 y =–4
2
sol: x = 4 y=
20

EJERCICIO 179
1. 8x –5 =7 y –9 2. x –1= y +1 3. 3(x + 2)= 2 2 (y +5)=
8x –7 y =–9 +5 y 7x
8x –7 y =–4
x–y=2 3x + 6 = 2 2 y +10 =
6x =3y +6 x –3= 3y –7 y 7x
6x –3y =6 x–y=2 3x = –6 + 2 y 7x =10 + 2 y
8x – 7 y =–4 (3 x – 3y = – (– 6–2y 10 + 2 y
x=– x=
) 4 1)
3 7
6x – 3y = 6 (–4) x– y=2
–9 y =–36 6x –12 =6
24x –21y =–12 y =4 6x =18
–24x +12 y =–24 x =3
6x –3 ( 4) =6
 +14 y = 30 + 6y
– x + 3y = 4 –7 (6 – 2 y)= 3(10
2y=6 8y = 72
+ 2 y) y=9
y=3
– 7x –2 (9) =10
x –3= 2
4 7x –18 =10
x =5
2 7x = 28
sol : x =5 y = 3
x=4
sol: x =3 y =4 sol : x = 4 y = 9
4. (
x –1= 2 y + 6. Continuación 8. Continuación
{ 30 ⎝
–5y =–40 –2x + 7 | – | = –6
6 )
x –1= 2 y
+12
x –2 y y =8 ⎝ 89J
=13
x =13+ 2 y –178x –210 =–534
x –2 (8) =–
x + 6 = 3 1–2 ( 10
–178x = –324

y) x –16 =–10 x=
324
=1
73

x + 6 = 3–6y x =6
89
x + 6y =–3 178
= =
x =–3–6y sol : x 6 y 8 30
13+ 2 y =–3–6y sol: 89 y =–
x =1 89
73

–3–6y –2 y =13 ( )( ) (
7. x – y – 6x +8y = – 10x +5y +
( )
9. 2 x +5 = 4 y –4x ( )
–8y =16
y =–2 3) 2x +10 = 4 y –16x
x – y – 6x –8y = –10x –5y – 18x –4 y = –10
( )
x –2 –2 =13
3 10( y – x)=11y –12x
x + 4 =13 –5x – 9 y = –10x –5y
x=9 – 3 5x – 4 y = – 3 10y –10x =11y –12x
2x – y = 0
sol : x = 9 y =–2 (x + y)–(9 y –11x)= 2 y – 2x
18x –4 y = –10
(
5. 30 – 8 – x = 2 y + ) x + y – 9 y +11x = 2 y – 2x
12x –8y = 2 y –2x 2 x – y = 0 (–9)
30 18x –4 y = –10
14x –10y = 0
30 –8 + x = 2 y +
30 5x – 4 y = – 3 (5)
x –2 y =8
14x –10y = 0 (– 2) –18x + 9 y = 0
5x –29 = x – 5 –4 ( 25x – 20y = –15 5y = –10
y)
5x –29 = x –5 + 4 – 28x + 20y = 0 y = –2
y
4x –4 y = 24 = –15
– 3x 2x – ( –2 ) = 0
x–y=6 x =5 2x + 2 = 0
x –2 y =8 5(5) – 4 y = – 3 2x = –2
x–y=6 ( – 25– 4 y = – 3 x = –1

1 )
x –2 y =8 –4 y = – sol : x = –1 y = –2
– x + y =–6 28
y=7

–y=2 y) sol : x =5 y = 7
y =–2 3x – 9x – y
=5y – 2x –
( )
x –2 –2 =8 ( )(
8. 5 x + 3y – 7x +8y = –6 )
9y
x +4 =8 – 4x + 5x +15y –7x –8y = –6
x=4 3y = 0 –2x + 7 y = –6
sol : x = 4 y =–2
7x –9 y –2 ( x –18y) = 0
(
6. 3x – 9x + y =5y – 2x + 9) ( 7x –9 y –2x + 36y = 0
 5x + 27 y = 0 10. 3x – 4 y – 2 –4 y = –14
x + 4 y =14
5 ( x –1)–(2 y –1)= 0
(2x – 7)= 0
5x –5 –2 y +1= 0
3x – 4 y –
5x – 2 y = 4
4x +14 = 0
x + 4 y =14
–x
5x – 2 y = 4 (2)
–2x + 7 y = –6 x + 4 y =14
4x –(3y + 7 ) =5y – 10x –4 y =8
47 (5)
4x – 3y – 7 =5y – 5x + 27 y = 0 (2)
47
4x –8y = – –10x + 35y = – 11x = 22
40 30
x–2y=– 10x +54 y = 0 x=2
10 89 y = – 2 + 4 y =14
–4x + 3y = 0 30
x –2 y = – (4) 30 4y
y = –89 =12
10 y=3
–4x + 3y = 0 Continúa Continúa y=3
4x –8y = – 40 sol : x =
2
 ( ) (
11. 12 x + 2 y –8 2 x + y =2 5x – ) (
( ) (
12. x y –2 – y x –3 =–14 )
6y ) xy –2x – xy +3y =–14
12x +24 y –16x –8y =10x –12 y –2x +3y =–14
–4x +16y =10x –12 y
y ( x –6 ) – x ( y + 9 ) =54
–14x +28y =0
xy –6y – xy –9x =54
– x +2 y =0
20 (x –4 y ) =–10 –9x –6y =54
20x –80y =–10 3x +2 y =–18
2x –8y =–1
– x + 2 y = 0 (2) –2x +3y =–14 (3)
2x –8y =–1 3x +2 y =–18 (2)
–2x +4 y = 0 –6x +9 y =–42
6x +4 y =–
2x –8y =–1 36
–4 y =–1 13y =–78
1 y =–6
y=
4
{ 1⎝ –2x +3 ( –6 ) =–14
– x +2 | | =0
⎝ J4 –2x –18 =–14
–4 x + 2 =0 –2x =4
x =–2
–4x =–2 sol: x =–2 y =–6
1
x=
2
sol : 1
x= y=
1 4
2

EJERCICIO
5x 3y
180 2. – y =9 ; x– =15

3x
1. +y =11 12 4
2
3x +2 y =22 5x –12 y =108 ; 4x –3y =60
y
x + =7 5x =108 +12 y
2 x 7y
108 +12 y
2x
3x +2y y=14
+ =22 ; x= 5 =5+ ; 3y –
x
=26
{ 108 +12 y⎝ 3. 3 14
2x + y =14
= – = –
4| | –3y 3x + 7 y =105 ; 42 y – x = 364
⎝
=60 5 J
3x 22 2 y 2x 14 y 4 (108 +12 y) –15y 3x + 7 y =105
=300
22 – 2 y 14 – – x + 42 y = 364 (3)
yx = x= + – =
3 2 432 48 y 15y 300 3x + 7 =105
y
2x +2 =14
2 (22 –2 y)=3(14 – 33y =–132
2 x =12 y =–4
y) x =6
4 x –3(–4)=60
44 –4 y =42 –3y sol: x =6 y =2
4x +12 = 60
– y =–2
4x =48
y =2
x =12
sol : x =12 y =–4 3x + 7 (9) =105
– 3x +126y 3x + 63=105
=1.092 3x = 42
133y =1.197 x =14
y=9
sol: x =14 y = 9
 x y y x
= ; = –1 x y x y –59 9. Continuación
11
4. 7. – =– ; + =
5 4 3 3 8 5 10 5 4 40 8x –5(4) = –4
4x =5y y=x– 5x –8y = –44 ; 8x +10y = –59 8x –20 = –4
3 5x –8y = –44 (5) 8x =16
4x –5y = 0 x–y=3 8x +10 y = –59 (4) x=2
4x –5y = 0
x – y = 3 (–4)
4x –5y = 0 25x –40y =– sol : x = 2 y = 4
–4x + 4 y = –12 220
– y = –12
y =12 32x + 40y = – 10. 12x +5y + 6 = 0
x –12 = 3 12 x +5y = –6
236
57x = –456 5x 7y
– = –12
x = –8
5(–8)–8y = –44
x =15 –40 –8y = –44 3 6
sol : x =15 –8y = –4 10x – 7 y = – 72
y =12
1 12x + 5y = –6 (5)
y=2
3x 1 5y 10x – 7 y = –72 (– 6)
5. – y=2 ; 2x = 1
5 4 2 sol : x = –8 y= 60x + 25y = –30
12x –5y =40 4x =5y 2
–60x + 42 y = 432
4x –5y = 0
67 y = 402
4x –5y = 0 x y 1 3y y=6
+ = ; 7x – 4 = 7
8. 07 8
12x –5y = 40
(– 8x + 7 y = 0 4x –21y 10x – 7 (6) = – 72
1) ; =196
4x –5y = 0 8x = –7 y 4x =196 + 10x – 42 = –72
21y
–12x +5y =–40 –7y 196 + 10x = – 30
21y
x= x=
–8x =–40 8 4 x=–3

x =5 –28 y =8(196 + sol : x = –3 y = 6

4(5) –5y =0
21y)
20 –5y = 0
–5y = –20 –28 y =1.568 x
+168y
5
–196y =1.568
y = –8

y=4 – 7(–8) = 3(y + 2)

11.
x=
sol: x =5 y= 8
4 x =15 ( y + 2)
x=7 x =15y + 30
2x 3y sol : x = 7 y = –8
– = 1 5x y 224
– =
6. 1 ; y 2 + 3x =
3 4 8 6 5 5
8x –9 y =12 ; 3y –20x =48 2x +1 y ; 2 x –3y =–8 y +15x = 224
8x –9 y =12 9. 5 = 4 15x = 224 – y
–20x + 3y =48
(3 ) 4 (2x +1)=5y
 8x –9 y = 12 x=
8x + 4 =5y 15

– 60 x + 9 y =144 8x –5y =–4 15(15y + 30)= 224 – y

–52x =156 2x –3y =–
(–4)
225y + 450 = 224 –
x =–3 8 y
8x –5y =– 226y = –226
4
–8x +12 y = 32 y = –1
8 ( –3 ) –9 y =12
–24 –9 y =12 8x – 5y =– 4 x =15(–1)+ 30
–9 y =36 7 y =28 x = –15+ 30
y =–4 y =4 x =15
sol : x =–3 y =–4 sol : x =15 y = –1
Continúa
12. x y 1 x y 13 + – – –
5 – 6 =– 30 ; 3 – 20=12 15. x 1 y 4 x4 y2
6x –5y = – ; 20x –3y = 65 10 = 5 ; 5 = 10
1
6x – 5y = –1 (3) x +1= 2 (y –4) ; 2 (x –4)= y –2
20x – 3y = 65 (–5) x +1= 2 y –8 ; 2x –8 = y –2
18x –15y = –3 x – 2 y = –9 2x – y = 6
–100x +15y =–325 x = 2 y –9

–82x =–328 2 (2 y –9 ) – y = 6
x=4 4 y –18 – y = 6
6(4 ) –5y =–1 3y = 24
–5y =–25 y =8
y =5
x = 2 (8) –9
sol : x =4 y =5 x=7
sol : x = 7 y =8
x –3 y –4 x –4 y+2 3y + 3 1+5x
13. – =0; + = 16. x=– y=–
3 4 2 5 4 4
( ) (
4 x –3 –3 y –4 = ) ( )
; 5 x –4 + 2 y + 2 = ( ) 4x = –3y –3 ; 4 y = –1–5x
0 30
4x –12 –3y +12 = ; 5x –20 + 2y + 4 = 4x + 3y = – 3 4 y +5x = –1
0 30
4x –3y = 0 5x + 2 y = 4x + 3y = –3 (4)
46
3y
5x + 4 y = –1 (–3)
{ 3y⎝
x= 4 16x +12 y = –12
5 | | + 2y = 46
⎝4 J –15x –12 y = 3
15y +8y =184
23y =184 x = –9
y =8 4 (–9 ) + 3y = – 3
x (=8)
3
–36 + 3y = –3
4 3y = 33
x =6 y =11
sol : x = y =8
sol : x = – y =11
6 9
x –1 y –1 13 x +1 y +1 2x
14. 2 x+y x–y =y+3
– =– – =– 17. = ;
2 3 36 3 2 6 12 3
3
( ) ( )
18 x –1 –12 y –1 =–13 ; 2x + 2 –3y –3= –4 (
2 x+y =x–y ;) (
2x = 3 y + 3 )
18x –12y =–7 2x –3y = – 2x + 2 y = x – y ; 2x = 3y + 9
3 x + 3y = 0 ; 2x –3y = 9
18x –12y = –7

2 x – 3y = –3
(–4) x =–3y
18x –12y = –7 (
2 – 3y –3y =)
–8x +12y =12 9
10x –6y –3y = 9
=5
1 –9 y = 9
x=
2 y =–1
 { 1⎝
2 | | –3y = –
⎝J
( )
x =–3 –1
3 x=3
2 y =–1
sol: x = 3
–3y = –4
4
y=
3

sol : 4
x= y=
1 3
2
 y –3
18. 3x – =6
5
x – – xy – ( ) ( )
15x – y +3 20. = x –7 22. y x –4 = x y –6
2
=30
4 2
15x – y = 27 xy –4 y = xy –6x
15x = 27 + y x –2 –2 ( y – x ) =4 (x – 6x –4 y =0
27 + y 5 11
= 7) – =0
3x –4x –2 y =–28 +2
x
15 x +2 y = 26 x –3 y –1
x –2
3y – =9 3x – y 3y – x
– =y– 5(y –1)–11(x –3)=0
7 8 5y –5 –11x +33 = 0
6
21y – x +2 3(3x – y ) –4 (3y – x ) = 24 ( y 11x –5y =28
=63 –13)
– x + 21y = 61 13x –15y –24 y =–312
– x = 61–21y 13x –39 y =–312 6x – 4 y = 0 (5)
x =21y –61
x + 2y = 26 (–13) 11x – 5y = 28 (–4)
27 + y =15(21y – 13x –39 y =– 312
30x –20y = 0
– 44 x + 20 y =– 112
61) –13x –26y =–338
–14x =–112
27 + y = 315y –915 13x –39 y =–312
x =8
–314 y =–942 –65y =–650
11(8) –5y = 28
y =3 y =10
–5y =28 –88
27 +3 x +2 (10) =26
x= –5y =–60
15
x =2
sol : x = 2 y= x +20 =26 y =12
3
x=6 sol : x =8 y =12
19. x+y y–x
7 – = sol : x = 6 y =10
6 3 24
4 ( x + y ) –8 ( y –
3 ( x + 3y )
21 5x + 6y = 17
x ) =7 3x –2 y
23.

4 x +4 y –8y +8x =7
12x –4 y =7
xx–y5 21. 12 – =3y +2 51(x + 3y)= 21(5x + 6y)
6
2
+
6
=
12 72 –3x + 2 y =6 ( 3y 51x +153y =105x +126y
+2)
4
6x +2 ( x – –4 y =–2
1
12
⎝J y=
y ) =5 {
|
6x +2x –2 y =5 3 ⎝
|
8x –2 y =5 (–2)
12x –4 y = 7 –4
8x – 2 y = 5
12x –4 y = 7 y
–16x +4 y =–10 =7
–4x =–3 4
3 9
x=
 –3x +2 y –18y =12 –72 6 –54x + 27 y =0
3x +16y =60 ( =0 2x = 2
5y – 3 x 2x – y = 0 x =1
=x 4
–y 4x –7 y
3 ) =–2
2 y +1
5y –3x
4x –7 y = –2 (2 y +1)
=3 ( x – y) –
4x –7 y = –4 y – 2
–3x –3x +5y +3y =0 8
4x –3y = –2
6x –8 y =0 y
3x +16y =60
2x – y = 0 (–2)
4x –3y = 2
6x – 8 y = 0 (2 ) =
–4x + 2 y = 0
3x +16y =60 0 4x – 3y = –2
12x –16y = 0 –
y=2
15x =60
2x –2
x =4
2
y =3 sol : x =1 y=2
sol: x=
3
4 1y = sol : x =4 y =3
2
 28. x – y –1 3
=–
7 7 x + y +1 17
=– x 3y 33
–= –
24.
2x –3y + 6 3x – 2 y – 26. 8 17 (x – y –1)= – 3(x + y +1)
4 2 4
1
7 (3x – 2 y –1)= – 7 (2x – 3y + x –32 =6y –33 17x –17 y –17 = –3x –3y – 3
x –6y =–1 20x –14 y =14
6) y – x 2x + y
17 – =– 10x –7 y = 7
21x –14 y – 7 = –14x + 21y –
42
35x – 35y = – 35
x – y = –1 3 2 24 x + y –1
= –15
6 10 8 ( y – x)–12 (2x + y ) =– x – y +1
=
x–y+4 y+2
17 x + y –1= –15 (x – y +1)
6 ( y + 2 ) =10 ( x – y + x + y –1= –15x +15y –15
8y –8x –24x –12 y =–17
4)
32x +4 y =17

6y +12 =10x –10y + 40 x – 6 y =–1 (2

32x + 4 y =17 16x –14 y = –14
–28 =10x –16y ) 8x – 7 y = –7
2 x –12 y =–2
–14 =5x –8y (3 10x – 7 y = 7
96x +12 y =51
x – y = – 1 (–8) ) 8x –7 y = – 7 (–1)
5x –8y = –14 98x =49 10x – 7 y = 7
1
–8x +8y = 8 x= –8x + 7 y = 7
2
5x –8y = –14 1 2x =14
– 3x =–6 –6y =–1 x=7
2
x=2 1–12 y =–2 10(7) – 7 y = 7
2–y=– –12 y =–3 – 7 y = –63
1
1 = y=9
–y=–
3
y=3 y sol x = 7 y = 9
4
:
sol : x = y= 6x + 9 y –4 2
sol: 1x = 1y =
2 3 29. 4x – 6y + 5 =5
2 4
x+y
x –2 y –7 ( ) (
5 6x + 9y –4 = 2 4x –6y +5 )
25. = –7 = 30x + 45y –20=8x –12 y +10
x–y 27. x + 2 y –5
22x +57 y = 30
x + y = – 7 (x – (y –5)(x –2)=(y –7)(x 2x + 3y –3 6
=
y) +2)
xy –2 y –5x +10 = xy + 2 y –7x – 3x + 2 y –4 11
x + y = – 7x + 7 y
8x –6y = 0 14 ( ) (
11 2x + 3y –3 = 6 3x + 2 y –4 )
x + y +1 3 2x –4 y =–24 22x + 33y –33=18x +12 y –24
x + y –1 =4
4 ( x + y +1)= 3(x + y x –2 y =–12 4x + 21y = 9
–1) x =–12 +2 y 22x +57 y = 30 (2)
x +1 y –3
x –1 = y –5 4x + 21y = 9 (–11)
4x + 4 y + 4 = 3x + 3y
–3 8x –6y = 0 (y –5)(x +1)=(x –1)(y –3)
x+y=–7 xy + y –5x –5= xy –3x – y +3
–2x +2 y =8
44x +114y = 60 –44x –231y = –99 –117y =–39
x+y =– (6 ) 1
7
–x+y=4 y=
8x –6y = 0 { 1⎝
3
x=y– 4x + 21 | | = 9
6x + 6y = – ⎝ J3
4
42 –12 +2 y = y –4
14x = – 42 y =8 4x + 7 = 9
x = –3 4x = 2
x =8 –4
– 3+ y = – x =4 1
x=
7 2
y=–4 sol : x =4 y =8
1
sol : x = – y = –4 sol : x = y=
3 1 3
2
 3x + 2 y 3x + 4 y
30
EJERICIO 181
=–
=–
30. x + y –15 32. x –6y 23
9

1. x +y =a +b ; x –y =a –b
3x + 2 y = – 9 ( x + y – 23(3x + 4 y ) =–30 ( x –6y ) x +y =a +b
69x + 92 y = –30x x –y =a –b
15) +180y x =a
3x + 2 y = – 9x – 9 y 99x –88y = 0 a +y =a +b
+135 9x –8y = 0
12x +11y =135 9x – y 63

4x 5(y –1)
3 – 8 = –1 =–
3+ x – y y =b
37
32x –15(y –1)= – sol: x =a y =b
37 (9x – y)= –63(3+ x – y)
24 333x –37 y = –189 – 63x + 2. 2x + y =b +2
32x –15y +15= – 24 ( 63y 2
2bx +by =b +2b
8
32x –15y = – 39 396x –100y = –189
) 396x –100y = –189
12x +11y = 135
bx – y =0
32x –15y = – 39 (– 9x – 8y = 0 (–44)
96x +88y =1.080 –2bx +2 y =0
3) 396x –100y = –189
–2bx +2 y =0
– 96x + 45y =117
–396x + 352 y = 0 2
133y =1.197 2bx +by =b +2b
2
y=9 2 y +by =b +2b
252 y =–189 y (2 +b ) =b (b
12x +11(9) =135 –3 +2) y =b
12x =135– y=
4 2 x +b =b +2
{ – 3⎝
99 9x –8 | | =0
⎝ 4
12x = 36
x=3 9x + 6 = 0 2x=2
sol : x = y=9 9x = –6 x =1
3
–2 sol : x =1 y =b
x= 3
2 x +5
–( 5– y =–60 –2 –3
31. 17 sol : x = 3 y = 4 3. 2x –y =3a ;
2x +5–17()5– y)=–1.020 x –2y =0
2x –85+17 y =–1.025 33. –
4 x +1 2 y –2x +4y =0
x 5– =
2x +17 y =–940 9 3 2x – y =3a
+ ( ) (
9x – 4x +1 = 3 2y –5 ) –2x +4y = 0
y 62
–(1– x ) =40 9x –4x –1= 6y –15
3y =3a
2 5x –6y =–14 y =a
y +62 –2(1– 3y + 2 x
+
y 18
– =
x)=80
y –2 + 2x =18 y 2x –60 =20
2x =80
2x + y =20
2x +17 y =–940 =
2 x + y = 20
(–1)
–
2x +17 y =–940
6
–2x – y = –20
0
16y = –960
 7 10 2x 4a x
– =2a
( ) (
70y –10 3y +2 = 7 x +18 ) a
70y –30y –20 = 7x +126
sol: x =2a y =a
–146 =7x –40y 5
=
3 4. x –y =1–a ; x +y =1+a
6
7x – 40y = –146 –5 ( ) a x –y =1–a
5x – 6 y = –14 (7 ) 2 x +y =1+a
–35x + 200y = 730 35x – x 2x =2
42y =–98
= x =1
5x =–14 +24 1+y =1+a
x=
y=– x=2 y =a
sol: 40 y =a
60 sol: x =2 y = 4 sol: x
x =40
=1
 x a +b
5. + y =2b x 8. ax –by = 0 ; x+y=
; – y = a –b ab
a+ = b 2
– = – abx –b y = 0 2
abx + aby = a +b
x ay 2ab ; x by ab b
x +ay = 2ab abx –b2 y =0
2
–x +by =b –ab –abx – aby = –a –b
ay +by =b 2 +
ab ( )
–b2 y –aby = – a +b
y (a +b ) =b (b + – yb (b + a ) = –(a +b)
a)
y =b 1
x + ab =2ab – yb = –1 ‹ y =
{ 1⎝ b
x = ab ax –b | | = 0
sol : x = ab y =b ⎝ Jb
1
ax =1 ‹ x =
x y 1 1 a
6.
b +a = 2
sol : x = y=
ax +by = 2ab a b
2 2
abx +b y = 2ab 9. mx –ny =m +n
2 2
; nx +my = m +n
2 2

x y a2 +b2 2 2 3
+ = mnx –n y =m n + n mnx +m 2 y = m3 + mn2
a b ab
bx + ay = a2 –mnx +n 2 y = –m2n –n3
+b2 mnx +m y = m3
2
+mn2
–abx – a2 y = –a 3 –
ab2
2
abx +b y = 2ab2 n2 y +m2 y = m3 –m2n +mn2 –n3
–abx
– a 2 y = –a 3 –ab2 y (m2 + n 2 ) =m2 (m –n) + n 2 (m –n)
b2 y – a 2 y = – a 3 + y (m2 + n 2 ) = (m2 + n 2 )(m –
ab2 n) y =m –n
(
y b2 –a2 = a b2 – ) ( m2 x
–mny = m3 +mn2

a2 )
y=a
ax +ba = 2ab n2 x +mny m2n +n3
=
(
a x +b = ) m2 x +n2 x =m 3 +mn 2 +m 2n + n3
2ab
x +b = 2b
x==b = x (m2 +n 2 ) = m (m2 +n 2 ) +n (m2 + n 2 )
x (m2 +n 2 ) = (m +n)(m2 +n 2 )
sol : x b ya
x=m+
x + y = a +b
7. sol : n x =m + ny =m –
n
–ax –ay =–a2 –ab x y
ax +by = a2 +b2 + =2m ; mx –ny = m3 –mn2
m n
–ax –ay =–a2 –ab
ax +by
= a2 +b2 xn + ym =2m2n xmn –n 2 y = m3n –mn3
xmn + ym2 = 2m3n
by –ay = b2 –ab +b =a +b x =a
y (b –a ) =b (b –a )
x
– xmn + ( ) (
y m2 + n2 = mn n2
n
2 +m2 )
y

m
n
3

–
m
3

y
m
2

y
n
2

m
n
3

m
3

n
y= m n
sol : x=a y =b
2 2
xn + m n = 2m n
x = m2
xn = m2n
‹
sol: x =m2 y =mn
 14. x + y = 2c ; a 2 (x – y ) = 2a3
x + y =a 2 2 2
–a x –a y = – 2a c ; a 2 x –a 2 y = 2a3
2 2 2 2
ax + ay =a –a x –a y = –2a c
2 2 3
a x –a y = 2a
ax –by = a (a +b) 2 3 2
– 2a y = 2a – 2a c
+b 2
2 2
ax –by = a +ab +b
2
–ax –ay =–a – 2a2 y = 2a2 (a – c)
ax –by = a + ab +b
2 2
–y=a–
2 cy = c –
–ay –by = ab + b a
– y (a +b)=b(a +b) x + c – a = 2c
– y =b x –a = c ‹ x = c + a
y = –b sol : x = a + c y=c–a
x –b =a a2 –b2
15. ax – by = 0 ; ay – bx =
x = a +b b2ax –b 3 y = 0 a2by –b2ax =a 2 ab
–b 2
sol: x = a +b y =–b
b2ax –b3 y = 0
–b 2ax + a2by =a 2 –b 2

x – y = m –n a 2by –b 3 y = a 2 –b 2
mx –my =m2 –mn (
by a 2 –b2 = a2 –b2 )
2 2
mx –ny = m –n by =1
2
1
–mx + my =–m +mn y=
my –ny = mn –n 2 b

(
y m –n =n m – ) ( { 1⎝
ax –b | | = 0
n y =n ) ⎝ Jb
x –n =m –n
x=m ax =1
1
x=
a
sol : x =m y =n
sol : 1 1
x =a y =b
16. x y
+ = a +b ; (
x – y = ab b –a )
2 2
b a
x y 2 2 a 2 x +b2 y = a3b2 + a2b3
+ =0 ; x 2 y 2b –a x – y = ab2 – a2b
a b + ab ; a x – a2 y = a3b2 – a4b
2
=
b a
bx + ay = ax + 2by = 2b2 – –a 2 x –b 2 = –a3b2 –a2b3
2
0 a y
–abx –a2 y = 0
abx + 2b2 y a2 x –a 2 y a3b2 –a4b
=2b3 –a2b
=
2b a
2 2
– =b 3– 2 –b y –a 2 y =
2
–a 2 b 3 – a4b
2b y a y
(
y 2b2 –a2 =b 2b2 –a2 ) ( ( )
– y a2 +b2 = –a2b b2 + a2 (
) )
– y = –a2b
y =b
x b y = a2b
 + =0
a b x a2b = a +b
+
bx + ab = 0 b2 a2
bx = – x +b3 = ab2 +b3
ab x = x = ab2
–a
sol: x =–a y =b sol : x = y = a2b
ab2
 y –b a +b
x a+b
m3 – n3 19. + b = b
17. mx –ny =
nx + my = m + n ; mn ( ) (
b x +b +a y –b =a a +b ) ( )
2 3 3 2 2 2 3
m nx + m y = m + m n m nx – mn y = m – bx +b2 +ay –ab =a2 +ab
3
n
– m2nx –m3 = – m3 – abx +ab2 + a2 y –a2b =a3 +a2b
2
y mn
2
m nx – mn2 y = m3 – –abx –ab2 –a2 y =–a3 –2a2b
n3
3
–m y –mn y = 2 2
–m n– – y– a +b
x– a
n3 a =– a
(
– ym m2 + n2 = –n m2 + n2 ) ( a
b
)
( ) (
a x –a –b y –a =–b a ) (
ym = n
)
+b ax –a 2 –by +ba =–
n ba –b 2
y=
{ n⎝ m abx –a2b –b2 y +b2a =–b2a –b3
nx + m | | = m + n abx –a2b –b2 y =–2b2a –b3
⎝ mJ –abx –ab2 –a 2 =–a3 –2a2b
y
nx + n = m + abx –a2b –b 2 y = –2b2 a –b3
n
nx = m
–ab2 –a2b –a2 y –b2 y =–a3 –2a2b –2b2a –b3
x= ab2 +a 2 b +a 3 +b 3 =a 2 y +b 2 y
m ( )
b2 a +b + a2 b + a = y a2 ( ) (
sol : n n +b2 )(
b2 + a2 a +b = y a2 )( ) (
m =m
a +b = y
+b2 )
x=
yn

(
18. a –b ) x –(a +b) y =b 2
– x a +b – a b+b
+b +
b b
=
3ab a
2
ax –bx –ay –by =b –
3ab
(a +b) x –(a –b) y =ab ( )
b x +b + a2 =a a +b ( )
–b
2
bx +b2 +a 2 =a 2 +ab
2
ax +bx –ay +by = ab –b
2
–ax +bx + ay +by = –b + 3ab
2
ax +bx –ay +by = –b +ab bx =ab – ‹ bx =b a –b ( )‹ x =a –b
b2
sol: x = a –b y =a +b
2
2bx + 2by = –2b x y 1 x y
2
a +b
2

+4ab 20. + = ; + =
+ + b a a2b2
2b(x + y)=2b(2a –b) a b a b ab
2 2 2 2
x + y = 2a –b abx + aby = a +b a bx + ab y = a +b
ax –bx –ay –by b2 –3ab 2 2
– a bx – a by =– a – ab
2

=
2
ax +bx –ay +by = –b + ab 2
– a bx – =– – ab
2 2
a by a
2ax 2
a bx + ab
2
y
–2ay = – 2ab
 2 2
= a +b
2a (x – y)=–2ab 2 2
ab y – a by =
2
b – ab
ayb (b – a ) =b (b
x – y =–b – a) ayb =b
x + y = 2a –b
x – y =–b ay =1
1
2x =2 (a – y=
b) x =a – { 1⎝ a
abx + ab | | = a
b ⎝J
+b a
a –b – y =–b abx +b = a +b 1
– y =–a x =b
y =a abx = a ‹ bx =1
‹
sol: 1
x = a –b y =a sol : x = y=
1 a
b
EJERCICIO 182
1 2 12 5
13 18 7 19 6. Continuación
4 7 2 1
4. =– ;
+ + =–
1. + = ; += x y 2 x y 2 4 11
x y 6 x y 36 15 39 + =50
23 4 – – = x 1
7
– – =– x y 2 4
x y 3 4
36 14 +44 =50
2 1 4 38 + =–
x
+ = x y 2
x y 3 1 1 4 +44x =50x
–3 – =
=–1 y 2 4 =6x
y –2 =y 2
=x
3 =y 36 –7x =–19x 3
1
2 1 4 12x =–36 sol: x = y=
+ = 2
3 4
x 3 3 x =–3
6 +x =4x sol: x =–3 y =–2 9 10 7 15
6 =3x ‹ 2 =x 9 3 5 4 7. + =–11; – =–4
5. + =27 ; + =22 x y x y
sol: x =2 y =3

3 2 1 2 5 23 x y x y 27 30
2. – = ; + = 3 1 + =–33
+ =9 x y
x y 2 x y 12 x y 14 30
6 4 – =–8
– + =–1 12 4 x y
– – =–
x y 41
36 =–41
6 15 23 x
+ = x y
5 4
+ = 22
x y 4 x y –1=x
19 19 7 7 15
= – =– – – =–4
y 4 14
x 1 y
4 =y 1 –7y –15 =–4y
3 2 1 –15 =3y
x – 4 =2 =x ‹ 10 + 4 –5 =y
5 4 2 =22
+ =22
6 –x = x 1 y
sol: x =–1 y =–5
6 =2x y
2
3 =x 10y +4 =22y 1 3 3
8. – = ; 1 5 4
sol: x y =4 4 =12y 2x y 4 x + 2 y=– 3
1
7 6 =y 1 3 3
=3 – =
5 4 3
3. + =7 ; – =4 2x y 4
x y x y 1 1
15 12 sol: x = 2 y = 3 1 5 2
+ =21 – – =
x y 6 8 4 11 2x 4y 3
14 12 6. – =–23 ; + =50 17 17
– =8 x yy x – =
x y 4 y 12
12 16
29 – + = 46 –12 = 4 y
=29
x x y
1=x 12 33 –3= y
+ =150 1 5 4
x y + =–
5 4 x 2 (–3) 3
+ =7 49
1 y =196 6 –5x = –8x
 y
5y +4 =7y 1 3x = –6
4 =2y =y x =–2
2 =y 4
Continúa sol: x = –2 y =–3
sol: x =1 y =2
 2 1 11
9. – =– ;
1 3 4 3 7 2 1 8 103 3 1 107 6 1 14
5x 3y – = – = ; + = 11. + = ; + =
45 10x 5y 5 10. x 3 4x y 84 10x 3y 60 5x 4y 5
3y
2 1 11 3 7 1 3 2 107
– 10x
+ = 6y 90 + =
– =
2 6 8 4x 12 y 6 5x 3y 30
– =
10x 5y 5 3 24 309 3
1 7
– – =– – – =–
31 31 4x y 84 5x 8 y 5
– 30y
= 18
–295 –295 13 13
= =
–18 12 y 84 24 y 6
=30y 1
3 y=7 4 =y
– =y 3 7 2
5
=
1
+
3
=
4
– () 6 1+ = 14
10x { 3⎝ 5 x 3 7 3 5x { 1⎝ 5
5 | |
⎝ 5J 9 – x = 2x 4| |
⎝4J
3+ 30x = 24x
9 = 3x 6 +5x =14x
1
6x =–3 ‹ x =–
2 3= x 6 = 9x
sol : x = y=7
sol : x =– y =– 2
1 3 3 3 =x
2 5
12. 2 1
1 1 1 1 2 3b 2 –3a a sol: x = y=
+ =a ; – b
– =
x y
y x 13. x y ; + =2 3 4
x y
a
1 1 2a 3ab 2 2 m+n m n
x+ y=a + = ; – 0=
– =2– 14. x mn x y
3a
y
1 1 x y
– + = –b m+n
x y 2b 2n 2n m
2 2 2a = –4 x + y =
= a –b ‹ =y – y 2m 2n
–
x
y a– – =0
1 1 b (–3a – =– 3a – x y
– =b b
2
2)
y
x 2 2(m + n) m+n
a –b b=y =
(
2 – x a –b = 2bx ) x m
2 –ax + xb = 2bx a b
x +b = 2 2m = x
– ax =bx – 2 m n
–ax –bx = – 2 a + x = 2x – =0
2m y
2 a=x
(
x a +b = 2 ‹ x =+a ) y –2n = 0
sol : x = a y =b
b y = 2n
2
sol : x = 2 sol: x = 2m y = 2n
a +b y = a –b

2 7
EJERCICIO 183 2.
–3·7
=10
21=––
11
4 5
4·3 –2·5 =12 –10 =2 5
1. 2 3 5.
–2
 9 –3
–8 5(–8)–(–3)(–2)= –40 –6 = –46
–11
9·7 –(–11 )(–3 )=63–33= 30
3 5 6. –3 7
–2 5 –15 –1
3. –2·3 –4·5 =–6 –20 =– –15·2 –13(–1)= –30 +13= –17
26
4 3 7.
13 2
7 9
4. 7 ( –2 ) –9·5= –14 –45= – 12 –1
12 (–9)–(13) (–1)= –108 +13= –95
59
8.
5 –2 13 –9
 10 3
9.
17 13 10·13 –3·17 =130 – 10.
–5 –8 (–5)(–21)–(–8)(–19)=105–152 = –47
51=79 –19 – 21
8 2 31 –85
11. 8(0) –2 ( –3 ) = 0 + 6 12. 31·43–(–85)(–20)=1.333–1.700 = –367
=6
–3 0 –20 43

EJERCICIO 184
6. ax –by =–1
1. 7x +8y =29 4. 15x –44y =–6 ax +by =7
5x +11y =26 –27x +32y =–
–1 –
1
x b
29 8 –6 –44 b
= –
26 11 319 – –1 32
x =7 8= 111 x= 7 b
= 37 =3 15 –44 b
208 a
77 –40 a
5 1 –27 32 –b +7b 6b 3
1 –192 –44 x= = =
7 2 x= –236 ab +ab 2ab a
1
= =
9 480 –1.188 –708 a –1
5 2 182 –145
6 3
y= =1 15 –6
37 –27 –1 a 7
y= y=a –b
sol: x y =1 15 –44
a b
=3
2. 3x –4y =13 –27 32 7a +a 8a 4
8x –5y =– –15 –162 –177 y= = =
1
y= = = 2ab 2ab b
5
–708 –708 4
13 – 1 sol: x = y=
–65 – 3 b
x= 4 = – =–5 sol: x =
20 =
85 3 4
–5 – 17 1 a
5 –15 y=
4
3–4 +32
8–5
3 13
8
y= –5 –15 – –119
= 17 =–7
5. 8x =–9y ( )
7. 3x – y + 2 = 2 y +1
3 –4 =
8x +9y =0 3x – y –2 = 2 y +1
8 –5 104
sol: x =– 17 2x +5 +3y = 3x –3y = 3
5 7 x – y =1
y =–7 2

3. 13x –31y =–326 4x +10 +6y =7

5y – ( x + 3)= 3x +1
25x +37y =146 4x +6y =–3
5y – x –3= 3x +1
09
–326 –31 –4x +5y = 4
146 37 x= –
x= 1– 5+ 4
13 –31 36 x= 1x = =9
25 37 89 27 9 5–4
–12.062 +4.526 –7.536 0 +27 = = =2 1 45
x= 46
1–
 –
45

x= = =–6 48 –36 12 4 4
481+775 1.256 8 0 11
13 –326 4+4
y= –y=
25 146 y=4 –3 1 =8
y= 8 9 44
13 –31
4 6 1–
25 37 sol: x = 1 y =8
–24 – – =–2
1.898 +8.150 10.048 y=
0 =
24
y= = =8 12 12 9 –
4
45
sol: 1.256 1.25 1
y =–2
sol: x =2
y 6
x =–6
=8
8. ax +2y =2
y –3 2y +
ax x+
–3y =–1 11. – 8 = 56 13. 2x =y+2
2 – 3
2 17
3
ax –6y =–2 8 ( x + 2 ) –3 ( y –3 ) = 34x –(2 y + 3)=17 ( y + 2)
22 20 34x –2 y –3=17 y + 34
–12 +4 –8 8x +16 –3y + 9 = 20 34x –19 y = 37
–2 –6 1 8x –3y + 25= 20
a2 =
x= = = 4x +1
8x –3y = –
a – –6a –2a –8 a 3y – = 3x +5
21
5
6 y –5 2x –3
6
–
5
=0 63y –(4x +1)= 21(3x +5)
a2
y= = –2a – –4a 5 ( y –5 ) –6 ( 2x –3 ) = 63y – 4x –1= 63x +105
a–2 1
= =
a2 –8a – 67x + 63y =106
2a 0
a – –8a 2 5y –25–12x +18 =
6 0
–12x +5y –7 = 0 37 19
1 1 –12x +5y = 7 106 63
sol: x = y = x=
a 2 –5 34 –19

–3
x y – 67 63
9. 4 + 6 =–4 x= 75
2.331+ 2.014
3x +2y =–48 8 – x= = 4.345
– 25 + 21 – 4 869 =5
x y x= 3 = = 2.142 –1.273
– =0
34 37
–
8 12 40 –36 –67 106
125 y=
4
3x –2y =0 8 –5 34 –19
–48 2 –12 7 –67 63
0 y=
x= –2 96 –0 96 =–8 8 –3 3.604 + 2.479
= = y= =7
3 2 –6 – –12 5 869
–12
6 56 – 60 – 4 sol: x =5 y = 7
= = =
3 – y– 1
2 40 –36 4
3 – sol : x = –1 y=– x+y
y=3 2 =
1 14. x–y=4
– =–12
12

3 –2 12. 3x –2 y =5 x + y = 4 (x –
y) x + y = 4x –
sol: x =–8 y =–12 mx + 4 y = 2 (m 4y
10. 3x + ay = 3a 5 – – 3x +5y = 0
+1
x +1) 2 (m2 x – y –1 1
a + ay = 2 x= x + y +1 =9
2 +1)4
x + a y = 2a
m)
3a +1 a 3–2
2a a
2 (m
20 + 4m4 4 (6 +
x=
3 a +1) m)
2
1 a x= =
12 + 2m 2 (6 +
 9x –9 y – 9 = x + y +1
9 ( x – y –1)= x + y +1 y =10
=2 8x –10 4x –5y =5
35 0 5
(3a –1)
3 2 2
3a + a –2a a
2 m2 5 0– – =5
x=
3a – a
2 =
a (3a –1)
=a y= (m x = –5 25 2
33a +1 +1) 5
( )
6 m +1 –5m m + 6 1 2a 32 –3
–3 0 5
= =
y= = y ==13 15–20 –
a m4 45
2 (6 + m ) 2 (6 + m) 2
1
a2
1 y= = –15– 0
=3
6a –3a –1 3a –1 1 sol : x = 2 y –3 5 –5
=
y= = =
a (3a – a (3a –1) 2 4 –5
1) a

1 sol : x =5 y=3
sol : x = a y=
a
 x +9 y +21
16. x –9 = y +39
15. x – y =2b
x
(x + 9)(y +39)=(x –9)(y +21)
xy +39x +9 y +351= xy +21x –9 y –189
a + y =2
+b a –b 18x +18y =–540

x + y =–30
x (a –b) + y (a +b) = 2 (a –
2

x +8 y +19
2b1 =
b )
2

2(a2 –b2 )a x –8 y +11

x= +b (y +11)(x +8)=(y +19)(x –8)
11
= (a +b) + 2 (a –b xy +8y +11x +88 = xy –8y +19x –152
2 2
x 2b
) a –ba +b
a +b + a –b –8x +16y =–240
2a (b + – x +2 y =–30
=b +
a)
=
2a a –30 1
12b –30 2 –60 + 30 –30
x = 1 =1 = =–10
2 +1 3
y= a –b 2(a2 – –1 2

b2 ) 1 –30
2 (a 2 –b 2 11
) –2b (a – –1 –30 –30 –30 –60
2a (a – y= = 3 = 3 =–20
y = b) a –ba +b
b)
= =a– 1 1
2a 2a
b
sol: x = a +b y = a –b –1 2
y =–20
sol: x =–10
EJERCICIO 185
2. x –2y =10 2x +3y =–8
1. x –y x +y
=7 x x =0 y =–5 x =0 y =–2 23
=1 y =– =0 y =7 y =0 x y =0
x =7 =10 x =–4
x y =0
=0 1
y x=1
=0

x+y
=7
x-y
=1

(4,3)
 sol : x=2 Y=- 4
sol : x=4 y= 3

2x + 3y = x - 2y =
-8 10
 3. 5x –3y 7x –y =–16 5. 3x y =15 2x =5
=0 +4 +y
x =3 y =5 x =0 y =16 7 x y=0 x= y
Escala 1:2 y =0 x =–2 2 =5 y =3 0 =5

3x + 4y =
5x - 3y 15
=0

7x - y =
(1,3)
-16

2x + y
=5
(-3,-5)

sol : x=-3 y= -5 sol : x=1 y= 3

4. 3x =–4y
5x –6y 6. 5x +2 y =16 4x +3y =10
=38
3x +4y x =– y =–8 x =2 y x=– y =6
=0 2 =3 2
x =4 y =–3 y =– x =4 x =4 y = – x y =2
3 2 =1

3x + 4y =
0 5x + 2y =
4x + 3y = 16
10

(4,-3)

5x - 6y =
38

(4,-2)

sol : x = 4 y= -3 sol : x=4 y=- 2

 x y 1
7. x +8 =y +2 y –4 =x 9. – =– 3x –2y =–1
+2
2 3 6
x –y =–6
–x +y x =3 y =5 ; x =–3 y =–4
=6
x =0 y x =0 y =6 x y 7 4x +3y =–7
+ =–
=6
x =–6 y x =–6 y =0 4 12
=0 3 x =–1 y =–1 ; x =–4 y =3

3x - 2y = -
y-4=x+ 1
2 4x + 3y =
x+8=y+ -7
2

(-1,-1)

sol : Equivalentes
sol : x = - 1 y=- 1

3x y
8. + =2 x –5y =25 10. x +3y =6 3x +9y =10
5 4
x =0 y x =0 y =119
12x +5y = x =0 y =–5 y =0
x =6 =2 x =3
40 1
y
=0
x= y =8 x =5 y =–4 3
0
y =–4
x =5

sol : x = 5 y=- 4
3x +9y
=10
12x + 5y =
40
x + 3y
=6
 (5,-4)

x - 5y =
25

s
o
l

I
n
c
o
m
p
a
t
i
b
l
e
s
 11. 2x +3y =–13 13. x +y =9
x =–2 y =–3 ; x =–5 y =–1 x =6 y =3 ; x =4 y =5
6x +9y =–39 x –y =–1
x =–2 y =–3 , x =– y =–1 x =0 y =1 ; x y =3
5 =2
x –2y =–6
x =0 y =3 ; x =2 y =4

(4,5) x - 2y = -
2x + 3y = - 6
13
6x + 9y = -
39 x-y=-
1 x+y
=9

sol : Equivalentes
x–2 y –3
12. – = y–2 x–3
11 + =– sol : x = 4 y= 5
2 3
3
2 3 14. x +y =5

3 ( x –2 ) – 2 ( y – 3)= 24 3 ( y – 2 ) + 2 ( x –3 ) = – x =0 y =5 ; x =2 y =3
3x +4y =18
22 x =2 y =3 ; x =–2 y =6
3x –6 –2 y + 6 = 24 3y –6 + 2x – 6 = –
22
3x –2 y = 24 2x + 3y = –
10
x=6 y=– x=4 y=– 2x +3y =13
3 6
x=4 y=– x = –2 y = – x =2 y =3 ; x y =1
6 2 =5

3x + 4y =
18 x+y
=5

2x + 3y = 3x - 2y =
-10 24
 2x + 3y =
13

sol : x = 4 y=- 6
sol : x = 2 y=3

(2,3)

(4,-6)
15. 2x +y =–1 EJERCICIO 186
x =0 y =–1 ; x =–1
y =1 1. x + y + z = 6
x –2y =– x – y + 2z =5
13
x =–5 y ; x =–3 y =5 x – y –3z = –10
=4
3x –2y =– x+y+ z=6
19
x =–5 y =2 ; x =–3 y =5 x – y + 2z =5
2x + 3z =11
x+y+z=6
2x + y =
-1 x – y –3z = –10
2x –2z = – 4
x – z = –2
x - 2y = - (-3,5) 2x + 3z =11
13
3x - 2y = - x – z = –2 (3)
19 2x + 3z =11
3x –3z = – 6
5x =5
x =1
2 (1) + 3z =11
2 + 3z =11
3z = 9
sol : x = - 3 y= 5 z=3

16. x –y =1 2y –x =–4 4x –5y =7 1+ y + 3 = 6

y+4=6
x =0 y =–1 x =0 y =–2 x =–2 y =–3 y=2
x =1 y =0 x =2 y =–1 x =3 y =1 sol : x =1 y = 2 z = 3
2. x + y + z =12
2x – y + z = 7
x+2y–z=6
x+y+
1 z
2y - x = - =12
x- 4 2x – y + = 7
y= z
3x + 2z =19
2x – y + z = 7 (2)
x+2y–z=6
4x - 5y =
7 4x –2 y + 2z =14
(-2,-3) x+2y–z=6
5x + z = 20
3x + 2z
=19 (–2)
5x + z = 20
3x + 2z = 19
 –10x –2z = – 40
– 7x = – 21
sol : x = - 2 y=- 3 x=3
Continúa
2. Continuación
4. Continuación
5(3) + z = 20 6. 5x –2 y + z =24
2x + y –3z = –1
(3)
15+ z = 20 2 x +5y –2z =–14
z =5 x – 3y – 2 z = – x –4 y +3z =26
3+ y +5=12 12 5x –2 y + z = 24 (2)
2 x +5 y – 2 z =–14
y +8 =12 6x + 3y –9z = –3 10x –4 y + 2z =
y=4 48
x –3y –2z = –12
7x –11z = –
15

sol: x = 3 y = 4 z x–z=–1 (–7) 2 x + 5y – 2 z =– 14

=5 7x –11z = – 12x + y = 34
3. x – y + z = 2
15
x+y+z=4 –7x + 7z = 7 5x –2 y + z =24 (–3)
2x + 2 y – z = – 4 7x –11z = – x –4 y +3z =26
x–y+z = 15 –15x +6 y –3z =–72
2 – 4z = –8 x –4 y +3z = 26
x+y+z = z=2
4
2x
+ 2z = 6 x –2 = –1 –14x + 2 y =–46
x+z=3 x =1
12x + y = 34 (–2)
x–y+z= 2
2 (1) + y –3 ( 2 ) = –1
–14x + 2 y =– 46
(2) 2 x + 2 y – z = – 2 + y – 6 = –1
4 –24x –2 y =–68
y – 4 = –1
2x –2 y + 2z = 4 –14x +2 y =–46
y=3
2x + 2 y – z = –4 –38x =–114
sol : x =1 y = 3 z = 2
4x +z=0 x =3
5. 2x + 3y + z =1 12 (3) + y =34
x + z = 3 (– 6x – 2 y – z = –14
36 + y =34
1) 3x + y – z =1 y =–2
2x + 3y + z
4x + z = 0 =1 3x + y – z 3 –4 ( –2 ) + 3z =26
=1 5x + 4 y
– x – z = –3 =2 3 +8 +3z =26
4x + z = 0 3z =15

3x =–3
x = –1
–1+ z = 3 2x + 3y + z = 1 z =5
z=4 6x – 2 y – z = – sol: x =3 y =–2 z =5
14
–1– y + 4 = 2 8x + y = –13
7. 4 x +2 y +3z =8
– y + 3= 2 5x + 4 y = 2
– y = –1 8x + y = –13
(–4) 3x +4 y +2z =–1
y =1 5x + 4 y = 2 2 x – y +5z =3

sol : x = –1 y =1 z = 4 4 6z = –2
4. 2x + y –3z = –1 x 3x – 2 y – z = – 5
x –3y –2z = –
12 3x –2 y – z = +
–5 2
2x + y –3z = –1 (2) y
3x –2 y – z = –5 –
 – 32x – 4 y =52 2 (– 2)+ 3(3) + z =1 4x +2 y +3z = 8 (–2)
– 27x =54 – 4 + 9 + z =1
3x +4 y +2z =–1
x=–2 –8x –4 y –6z =–16
3x +4 y +2z =– 1
8 ( –2 ) + y = –13 –5x –4z =–17
–16 + y = –13 5x +4z =17
y=3 4 x +2 y +3z =8
2x – y +5z =3 (2)

7x –7z = –7 4 x +2 y + 3z =8
5 + z =1
x – z = –1 z=– 4 x –2 y +10z =6
4
sol : x = – y = 3 z = – 8x +13z =14
Continúa 2 4 Continúa
7. Continuación 9. 2x + 4 y + 3z = 3 10. Continuación 12. 4x – y +5z = –6
10x –8y –9z = 0 2x + 4 y –2z =
8x +13z =14 (–5) 3x + 3y –4z = 30
5x + 4z =17 (8) 4x + 4 y –3z = 2 2 6x + 2 y –3z = 33
x + y +2z =–17

–40x –65z =–70 2x + 4 y + 3z = 3 3x = –15 4x – y +5z = – 6(3)

40x + 32 z =136 (3) +5y 4x + 3y = 2 (3)
3x + 3y –4z = 30
–33z =66
z =–2
10x –8y –9z =0 3x + 5y =–15 (– 12x –3y +15z = –18
6x +12 y + 9z = 9 12x + 9 y = 6 3x + 3y – 4z = 30
4)
10x – 8y –9z = 0
5x +4 (–2)=17 16x + 4 y =9 –12x – 20 y = 15x +11z =12
60

5x –8 =17 10x –8y –9z = 0 –11y = 66 4x – y +5z = – 6 (2)

5x =25 4 x + 4 y –3 z = 2 (– y =–6 6x + 2 y –3z = 33
x =5 4x + 3 ( –6 ) = 2 8x – 2 y +10z = –
3)
2 (5) – y +5 (– 4x –18 =2 12
10x – 8y –9z = 0
4x = 20 6x + 2 y – 3z =
2 ) =3 –12x –12 y + 9z =– 6
10 – y –10 33
=3 – 2x – 20y =–6
14x + 7z = 21
x +10y = 3 x =5 2x + z = 3
– y =3
sol: x =5 y =–3 16x + 4 y =9 (5) 5–6 + 2z =–17 15x +11z =12
y =–3 z =–2 x +10y = 3 (– 2z =–16 2x + z = 3 (–11)
2)
z =–8
80x + 20y = 15x +11z = 12
8. 6x + 3y + 2z sol: x =5 y =–6 z =–8
45 –22x –11z = –33
=12
– 2 x – 20y =– 6
9x – y + 4z = 37 =39
10x +5y + 3z = 78 11. 7x + 3y – 4z = – –7x = –21
21 1
x x= 35 x=3
6x + 3y + 2z 2
=12 (3 3x – 2 y +5z = 38 2 (3) + z =
9x – y + 4z = ) x + y –6z = – 27
37 3
{ 1⎝
6x + 3y + 2z =12 16 | | +4 y = 7x + 3y –4z = – z = –3
9 35
27x – 3y +12z
=111 ⎝ 2J x + y – 6z = – 27 (– 3) 4 (3) – y +5(–3)= –6
8+4y=9
33x +14z =123 7x + 3y – 4z = – 35 12 – y –15= –6
4 y =1 – 3x – 3y +18z =
9x – y + 4z = 37 (5) – y –3= –6
1 81 4x +14z =
10x +5y + 3z = 21 y= 46 – y = –3
45x –5y + 20z =185 { 1⎝ { 41 ⎝ 2x + 7z = 23 y=3
10x +5y + 3z = 21 2 | | + 4 | |+ sol
: x=3y=3z=–3
3z =3
⎝ 2J ⎝ 4J 1+1+ 3x – 2 y +5z = 38
55x + 23z = 3z =3
206 33x +14z x + y – 6z = – 27 (2)
=123 (5) 3z =1 3x – 2 y + 5z = 13. 9 x +4 y –10z = 6
55x + 23z = (– 38 6x –8 y +5z =–1
206 3)
1
165x + 70z = 615 z= 2x + 2 y –12z = – 12x +12 y –15z =10
54
z 3
–165x – 69z =– 1 1 1 5x – 7z = –16
618 so x = y= z= 2x + 7z = 23
l:
 1 12 z
2 y =
6x – 8y + 5z =–1 (–2) x
+
–
15
1
0
2 4 3
( )
33x +14 – 3 =123 5x – 7z = – –12x +16y –10z = 2
10. 3x + y + z =1
33x –42 =123 16 12x +12 y –15z =10
33x =165 x + 2 y – z =1
7x = 7 28y –25z =12
x =5 x + y + 2z = –17
x =1 9 x +4 y –10z = 6 (2)
(
6(5) + 3y + 2 – 3x + y + z =1
2 (1) + 7z = 23
x + 2 y – z =1 6x – 8 y + 5z =– 1 (–3)
)
3 =12 7z = 21
4x + 3y = 2 18x + 8y –20z =12
30 + 3y – 6 =12 z=3 –18x +24 y –15z = 3
x + 2 y – z = 1 ( 2)
3y + 24 =12
1+ y –6(3) = –27
3y =–12 – =–
y =–4 x + y + 2z = – y 17 27 – =
17
sol: x =5 y =– z = –3 y = –10 32 y 35z 15
4
Continúa
sol: x =1 y = –10 z = 3 Continúa
13. Continuación 14. Continuación
17. y + z = –8 19. 3z –5x =10
28y –25z =12 (– 10x – ( –2 ) + z
2x + z = 9 5x –3y =–7
3y + 2x = – 3y –5z =–13
8) =10
{ 1⎝
3
32y –35z =15 (7) 10 | | – ( –2) + z
=10

–224 y + 200z = – ⎝ 5J
2+2+z y + z = –8 –5x + 3z =10
96
=10 2x + z = 9 (–1) 5x – 3 y =– 7
224y –245z =105 +=– –3y +3z = 3
–45z = 9 z=6
1 sol:
y =–2 z = y z 8 – + =
x=
1 6
z=– – 2x –z=– y z 1
5 5
15. x + y =1 3y –5z =–13
9
– 2x + y = –
{ 1⎝ 17
28y –25 | – | y+z=– –2x + y = –17 – y + z = 1 (3)
=12 1
⎝ 5J z + x = –6 2x + 3y = – 3y –5z =–13
3
28y +5=12
x+y = 1 4y=– – 3 y + 3z = 3
28y = 7
1 y + z = –1 (– 20 –2z =–10
1) y = –5
y= x + y =1 –5+ z = – z =5
4
{ 1⎝ { 1⎝ –y–z 8 – y +5=1
6x –8 | | +5 | – | =
⎝ J ⎝ =1 x – z = z=– – y = –4
2
–1 J 3 y=4
4 5 x–z=2
x + z = –6 2x –3= 9 5x –3 ( 4 ) =–7
6x –2 –1= –
2x =12
1 x=6
6x = 2
sol : x = 6 y = –5 z = – 5x –12 =–7
x= 2x = –4
1 3 5x =5
1 1 x = –2 x =1
= = 3
=–
1
sol : x y z
3 4 –2+y 18. 3x – 2 y = sol: x =1 y = z =5
=1 0 4
5

5x +3y – z =– y=
14. 11 10x – y + z 3y – 4z = 25
3+ z = – 3
=10 z –5x = –14
1
z = –4
15x + 2 y – z =–7 sol : x = – 2 y = 3 z = 3y – 4z = 25 20. x –2 y = 0
5x +3y – z =– –4 –5x + z = –14 (4) y –2z =5
11 10x – y + z =
10 16. x + 2 y = –1 3y – 4z = 25 x + y + z =8
15x +2 y =–1 –20x
2y+z=0 + 4z = –56 x + y + z =8

10x – y + z x + 2z –20x + 3y =– x –2y = 0(–1)

=10 31
=11

15x + 2 y – z =– 7 2 5 5 x+2y = –1
1 25
 2y+z= 0 3x – 2 y = 0
(–1) – (2 x + y + z =8
x+2y = –1 –40x + 6y = – –x+2y =0
+ )
–2 y – z = 0 62 3y + z =8
= 9x –6y = 0 (3
3y + z =8 (2)
31 )
x –z=–
15x +2 y =– 1
1 x – z = –1 (–1) –31x = –62 y –2z =5
–50 x – 2 y =– 6 6y + 2z =16
x=2
–35x =–7 x + 2z = 11 y –2z = 5
1 3 (2 ) – 2 y =
= –x+z=1 7y = 21
0
6–2y=0
x x + 2z =11 y=3
5 –2 y = – 6
{ 1⎝ 3z =12 x –2 (3) = 0
15 | | + 2 y = – y=3
⎝J z=4 x –6 = 0
1 3(3) – 4z = 25
5 x –4 = –1 ‹ x = 3 x=6
–4z =16
3 +2 y =–1 3+ 2 y = –1 6 + 3+ z =8
z=–4
2 y =–4 2 y = –4 ‹ y = sol : x = 2 y = 3 z = z = –1
–4
y =–2 –2
Continúa sol : x = 3 y = –2 z = sol : x = 6 y = 3 z = –1
4
21. 5x –3z = 2
23. Continuación 24. Continuación
2z – y =–5
x +2 y –4z =8 x –y +z 6 –2y +18 =0
x +2 y –4z =3 x +y – –2y +24 =0
=8
z =1
– y + 2z =–5 (2) 2x =4 –2y =–24
x =2 y =12
x +2 y –4z = 8 2 –y +5 sol: x =6 y =12 z =18
–2 y +4z =–10 =3
x =–2 –y +7 =3
5 ( –2 ) –3z =2 –y =–4
–10 –3z y =4 x y z
=2
+ + = 21
sol: x =2 y =4 z 3 4 3
–3z =12
z =–4 =5 4x +3y +4z = 252
x y z
+ – =0
2 ( –4 ) – y =– 5 6 3
24. x y z
2 + 2 – 3 =3 6x +5y –10z =0
5 x y z
3x +3y –2z =18 + – =3
–8 – y =–5 x y z 10 3 6
+ – =–5
– y =3
y =–3

sol: x =–2 y =–3 z =–4 3 6 2 3x +10 y –5z =90

6x + 5y –10z = 0
(–2)
22. 2 x – z =14 2 x + y –3z =–
4 x + y – z =41 3x +10y – 5z =90
3x – y +5z =53 30
4x + y – z = 41 x y z –12x –10y + 20z =0
– + =0
6 3 6 3x +10y – 5z = 90
x –2 y + z =0
3x – y +5z
=53 3x +3y –2z =18 (2 –9 x +15z = 90
)
x –2 y + z = 0
7x +4z = 94
6x +6y –4z =36
(3 –3x +5z =30
7 x +4z = 94 4 x + 3y +4z = 252 (10)
3x – 6 y +3z = 0 ) 3x +10 y –5z =90 (–3)
2 x – z =14 (4)
9x – z =36
40x + 30y +40z = 2.520
7 x +4z = 2 x + y –3z =–30 (2) – 9 x – 30 y +15z =– 270
94
8x – 4 z =56

15x =150 x –2 y + z = 0 +55z = 2.250

31x
x =10 4 x +2 y –6z =–60
2 (10) – z =14 x –2 y + z = 0
–3x + 5z =30 (–11)
31x +55z = 2 .250
20 – z =14 5x –5z =–60
x – z =–12 33x –55z = –330
– z =–6
9 x – z = 36 31x +55z = 2 .250
z =6
4 (10) + y –6
x – z =–12 –1 ( ) 64 x =1.920
x =30
=41
+ –y =3 z –x +y =7
y +34 = 41 y
y =7 =
7
sol: x =10 y =7 z =6 z
23. x +y –z
=1 z +x – +
y =3 z –x x
 9x – z =36 +5z =30
–
– x + z = 12 8x =48
3 –90 +5z =30
x =6
5z =120
6 – z =–12 (
z =24
– z =–18
3 6 (30) +5y –10(24) =0
z =18
0 180 –240 +5y =0
5y = 60
)
y =12
2z =10 ‹ z Continúa sol : x = 30 y z =24
=5 =12
Continúa
 y+z 28. Continuación
x – 3 =4 27. Continuación
y – z x +2 5x – y –5z = 22
=
3x – y – z 3 10 – x –3y + 3z =–22 (5)
=12
x+z
y– 10 ( y – z ) = 3 (x + 5x – y – 5z = 22
=10
8 2)

8y – x – z 10y –10z = 3x –5x –15y +15z =–110

=80 +6 –16y +10z =–88

y–x 3x –10y +10z =–6

z– =5
2 2x – 5y – 2z =–20 (5 –8y +5z =–44
)
2z – y + x =10 3x –10y +10z = –6 –8y +5z = –44
3x – y – z =12 (– 10x –25y –10z =–100 8 y – 7 z = 36
3x –10y +10z = –6 –2z = –8
1)
– x +8y – z =80
–3x + y + z =–12 13x –35y =–106 z=4
– x +8y – z = 80 13x –35y =–106 (2) 8y –7 (4) = 36
–4x +9 y =68 8y –28 = 36
5x – 2 y = 28 (–35)
– x +8y – z =80 (2) 8y = 64
26x –70y =–212
x – y +2z =10 –175x +70y =–980 y =8
–149x =–1.192 – x –3(8) + 3(4) =–22
–2x +16y –2z =160
x – y +2z = 10 – x –12 = –22
x =8
– x +15y =170 5(8) –2 y = 28 x =10
–4x + 9 y = 68 40 –2 y = sol: x =10 y =8 z = 4
28
– x +15y =170 (–4) –2 y =–12 y–z

–4x + 9 y = 68 y =6 x–y+ =3
2x – 2 y +2y – z
4x –60y =–680 2 (8) –5(6) –2z =–20 = 6 2x – y –
–51y =–612 z=6
16 –30 –2z =–20 x – 2y x4– z
y =12 –2z =–6 – =
–4x +9 (12) =68 0
z=3
– + =
sol : x =8 y =6 z =3 2x –2 y – x + z = 0
4x 108 68 x–2y+z=0
–4x =–40 z+4 y–z
28. y– = x –6 –x=–
x =10 2 52
2 y – z – 4 = 2x – y – z –2x = –10
3(10) –12 – z =12
12 2x – y – z = 6
30 –12 – z =12
–2x + 2 y – z = –8
x –7
18 – z =12 z– = y –5 –2x + y – z = –10
3
z =6 – 2z = – 4
z =6 3z – x + 7 = 3y –
sol: x =10 y =12 z=2
15
x + y y +4 x–2y+z=0
27. 7 = 5 – x –3y + 3z = –22 2x – y – z = 6 (–2)
y+2
x– 5 =z+4
5 ( x + y ) =7 ( y + 5x – y – 2 =5z + 20 x–2y+z=0
5x – y –5z = 22
4) –2x + 2 y – z = – – 4x + 2 y + 2z = –12
5x +5y =7 y +28 8 – 3x + 3z = –12
5x –2 y =28
x – z y –4
= – x –3y + 3z = – 22 (– – 3x + 3(2) = –12
5 2 2)
–2x + 2 y – z = –8 –3x + 6 = –12
2 ( x – z ) =5 ( y –
2 x + 6 y – 6z = 44 – 3x = –18
4) 8y – 7z = 36 x=6
2x –2z =5y – Continúa Continúa
20
2 x –5y –2z =–20
Continúa
29. Continuación 3 2
31. + = 2
6 –2y +2 =0 x y 32. Continuación
–2y +8 =0 2 2 3 1 4 2 (5)
–2y =–8 y +z = 2 (– x + y + z= –6

1 )
y =4 6 5 6 (4)
sol: x =6 y =4 3 2 x – y – z = 31
z =2 x +y = 2
2 2 3 5 20 10
–
– =– x + y + z =–30
y z 2
11 3 2 1 24 20 24
– = – – =124
30. + =5 x y z
x1 y x z 2
1 3 2 1 29 14
– = (2) x – z =94
+ = 6 –1 ( ) x z 2
x z 5 6
1 1 1 4 4 – – =12 (–7 )
+ = 5 x +z =3 x z
x y 6 4 29 14 (3)
1 1 – =94
– – =– – =1 x z
x z
6
x z 35 42
1 1 1 4 4 x + z =84
– = –1 x+ z = 3 87 42
y z 7 7 – =282
1 1 =
x 3 x z
– = –1
y z x= 122 = 366
1 1 3 x 122 =366x
y +z = 7 3 2 1
+
=2
3 y
2 2 =x
y =6 1+ =2 3
y 5 6
– – =–12
1 1 z
3= y 2
y =1 3 6
2=y – – =–
1 1
+ =5
x 1 15 12
3 3 2 1 6z
1 3– z = 2 – =3
+ 3=5 2 1 z
x 1– z = 2
1 –2 = z
=2 2 1 1 4 2
1x – z =– 2 1 + y + –2=–6
=x 4=z 3
2
sol x = y =2 z =4 4
1 1 3 + –1=–6
:
3
+ =6 y
1 z
2 4
=–8
1 y
2+ =
32. 1 + 4 + 2 =– (– 1
6 6 – =y
z x y z 2) 2
1 3 2 4 sol: x = y =– z =–2
z= 4 x +y +z = 3 1 1
1 4 2
1 + + =– 3 2
=
 4 6
x y z
sol :
1x = 1y = 1z = 6 4 8
– x – y – z =–6
2 3 4
5
– 6 = –12
–
x z
Continúa