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Uuientes sistemas por la regla de Cramer \[ \left\{\begin{array}{l}2 x+y=5 \\ 3 x-4 y=-2\end{array}\right. \]

Ask by Ray Carter. in Colombia
Mar 14,2025

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La solución del sistema es \( x = \frac{18}{11} \) y \( y = \frac{19}{11} \).

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Para resolver el sistema de ecuaciones usando la regla de Cramer, primero necesitamos calcular el determinante de la matriz de coeficientes. Para las ecuaciones \(2x + y = 5\) y \(3x - 4y = -2\), la matriz de coeficientes es: \[ D = \begin{vmatrix} 2 & 1 \\ 3 & -4 \end{vmatrix} = (2)(-4) - (1)(3) = -8 - 3 = -11 \] Luego, calculamos los determinantes \(D_x\) y \(D_y\) sustituyendo las columnas de la matriz de coeficientes por el vector de términos independientes: \[ D_x = \begin{vmatrix} 5 & 1 \\ -2 & -4 \end{vmatrix} = (5)(-4) - (1)(-2) = -20 + 2 = -18 \] \[ D_y = \begin{vmatrix} 2 & 5 \\ 3 & -2 \end{vmatrix} = (2)(-2) - (5)(3) = -4 - 15 = -19 \] Finalmente, se calculan las variables: \[ x = \frac{D_x}{D} = \frac{-18}{-11} = \frac{18}{11} \] \[ y = \frac{D_y}{D} = \frac{-19}{-11} = \frac{19}{11} \] Así que la solución del sistema es \(x = \frac{18}{11}\) y \(y = \frac{19}{11}\).

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