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Ecuaciones 1° grado

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Ecuaciones 1° grado

  1. 1. Resolución de ecuaciones de primer grado
  2. 2. Índice <ul><li>Definiciones </li></ul><ul><li>Resolución de ecuaciones de primer grado sencillas </li></ul><ul><li>Resolución de ecuaciones con paréntesis </li></ul><ul><li>Resolución de ecuaciones con denominadores </li></ul><ul><li>Resolución de problemas </li></ul>
  3. 3. Identidades y ecuaciones <ul><li>Una identidad es una igualdad que se cumple siempre. </li></ul><ul><li>Por ejemplo: 3a = a + a + a se cumple para cualquier valor de a. </li></ul><ul><li>En cambio, una ecuación es una igualdad que sólo se cumple para algún o algunos valores . </li></ul><ul><li>Por ejemplo: a + 4 = 6 sólo se cumple para a =2 . </li></ul>
  4. 4. Ecuaciones de primer grado segundo miembro primer miembro Una ecuación de primer grado es una igualdad formada por uno o más polinomios de primer grado y en la que la variable es una letra llamada incógnita . Términos de la ecuación
  5. 5. Son ecuaciones de primer grado? NO SI NO
  6. 6. Resolución de ecuaciones de primer grado Ejemplo : <ul><li>2x +3 = 5 – x </li></ul><ul><li>Pasamos cambiando de signo 2x + x = 5 - 3 </li></ul><ul><li>Hacemos las operaciones con números enteros 3x=2 </li></ul><ul><li>El 3 pasa dividiendo x=2/3 </li></ul>
  7. 7. Mas ejemplos     3x – 1 = 2 3x = 2 + 1 => 3x = 3 => x = 3/3 => x = 1 2x – 5 = x + 2 2x - x = 2 + 5 => x = 7 7x – 6 + 6 = 5x + 3 + 6 7x - 5x = 6 + 3 + 6 - 6 2x = 6+3 => 2x = 9 => x = 9/2     8 –x = 4 + 2 - x = 4 + 2 - 8 =>- x = 6 - 8 =>- x =- 2 => x=2
  8. 8. En definitiva... 5x + 2x – 3x + 4x – 6 + 8 – 3x + 1 = 2x – 5 + 4x – 6 + 2 5x + 2x - 3x + 4x - 3x - 2x - 4x = - 5 - 6 + 2 + 6 - 8 - 1 11x - 12x = 8 - 20 -x = - 12 => x = 12 No olvides cambiar de signo la x cuando ésta sea negativa, CAMBIANDO DE SIGNO EL NÚMERO DE LA DERECHA
  9. 9. Ecuaciones con paréntesis <ul><li>Quitamos los paréntesis con la regla del producto. </li></ul>- 3 ( 2x + 1 ) + 5 ·( - x + 6 ) = 7 - 6x – 3 – 5x + 30 = 7 - 6x – 5x = 7 - 30 + 3 - 11x = -20 =>
  10. 10. EJERCITEMOS <ul><li>1) x + 2 = 5 Despejar la incógnita </li></ul><ul><li>x = 5 – 2 </li></ul><ul><li>x = 3 </li></ul><ul><li>2) - 4x + 3 - 3x + 2 = 5x + 6 </li></ul><ul><li>3) 6x – 5 = 7x + 4 </li></ul><ul><li>4) 2(6x + 1 ) = 3x – 5 </li></ul><ul><li>5) 6 + 3x = 3(2x + 1) </li></ul><ul><li>6) 3(-2x - 3) = -6(3x - 2) </li></ul><ul><li>7) x + 2(3x - 6) = 2(x + 3) - 6( x + 1) </li></ul><ul><li>8) 2(x - 1) + 2(x + 1) = 6(x + 3) -5(2 - 7x) </li></ul><ul><li>9) -3(3x+1+4x)=-9x-1 </li></ul><ul><li>10) 2(-x-1)-3(x-9)=2 </li></ul><ul><li>11) 2(x+1)+2(6x+5)+9(-3x-2)=4 </li></ul>
  11. 11. <ul><li>2) - 4x + 3 - 3x + 2 = 5x + 6 </li></ul>
  12. 12. <ul><li>3) 6x – 5 = 7x + 4 </li></ul>
  13. 13. <ul><li>4) 2 ( 6x + 1 ) = 3x – 5 </li></ul>
  14. 14. <ul><li>5) 6 + 3x = 3( 2x + 1) </li></ul>
  15. 15. <ul><li>6) 3 (- 2x - 3 ) = - 6 ( 3x - 2 ) </li></ul>
  16. 16. <ul><li>7) x + 2 ( 3x - 6 ) = 2 ( x + 3 ) - 6 ( x + 1 ) </li></ul>
  17. 17. <ul><li>8) 2(x - 1) + 2(x + 1) = 6(x + 3) -5(2 - 7x) </li></ul>
  18. 18. <ul><li>9) -3(3x+1+4x)=-9x-1 </li></ul>
  19. 19. <ul><li>10) 2(-x-1)-3(x-9)=2 </li></ul>
  20. 20. <ul><li>11) 2(x+1)+2(6x+5)+9(-3x-2)=4 </li></ul>
  21. 21. Ecuaciones con denominadores <ul><li>Caso: una fracción a la izquierda y otra a la derecha </li></ul>3 ( x – 1 ) = 2 ( 4x – 5 ) 3x - 3 = 8x - 10 => 3x-8x = -10+3 -5x = -7 => x=7/5 <ul><ul><li>Podemos multiplicar en cruz de esta manera </li></ul></ul><ul><ul><li>Y resolvemos como hasta ahora </li></ul></ul>
  22. 22. Ecuaciones con denominadores <ul><li>Caso general :Más de una fracción a la izquierda y/o más de una fracción a la derecha </li></ul><ul><ul><li>Multiplicamos TODA la ecuación por el M.C.M. de los denominadores </li></ul></ul><ul><ul><li>Primero dividimos y después multiplicamos </li></ul></ul>m.c.m. ( 6, 4 ) = 2 2 ● 3 = 12 6 = 2 ● 3 4 = 2 ● 2 = 2 2
  23. 23. ¡Ejemplo importante! <ul><li>Si las fracciones contienen más de un número o incógnita, </li></ul>Tendremos que colocar paréntesis y aplicar la regla del producto. 1 ● x – 2 ● (4x – 5) = 3 ● 3x x – 8x + 10 = 9x - 16x = -10 => - 7x – 9x = -10
  24. 24. Y el ejemplo mas complicado... <ul><li>Si tenemos números que multiplican a paréntesis </li></ul>Multiplica Multiplica por el M.C.M. Quita los denominadores 9x + 18x – 40x = 10 + 25 – 18 – 30
  25. 25. Un ejemplo mas y ejercicios 1 · x – 2 · (4x – 5) = 3 · 3x x – 8x + 10 = 9x x – 8x –9x = -10 m.c.m. (6,3,2) = 6 ; - 16x = -10 ¡¡¡NO OLVIDES COLOCAR PARÉNTESIS!!! Ejercicios:
  26. 26. Más ejercicios.... ... ... ... ... ... ... ... ... ... ...
  27. 27. SIGAMOS EJERCITANDO <ul><li>1) 4x + 6 = 2x + 1 </li></ul>3
  28. 28. <ul><li>2) 6x + 6 = 3x - 23 </li></ul>2 3
  29. 29. <ul><li>3) 2 x+ 3( x - 5 ) = 2x </li></ul>3 2
  30. 30. <ul><li>4) - 3x + 2 + 5x = 2 </li></ul>4 2
  31. 31. <ul><li>5) 6x + -2(x – 3) - -2(x + 1)= 2 3x- 4x + 4 </li></ul>5 6 2 ) (
  32. 32. Un aspecto a recordar <ul><li>Podemos dejar la incógnita a la derecha de la ecuación. ¡Y sigue estando bien!. </li></ul>Ejemplo: x -5 = 6x => -5 = 6x-x => -5 = 5x => -1 = x <ul><li>Lo que pasa es que podemos dar la vuelta a la igualdad así : </li></ul>x = -1 <ul><li>¿Sabes por qué? </li></ul>-1 = x => -x = 1=> x = -1 Ejercicio: 6 = x => x = 6 -3 = -x => -x = -3 => x=3
  33. 33. Traducción a lenguaje algebraico <ul><li>Sea el número pedido la letra X </li></ul><ul><li>El doble de un número </li></ul><ul><li>El triple de un número </li></ul><ul><li>El quíntuplo de un número </li></ul><ul><li>La mitad de un número </li></ul><ul><li>La séptima parte de un número </li></ul>2X 3X 5X X/2 X/7
  34. 34. Traducción a lenguaje algebraico I <ul><li>El doble de un número más la cuarta parte del mismo número </li></ul><ul><li>El cuádruplo de un número menos la mitad del triple de éste número es ocho </li></ul><ul><li>La suma de dos números consecutivos </li></ul><ul><li>Si yo tengo X años, dentro de tres años tendré, el doble de los que yo tuve hace 15 años </li></ul>2x + 4x - = 8 X + X+1 X+3 = 2( X – 15 )
  35. 35. Resolución de problemas <ul><li>Identifica la incógnita </li></ul><ul><li>Plantea la ecuación. </li></ul><ul><li>Resuelve la ecuación. </li></ul><ul><li>Comprueba la solución. </li></ul><ul><li>Expresa con palabras la solución. </li></ul>
  36. 36. Primer ejemplo 1) Identifica X: El número pedido 2) Plantea 3) Resuelve 4) Comprueba 18-12 = 6 ; 18/3 =6=> 6 = 6 5) Expresa El número pedido es el 18 Si restamos 12 a un número, se obtiene la tercera parte. ¿Cuál es el número?
  37. 37. Segundo ejemplo 1) Identifica X: El número pedido 2) Plantea 3) Resuelve 4) Comprueba 8/2+20 =24 ; 3*8=24 =>24 = 24 5) Expresa El número pedido es el 24 x/2 20 3x A) x/2+20=3x B) x/2=3x-20 ? Calcular la mitad de un número que es 20 unidades menor que su triple.
  38. 38. Tercer ejemplo 1) Identifica: Precio helado : Precio cómic: Precio videojuego 2) Plantea: 3) Resuelve: 4) Comprueba: 11+2,2+1,1=14,3 5) Expresa: El videojuego costaba 11€, el cómic 2,20€, y el helado 1,10€ 2x 5·2 x = 10x x Por un video juego, un comic y un helado, Andrés ha pagado 14,30 €. El video juego es cinco veces mas caro que el comic, y este cuesta el doble que el helado. ¿Cuál era el precio de cada artículo?

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