¿Cuál de las siguientes operaciones NO produce siempre una ecuación equivalente?

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¿Qué es una ecuación? Una **ecuación** es una igualdad algebraica que contiene una o más incógnitas (variables). Resolver una ecuación consiste en encontrar el valor o los valores de la(s) incógnita(s) que hacen que la igualdad sea verdadera. Estos valores se denominan **soluciones** o **raíces** de la ecuación. Una expresión matemática de la forma $A = B$, donde $A$ y $B$ son expresiones algebraicas que contienen variables. La solución es el conjunto de valores que satisfacen la igualdad. El símbolo '=' indica que el valor de la expresión de la izquierda es idéntico al de la derecha para los valores que son solución. ### Clasificación de las ecuaciones Tipo | Característica | Ejemplo Lineal | La incógnita tiene exponente 1 | $2x + 3 = 7$ Cuadrática | La incógnita tiene exponente máximo 2 | $x^2 - 5x + 6 = 0$ Racional | Contiene fracciones con la incógnita en el denominador | $\frac{2}{x} = 3$ Irracional | La incógnita está dentro de un radical | $\sqrt{x+1} = 4$ Exponencial | La incógnita está en el exponente | $2^x = 16$ Logarítmica | La incógnita está dentro de un logaritmo | $\log_2(x) = 3$ ### Ecuaciones equivalentes Dos ecuaciones son **equivalentes** si tienen exactamente las mismas soluciones. Para obtener una ecuación equivalente a otra, podemos: 1. **Sumar o restar** la misma cantidad en ambos lados. 2. **Multiplicar o dividir** ambos lados por la misma cantidad **distinta de cero**. Al multiplicar o dividir una ecuación por una expresión que contiene la incógnita, podríamos introducir soluciones extrañas o perder soluciones. Es crucial verificar las soluciones obtenidas en la ecuación original. ¿Cuál de las siguientes operaciones NO produce siempre una ecuación equivalente? Multiplicar por una expresión que contiene la incógnita (y que podría ser cero) puede alterar las soluciones o introducir raíces extrañas. Las otras operaciones son válidas universalmente. --- ## Resolución de ecuaciones lineales La ecuación lineal general con una incógnita es de la forma: $$ax + b = 0$$ donde $a \neq 0$. Su solución única es: $$x = -\frac{b}{a}$$ ### Pasos para resolver una ecuación lineal 1. **Eliminar denominadores**: Multiplicar todos los términos de la ecuación por el mínimo común múltiplo (MCM) de los denominadores. 2. **Eliminar paréntesis**: Aplicar la propiedad distributiva para simplificar. 3. **Trasposición de términos**: Agrupar todos los términos con la incógnita en un lado de la igualdad y los términos independientes en el otro. 4. **Reducir términos semejantes**: Simplificar cada lado de la ecuación. 5. **Despejar la incógnita**: Dividir ambos lados por el coeficiente de la incógnita. Resolver: $\frac{2x-1}{3} = \frac{x+4}{2}$ **Paso 1**: MCM de 3 y 2 es 6. Multiplicamos ambos lados por 6: $6 \cdot \frac{2x-1}{3} = 6 \cdot \frac{x+4}{2} \implies 2(2x-1) = 3(x+4)$ **Paso 2**: Eliminar paréntesis: $4x - 2 = 3x + 12$ **Paso 3**: Trasponer términos: $4x - 3x = 12 + 2$ **Paso 4**: Reducir: $x = 14$ **Paso 5**: La solución es $x = 14$. Comprobamos en la ecuación original: $\frac{2(14)-1}{3} = \frac{28-1}{3} = \frac{27}{3} = 9$ $\frac{14+4}{2} = \frac{18}{2} = 9$ La igualdad se cumple, por lo tanto, la solución es correcta. ### Casos especiales en ecuaciones lineales Si al resolver una ecuación lineal se obtiene una afirmación que es siempre verdadera (ej. $0 = 0$), la ecuación tiene **infinitas soluciones** y se llama **identidad**. Si se obtiene una contradicción (ej. $0 = 5$), la ecuación no tiene solución y se llama **inconsistente**. --- ## Sistemas de ecuaciones lineales Un sistema de ecuaciones lineales es un conjunto de dos o más ecuaciones lineales que comparten las mismas incógnitas. La solución es el conjunto de valores que satisface **todas** las ecuaciones simultáneamente. ### Métodos de resolución para sistemas 2x2 1. Despejar una incógnita en una de las ecuaciones. 2. Sustituir la expresión obtenida en la otra ecuación. 3. Resolver la ecuación resultante (que ahora es lineal con una incógnita). 4. Sustituir el valor hallado para obtener la otra incógnita. 1. Despejar la **misma** incógnita en ambas ecuaciones. 2. Igualar las expresiones obtenidas. 3. Resolver la ecuación resultante. 4. Sustituir el valor hallado para obtener la otra incógnita. 1. Multiplicar las ecuaciones por números adecuados para que los coeficientes de una incógnita sean opuestos. 2. Sumar las ecuaciones para eliminar esa incógnita. 3. Resolver la ecuación resultante. 4. Sustituir el valor hallado para obtener la otra incógnita. El error más común al usar el método de reducción es olvidar multiplicar **todos** los términos de la ecuación por el factor elegido, no solo el término que se quiere eliminar. Resolver el sistema: $$\begin{cases} 2x + y = 8 \\ x - 3y = -3 \end{cases}$$ **Usando el método de sustitución**: 1. Despejamos $y$ de la primera ecuación: $y = 8 - 2x$ 2. Sustituimos en la segunda: $x - 3(8 - 2x) = -3 \implies x - 24 + 6x = -3 \implies 7x = 21 \implies x = 3$ 3. Sustituimos $x=3$ en $y = 8 - 2(3) = 8 - 6 = 2$ La solución del sistema es $x = 3$, $y = 2$. ### Clasificación de sistemas lineales Un sistema puede ser: * **Compatible determinado**: Tiene una única solución. * **Compatible indeterminado**: Tiene infinitas soluciones. * **Incompatible**: No tiene solución. En un sistema 2x2, la representación gráfica de cada ecuación es una recta. * Si las rectas se cortan en un punto → **Compatible determinado**. * Si las rectas son coincidentes → **Compatible indeterminado**. * Si las rectas son paralelas → **Incompatible**. ```mermaid flowchart TD A["Clasificación de sistemas 2x2"] A --> B["Compatible"] A --> C["Incompatible"] B --> D["Determinado\n(1 solución)"] B --> E["Indeterminado\n(Infinitas)"] D --> F["Rectas secantes"] E --> G["Rectas coincidentes"] C --> H["Rectas paralelas"] ``` Para recordar la clasificación, asocia "Determinado" con "Decidido" (una única respuesta) e "Indeterminado" con "Indeciso" (muchas opciones). --- ## Ecuaciones cuadráticas Una ecuación cuadrática o de segundo grado es aquella que puede expresarse en la forma: $$ax^2 + bx + c = 0$$ con $a \neq 0$. Los coeficientes $a$, $b$ y $c$ son números reales. ### Métodos de resolución #### 1. Factorización Si la ecuación se puede factorizar como $(x - r_1)(x - r_2) = 0$, entonces las soluciones son $x = r_1$ y $x = r_2$. Se basa en la propiedad: "Si el producto de dos números es cero, al menos uno de ellos es cero". #### 2. Completando el cuadrado Consiste en transformar la ecuación en un trinomio cuadrado perfecto. Por ejemplo, para $x^2 + bx + c = 0$, se suma y resta $(b/2)^2$ para obtener $(x + b/2)^2 = (b/2)^2 - c$. #### 3. Fórmula general (Ecuación cuadrática) La solución de $ax^2 + bx + c = 0$ se obtiene con la **fórmula general**: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ La fórmula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ proporciona las raíces de la ecuación $ax^2 + bx + c = 0$. Su validez es universal para todos los casos. #### Discriminante y naturaleza de las soluciones El discriminante es la expresión $\Delta = b^2 - 4ac$. Su valor determina la naturaleza de las raíces: * **Si $\Delta > 0$**: Dos soluciones reales y distintas. * **Si $\Delta = 0$**: Una solución real doble (o dos soluciones iguales). * **Si $\Delta < 0$**: Dos soluciones complejas conjugadas (no hay soluciones reales). Resolver: $x^2 - 5x + 6 = 0$ **Por factorización**: Buscamos dos números que sumen $-5$ y cuyo producto sea $6$. Esos números son $-2$ y $-3$. $(x - 2)(x - 3) = 0 \implies x - 2 = 0 \text{ o } x - 3 = 0 \implies x = 2, x = 3$ **Usando la fórmula general** ($a=1, b=-5, c=6$): $\Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1$ $x = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2} \implies x = \frac{5+1}{2} = 3, x = \frac{5-1}{2} = 2$ Un error común al aplicar la fórmula general es confundir el signo de $b$ cuando es negativo. Recuerda que $-b$ es el opuesto de $b$. Por ejemplo, para $2x^2 - 4x + 1 = 0$, $b = -4$, por lo que $-b = 4$. La fórmula correcta es $x = \frac{4 \pm \sqrt{(-4)^2 - 4(2)(1)}}{4}$. Las raíces de una ecuación son los valores numéricos que, al ser sustituidos por la incógnita, satisfacen la igualdad. También se les llama ceros de la función polinómica asociada. ¿Cuál es el discriminante de la ecuación $3x^2 - 2x + 1 = 0$ y cuántas soluciones reales tiene? $\Delta = b^2 - 4ac = (-2)^2 - 4(3)(1) = 4 - 12 = -8$. Como es negativo, no tiene soluciones reales, sino dos complejas conjugadas. ### Propiedades de las raíces (Teorema de Vieta) Para la ecuación $ax^2 + bx + c = 0$ con raíces $x_1$ y $x_2$: * Suma de raíces: $x_1 + x_2 = -\frac{b}{a}$ * Producto de raíces: $x_1 \cdot x_2 = \frac{c}{a}$ --- ## Ecuaciones con valor absoluto Una ecuación en valor absoluto tiene la forma $|f(x)| = k$, donde $k \ge 0$. Para resolverla, se plantean dos ecuaciones sin el valor absoluto: 1. $f(x) = k$ 2. $f(x) = -k$ Resolver $|2x - 1| = 5$. 1. $2x - 1 = 5 \implies 2x = 6 \implies x = 3$ 2. $2x - 1 = -5 \implies 2x = -4 \implies x = -2$ Las soluciones son $x = 3$ y $x = -2$. --- ## Resumen de conceptos clave * Una **ecuación** es una igualdad matemática con incógnitas. * La solución de una ecuación es el valor que la satisface. * Para **ecuaciones lineales** ($ax+b=0$), la solución es $x=-b/a$. * Un **sistema de ecuaciones** se puede resolver por sustitución, igualación o reducción. * Las **ecuaciones cuadráticas** ($ax^2+bx+c=0$) se resuelven por factorización o con la **fórmula general**: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. * El **discriminante** ($\Delta = b^2 - 4ac$) indica la cantidad de soluciones reales. * Siempre **verificar** las soluciones en la ecuación original, especialmente al manipular expresiones con la incógnita. Hemos cubierto los tipos fundamentales de ecuaciones (lineales, sistemas y cuadráticas), sus métodos de resolución y la importancia de verificar las soluciones. 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Examen sobre ecuaciones

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