http://perso.wanadoo.es/timonmate ÍNDICE A. ECUACIONES TIPO

ECUACIONES GRADO 1
http://perso.wanadoo.es/timonmate
ÍNDICE
x=6
7)
x – 2 = 30
A. ECUACIONES TIPO ax+b=c
x = 30 + 2
B. ECUACIONES TIPO ax+c=bx+d
x = 32
C. ECUACIONES CON DENOMINADOR
(monomios en el numerador)
D. ECUACIONES POLINÓMICAS
8)
x=
9)
E. ECUACIONES CON DENOMINADOR
(binomios en el numerador)
F. OTROS TIPOS
10)
11)
2)
x=5+1
12)
14)
Þ 4x = 8 Þ x =
15)
x + 6 = 11
x + 14 = 20
x = 20 – 14
10
=5
2
4x + 4 = 12
x=7+4
8
=2
4
13x + 4 = 11
13x = 11 - 4 Þ
Þ 13x = 7 Þ x =
x=5
Juan J. Pascual
2x + 2 = 12
4x = 12 - 4 Þ
x = 11 – 6
6)
80
23
x–4=7
x = 11
5)
23x = 80
Þ 2x = 10 Þ x =
x=3+2
4)
-14 14
=
=7
-2
2
2x = 12 - 2 Þ
x–2=3
x=5
14
= -7
2
–2x = –14
x=
13)
4
3
–2x = 14
x=
x=6
3)
3x = – 4
x =-
A. ECUACIONES TIPO ax+b=c
x+2=4
x=4–2
x=2
x–1=5
80
= 20
4
x =-
EJERCICIOS RESUELTOS
1)
4x = 80
16)
7
13
6x – 5 = 3
6x = 3 + 5 Þ
1/6
Colegio Virgen de la Peña
Þ 6x = 8 Þ x =
17)
Matemáticas Primer Ciclo ESO
9x – 4 = 14
9x = 14 + 4 Þ
18
Þ 9x = 18 Þ x =
=2
9
18)
11x = -2 - 5 - 3 - 13 Þ
23
Þ 11x = -23 Þ x = 11
8 4
=
6 3
6x – 3 = 2
B. ECUACIONES TIPO ax+c=bx+d
26)
3x = 5 Þ x =
6x = 2 + 3 Þ
Þ 6x = 5 Þ x =
19)
5
6
9x = 4 - 2 + 2 Þ
20)
4
9
3x – 4 = 9 – 2
Þ 3x = 11 Þ x =
11
3
11x + 3 – 4 = 15 – 2
29)
17x – 3 + 2 = –10 – 3
12x – 2 – 4 = –3 – 4 – 4
12x = -3 - 4 - 4 + 2 + 4 Þ
5
Þ 12x = -5 Þ x = 12
25)
Juan J. Pascual
30)
16x + 3 – 4 – 2 = – 4 – 2
16x = -4 - 2 + 4 + 2 - 3 Þ
3
Þ 16x = -3 Þ x = 16
24)
4x – x + x = 3 + 10
13
4
C. ECUACIONES CON DENOMINADOR
(monomios en el numerador)
17x = -10 - 3 + 3 - 2 Þ
12
Þ 17x = -12 Þ x = 17
23)
3x – 2x = 4
4x = 13 Þ x =
11x = 15 - 2 - 3 + 4 Þ
14
Þ 11x = 4 Þ x =
11
22)
5
3
x=4
28)
3x = 9 - 2 + 4 Þ
21)
27)
9x – 2 = 4 – 2
Þ 9x = 4 Þ x =
x + 2x = 5
11x + 13 = – 2 – 5 – 3
31)
x 2 1
+ =
3 7 3
m.c.m(3,7) = 21
x × 7 2 × 3 1× 7
7x 6
7
+
=
Þ
+ =
Þ
21
21
21
21 21 21
1
7x + 6 = 7 Þ 7x = 1 Þ x =
7
x 1
- =2
3 4
m.c.m(4, 3) = 12
x × 4 1 × 3 2 × 12
4x 3
24
+
=
Þ
+ =
Þ
12
12
12
12 12 12
21
4x + 3 = 24 Þ 4x = 21 Þ x =
4
3x 3 1
+ =
2
4 2
m.c.m(2, 4) = 4
3x × 2 3 × 1 1 × 2
6x 3 2
+
=
Þ
+ = Þ
4
4
4
4 4 4
1
6x + 3 = 2 Þ 6x = -1 Þ x = 6
2/6
Colegio Virgen de la Peña
32)
33)
34)
35)
4x 7 1
- = -1
9 6 3
m.c.m(3, 6, 9) = 18
4x × 2 7 × 3 1 × 6 1 × 18
+
=
Þ
18
18
18
18
8x 21
6 18
Þ
+ = - Þ
18 18 18 18
Þ 8x + 21 = 6 - 18 Þ
33
Þ 8x = -33 Þ x = 8
5x 1 3
- = -5
6 7 3
36)
37)
m.c.m(3, 6,7) = 42
5x × 7 1 × 6 3 × 12 5 × 42
=
Þ
42
42
42
42
35x 6
36 210
Þ
+
=
Þ
42 42 42 42
Þ 35x + 6 = 36 - 210 Þ
Þ 35x = 36 - 210 - 6 Þ 35x = 180 Þ
180 36
Þx=
=
35
7
4x 1
1
- = -1
5 20 10
m.c.m(5, 10, 20) = 20
4x × 4 1 × 1 1 × 2 1 × 20
=
Þ
20
20
20
20
16x 1
2 20
Þ
- = - Þ
42 42 42 42
Þ 16x - 1 = 2 - 20 Þ
Þ 16x = 2 - 20 + 1 Þ 16x = -17 Þ
17
Þ x=16
9x 14 1 1
+ = 2
6 3 9
m.c.m(2, 3, 6, 9) = 18
9x × 9 14 × 3 1 × 6 1 × 2
+
=
Þ
18
18
18
18
81x 42
6
2
Þ
+
= - Þ
18 18 18 18
Þ 81x + 42 = 6 - 2 Þ
Þ 81x = 6 - 2 + 42 Þ
46
Þ 81x = 46 Þ x =
81
Juan J. Pascual
Matemáticas Primer Ciclo ESO
-
11x 14
1 1
+ -1 = +
2
6
4 8
m.c.m(2, 4, 6, 8) = 24
11x × 12 14 × 4 1 × 24 1 × 6 1 × 3
+
=
+
Þ
24
24
24
24
24
132x 56 26
6
3
Þ+ - = - Þ
24
24 24 24 24
Þ -132x + 56 - 26 = 6 - 3 Þ
27
9
Þ 132x = -27 Þ x = =132
44
13x 16
1 12
+ -2 = +
2
6
4 8
m.c.m(2, 4, 6, 8) = 24
13x × 12 16 × 4 2 × 24 1 × 6 12 × 3
+
=
Þ
24
24
24
24
24
Þ
156x 64 48
6 36
+ - = - Þ
24
24 24 24 24
Þ 156x + 64 - 48 = 6 - 36 Þ
46
23
Þ 156x = -46 Þ x = =156
78
38)
39)
3x
1
+ x - 2 = + 2x
2
4
m.c.m(2, 4) = 4
3x × 2 x × 4 2 × 4 1 × 1 2x × 4
+
=
+
Þ
4
4
4
4
4
6x 4x 8 1 8x
Þ
+ - = +
Þ
4
4 4 4 4
Þ 6x + 4x - 8 = 1 + 8x Þ
Þ 6x + 4x - 8x = 1 + 8 Þ 2x = 9 Þ
9
Þx=
2
13 x
x x
+ -2 = +
2 6
4 8
m.c.m(2, 4, 6, 8) = 24
13 × 12 x × 4 2 × 24 x × 6 x × 3
+
=
+
Þ
24
24
24
24
24
156x 4x 48 6x 3x
Þ
+ - =
- Þ
24
24 24 24 24
Þ 156x + 4x - 6x + 3x = 48 Þ
3/6
Colegio Virgen de la Peña
Þ 157x = 48 Þ x =
40)
Matemáticas Primer Ciclo ESO
48
157
44)
-4 × 3 + 4 × x = -3 Þ -12 + 4x = -3 Þ
Þ 4x = -3 + 12 Þ 4x = 9 Þ
5x
2x 12
+ 2-x =
+
3
5
8
m.c.m(3, 5, 8) = 120
5x × 40 2 × 120 x × 120
+
=
120
120
120
2x × 24 12 × 15
=
+
Þ
120
120
200x 240 120x
Þ
+
=
120 120 120
48x 180
=
Þ
120 120
Þ 200x + 240 - 120x =
= 48x - 180 Þ
Þ 200x - 120x - 48x =
= -180 - 240 Þ
Þ 32x = -420 Þ
420
105
Þ x==32
8
Þx=
45)
42)
3x - 3 × 2 = 7 Þ 3x - 6 = 7 Þ
Þ 3x = 7 + 6 Þ 3x = 13 Þ
13
Þx=
3
4(3 - x) = -3
4 × 3 - 4 × x = 7 Þ 12 - 4x = 7 Þ
Þ 12 - 7 = 4x Þ 4x = 5 Þ
5
Þx=
4
43)
Þ -4x + 8x = -3 - 12 Þ
Þ 4x = -15 Þ
Þ x =46)
Juan J. Pascual
15
4
4(3 - x) + 3(x - 1) = -3
12 - 4x + 3x - 3 = -3 Þ
Þ -4x + 3x = -3 + 3 - 12 Þ
Þ -x = -12 Þ x = 12
4(3 − x) − ( − 3 − 5x) = − 1
12 - 4x + 3 + 5x = -1 Þ
Þ -4x + 5x = -1 - 3 - 12 Þ
Þ x = -16
48)
-4(3 - x) - (-3 - 5x) = -1 + x
49)
-12 + 4x + 3 + 5x = -1 + x Þ
Þ 4x + 5x - x = -1 - 3 + 12 Þ
8
Þ 8x = 8 Þ x = = 1
8
-7(1 - x - 2) - (-3 - x) = -(2x - 1)
-7 + 7x + 14 + 3 + x = -2x + 1 Þ
Þ 7x + x + 2x = 1 + 7 - 14 - 3 Þ
11
Þ 10x = 11 Þ x =
10
3(3 - x) = -3 - 2
3 × 3 - 3 × x = -5 Þ 9 - 3x = -5 Þ
Þ 9 + 5 = 3x Þ 3x = 14 Þ
14
Þx=
3
4(3 - x + 2x) = -3
Þ 12 - 4x + 8x = -3 Þ
D. ECUACIONES POLINÓMICAS
3(x - 2) = 7
9
4
4 × 3 - 4 × x + 4 × 2x = -3 Þ
47)
41)
-4(3 - x) = -3
50)
-4(3 - x) - (-3 - 5x) = 2(x - 3x - 1)
4/6
Colegio Virgen de la Peña
51)
Matemáticas Primer Ciclo ESO
-12 + 4x + 3 + 5x = 2x - 6x - 2 Þ
Þ 4x + 5x - 2x + 6x = -2 + 12 - 3 Þ
7
Þ 13x = 7 Þ x =
13
-2(x + 2) - 3(5 - x) = 2(3x - 4x) + 1
55)
m.c.m(2, 3, 6) = 6
(x + 1)× 2 (x + 1)× 1 3 × 3
+
=
Þ
6
6
6
Þ (x + 1) × 2 + (x + 1)× 1 = 9 Þ
Þ 2x + 2 + x + 1 = 9 Þ
2x + x = 9 - 2 - 1 Þ
6
Þ 3x = 6 Þ x = = 2
3
-2x - 4 - 15 + 3x = 6x - 8x + 1 Þ
Þ -2x + 3x - 6x + 8x = 1 + 4 + 15 Þ
20
Þ 3x = 20 Þ x =
3
E. ECUACIONES CON DENOMINADOR (binomios
en el numerador)
52)
56)
x - 1 12
=
2
8
(x + 1)× 2
=
x + 2 12
=
2
3
54)
57)
m.c.m(2, 8) = 8
x×1
Þ (x - 1)× 4 = x Þ
8
8
Þ 4x - 4 = x Þ 4x + x = 4 Þ
4
Þ 5x = 4 Þ x =
5
Juan J. Pascual
3
2
3(x - 4) 5(2x + 1) 2x
+
=
2
3
4
m.c.m(2, 3, 4) = 12
3(x - 4) × 6 5(2x + 1) × 4 2x × 3
+
=
Þ
12
12
12
Þ 18(x - 4) + 20(2x + 1) = 6x Þ
Þ 18x - 72 + 40x + 20 = 6x Þ
18x + 40x - 6x = 72 - 20 Þ
52
Þ 52x = 52 Þ x =
=1
52
=
=
=
Þ -2x = -3 Þ 2x = 3 Þ x =
12 × 2
Þ ( x + 2)× 3 = 24 Þ
6
6
Þ 3x + 6 = 24 Þ 3x = 24 - 6 Þ
18
Þ 3x = 18 Þ x =
=6
3
x-1 x
=
2
8
(x - 1)× 4
+
Þ 2x + 2 + x + 1 = 5x Þ
2x + x - 5x = -2 - 1 Þ
m.c.m(2, 3) = 6
( x + 2) × 3
(x + 1)× 1
5x × 1
Þ
6
6
6
Þ (x + 1) × 2 + (x + 1)× 1 = 5x Þ
12 × 1
Þ (x - 1)× 4 = 12 Þ
8
8
Þ 4x - 4 = 12 Þ 4x = 12 + 6 Þ
18 9
Þ 4x = 18 Þ x =
=
4
2
53)
x + 1 x + 1 5x
+
=
3
6
6
m.c.m(2, 6) = 6
m.c.m(2, 8) = 8
(x - 1)× 4
x+1 x+1 3
+
=
3
6
2
58)
3(2x - 4 + x) 5(2x - 1) 2(x - 3)
=
2
3
4
5/6
Colegio Virgen de la Peña
Matemáticas Primer Ciclo ESO
m.c.m(2, 3, 4) = 12
3(2x - 4 + x) × 6 5(2x - 1) × 4
=
12
12
2(x - 3) × 3
=
Þ
12
Þ 18(3x - 4) - 20(2x - 1) = 6(x - 3) Þ
Þ 54x - 72 - 40x + 20 = 6x - 18 Þ
Þ 54x - 40x - 6x = -18 + 72 - 20 Þ
34
17
Þ 8x = 34 Þ x =
=
8
4
F. OTROS TIPOS
59)
60)
61)
62)
63)
Juan J. Pascual
1
=2
x
1 2
1× 1 2 × x
= Þ
=
Þ 1 = 2x Þ
x 1
x
x
1
Þx=
2
2
3x
=4
2x
3x 2 4
3x 4
3x × 1 4 × 2
= Þ
= Þ
=
Þ
2x
1
2
1
2
2
8
Þ 3x = 8 Þ x =
3
1 2
=
2 x
1× x 2 × 1
=
Þx=2
2x
2x
3
5
=
2 7x
3 × 7x 5 × 2
=
Þ 21x = 10 Þ
14x
14x
10
Þx=
21
5
1
=
-1
2 7x
64)
65)
5 × 7x 1× 2 1 × 14x
=
Þ
14x
14x
14x
Þ 21x = 2 - 14x Þ
Þ 21x + 14x = 2 Þ
2
Þ 35x = 2 Þ x =
35
5
1
+3=
-2
2
3x
5 × 3x 3 × 6x 1 × 2 2 × 6x
+
=
Þ
6x
6x
6x
6x
Þ 15x + 18x = 2 - 12x Þ
Þ 15x + 18x + 12x = 2 Þ
2
Þ 45x = 2 Þ x =
45
2x
3
=
1- x 5
3 × (1 - x)
2x × 5
=
Þ 10x = 3 - 3x Þ
(1 - x) × 5 (1 - x) × 5
3
Þ 10x + 3x = 3 Þ 13x = 3 Þ x =
13
66)
1
1- 2 = 1
x
1
1
×1
1 2
1 1× x 2
1× x
- =- Þ
=Þ
1 x
1
x
x
x
1
1
1
Þ x - = -x Þ x + x = Þ 2x = Þ
2
2
2
1
1
Þx=
Þx=
2×2
4
6/6