Ford Cummings
05/20/2023 · Elementary School

Encontrar el número complejo \( z \) que es solución de la siguiente ecuación: \( 2-i=\frac{z}{3+i} \)

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Para encontrar el número complejo \( z \) que satisface la ecuación \( 2 - i = \frac{z}{3 + i} \), primero multiplicamos ambos lados de la ecuación por \( 3 + i \) para despejar \( z \): \[ z = (2 - i)(3 + i) \] Ahora, realizamos la multiplicación en el lado derecho: \[ z = (2)(3) + (2)(i) + (-i)(3) + (-i)(i) \] Calculamos cada término: 1. \( 2 \cdot 3 = 6 \) 2. \( 2 \cdot i = 2i \) 3. \( -i \cdot 3 = -3i \) 4. \( -i \cdot i = -(-1) = 1 \) Ahora sumamos todos los términos: \[ z = 6 + 2i - 3i + 1 \] Combinamos los términos reales y los términos imaginarios: \[ z = (6 + 1) + (2i - 3i) = 7 - i \] Por lo tanto, la solución es: \[ \boxed{7 - i} \]

Quick Answer

La solución es \( z = 7 - i \).
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