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