Warren Cummings
10/28/2023 · Senior High School
\( \begin{aligned} \text { f. } & \frac{2(x+4)}{3}-\frac{y}{2}=\frac{9}{2} \\ x+2 y-\frac{1}{3}(3 x-2) & =-\frac{4}{3}\end{aligned} \)
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Tutor-Verified Answer
Step-by-step Solution
Para resolver el sistema de ecuaciones:
1. \(\frac{2(x+4)}{3}-\frac{y}{2}=\frac{9}{2}\)
2. \(x+2y-\frac{1}{3}(3x-2)=-\frac{4}{3}\)
Primero, simplificamos cada ecuación.
**Ecuación 1:**
Multiplicamos toda la ecuación por 6 (el mínimo común múltiplo de 3 y 2) para eliminar los denominadores:
\[
6 \left( \frac{2(x+4)}{3} \right) - 6 \left( \frac{y}{2} \right) = 6 \left( \frac{9}{2} \right)
\]
Esto se convierte en:
\[
4(x+4) - 3y = 27
\]
Desarrollamos:
\[
4x + 16 - 3y = 27
\]
Reorganizamos:
\[
4x - 3y = 27 - 16
\]
\[
4x - 3y = 11 \quad \text{(Ecuación 1 simplificada)}
\]
**Ecuación 2:**
Simplificamos la segunda ecuación:
\[
x + 2y - \frac{1}{3}(3x - 2) = -\frac{4}{3}
\]
Multiplicamos por 3 para eliminar el denominador:
\[
3(x + 2y) - (3x - 2) = -4
\]
Esto se convierte en:
\[
3x + 6y - 3x + 2 = -4
\]
Simplificamos:
\[
6y + 2 = -4
\]
Reorganizamos:
\[
6y = -4 - 2
\]
\[
6y = -6
\]
\[
y = -1 \quad \text{(Ecuación 2 simplificada)}
\]
Ahora sustituimos \(y = -1\) en la Ecuación 1:
\[
4x - 3(-1) = 11
\]
\[
4x + 3 = 11
\]
\[
4x = 11 - 3
\]
\[
4x = 8
\]
\[
x = 2
\]
Por lo tanto, la solución del sistema de ecuaciones es:
\[
\boxed{(2, -1)}
\]
Quick Answer
La solución del sistema de ecuaciones es \( (2, -1) \).
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