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Sagot :
¡Claro! Vamos a resolver cada sistema de ecuaciones utilizando el método de igualación paso a paso.
### 1. Sistema de ecuaciones:
[tex]\(\left\{\begin{array}{c}x-2 y=11 \quad (1) \\ x+5 y=-17 \quad (2)\end{array}\right.\)[/tex]
Paso 1: Despejamos [tex]\(x\)[/tex] en ambas ecuaciones.
De la ecuación (1):
[tex]\[ x = 11 + 2y \][/tex]
De la ecuación (2):
[tex]\[ x = -17 - 5y \][/tex]
Paso 2: Igualamos ambas expresiones de [tex]\(x\)[/tex]:
[tex]\[ 11 + 2y = -17 - 5y \][/tex]
Paso 3: Resolvemos para [tex]\(y\)[/tex]:
[tex]\[ 11 + 2y + 5y = -17 \][/tex]
[tex]\[ 11 + 7y = -17 \][/tex]
[tex]\[ 7y = -17 - 11 \][/tex]
[tex]\[ 7y = -28 \][/tex]
[tex]\[ y = -4 \][/tex]
Paso 4: Sustituimos el valor de [tex]\(y\)[/tex] en una de las ecuaciones para encontrar [tex]\(x\)[/tex]:
Sustituimos en la ecuación (1):
[tex]\[ x - 2(-4) = 11 \][/tex]
[tex]\[ x + 8 = 11 \][/tex]
[tex]\[ x = 11 - 8 \][/tex]
[tex]\[ x = 3 \][/tex]
Solución del primer sistema:
[tex]\[ x = 3 \][/tex]
[tex]\[ y = -4 \][/tex]
### 2. Sistema de ecuaciones:
[tex]\(\left\{\begin{array}{l}-x+y=-1 \quad (3) \\ 4 x-2 y=5 \quad (4)\end{array}\right.\)[/tex]
Paso 1: Despejamos [tex]\(x\)[/tex] en la ecuación (3):
[tex]\[ -x + y = -1 \implies -x = -1 - y \implies x = 1 + y \][/tex]
Paso 2: Sustituimos [tex]\(x = 1 + y\)[/tex] en la ecuación (4):
[tex]\[ 4(1 + y) - 2y = 5 \][/tex]
[tex]\[ 4 + 4y - 2y = 5 \][/tex]
[tex]\[ 4 + 2y = 5 \][/tex]
[tex]\[ 2y = 1 \][/tex]
[tex]\[ y = \frac{1}{2} \][/tex]
Paso 3: Sustituimos el valor de [tex]\(y\)[/tex] en [tex]\(x = 1 + y\)[/tex]:
[tex]\[ x = 1 + \frac{1}{2} \][/tex]
[tex]\[ x = \frac{3}{2} \][/tex]
Solución del segundo sistema:
[tex]\[ x = \frac{3}{2} \][/tex]
[tex]\[ y = \frac{1}{2} \][/tex]
### 3. Sistema de ecuaciones:
[tex]\(\left\{\begin{array}{c}4 x+5 y=-3 \quad (5) \\ -7 x+3 y=-13 \quad (6)\end{array}\right.\)[/tex]
Paso 1: Despejamos [tex]\(x\)[/tex] en la ecuación (5):
[tex]\[ 4x + 5y = -3 \implies 4x = -3 - 5y \implies x = \frac{-3 - 5y}{4} \][/tex]
Paso 2: Sustituimos [tex]\(x = \frac{-3 - 5y}{4}\)[/tex] en la ecuación (6):
[tex]\[ -7\left(\frac{-3 - 5y}{4}\right) + 3y = -13 \][/tex]
[tex]\[ \frac{21 + 35y}{4} + 3y = -13 \][/tex]
Multiplicamos todo por 4 para eliminar el denominador:
[tex]\[ 21 + 35y + 12y = -52 \][/tex]
[tex]\[ 21 + 47y = -52 \][/tex]
[tex]\[ 47y = -52 - 21 \][/tex]
[tex]\[ 47y = -73 \][/tex]
[tex]\[ y = \frac{-73}{47} \][/tex]
Paso 3: Sustituimos el valor de [tex]\(y\)[/tex] en [tex]\(x = \frac{-3 - 5y}{4}\)[/tex]:
[tex]\[ x = \frac{-3 - 5\left(\frac{-73}{47}\right)}{4} \][/tex]
[tex]\[ x = \frac{-3 + \frac{365}{47}}{4} \][/tex]
[tex]\[ x = \frac{-3 \cdot \frac{47}{47} + \frac{365}{47}}{4} \][/tex]
[tex]\[ x = \frac{\frac{-141 + 365}{47}}{4} \][/tex]
[tex]\[ x = \frac{\frac{224}{47}}{4} \][/tex]
[tex]\[ x = \frac{224}{4 \cdot 47} \][/tex]
[tex]\[ x = \frac{224}{188} \][/tex]
[tex]\[ x = \frac{56}{47} \][/tex]
Solución del tercer sistema:
[tex]\[ x = \frac{56}{47} \][/tex]
[tex]\[ y = \frac{-73}{47} \][/tex]
Por lo tanto, las soluciones de los sistemas de ecuaciones son:
1. [tex]\( x = 3, y = -4 \)[/tex]
2. [tex]\( x = \frac{3}{2}, y = \frac{1}{2} \)[/tex]
3. [tex]\( x = \frac{56}{47}, y = \frac{-73}{47} \)[/tex]
### 1. Sistema de ecuaciones:
[tex]\(\left\{\begin{array}{c}x-2 y=11 \quad (1) \\ x+5 y=-17 \quad (2)\end{array}\right.\)[/tex]
Paso 1: Despejamos [tex]\(x\)[/tex] en ambas ecuaciones.
De la ecuación (1):
[tex]\[ x = 11 + 2y \][/tex]
De la ecuación (2):
[tex]\[ x = -17 - 5y \][/tex]
Paso 2: Igualamos ambas expresiones de [tex]\(x\)[/tex]:
[tex]\[ 11 + 2y = -17 - 5y \][/tex]
Paso 3: Resolvemos para [tex]\(y\)[/tex]:
[tex]\[ 11 + 2y + 5y = -17 \][/tex]
[tex]\[ 11 + 7y = -17 \][/tex]
[tex]\[ 7y = -17 - 11 \][/tex]
[tex]\[ 7y = -28 \][/tex]
[tex]\[ y = -4 \][/tex]
Paso 4: Sustituimos el valor de [tex]\(y\)[/tex] en una de las ecuaciones para encontrar [tex]\(x\)[/tex]:
Sustituimos en la ecuación (1):
[tex]\[ x - 2(-4) = 11 \][/tex]
[tex]\[ x + 8 = 11 \][/tex]
[tex]\[ x = 11 - 8 \][/tex]
[tex]\[ x = 3 \][/tex]
Solución del primer sistema:
[tex]\[ x = 3 \][/tex]
[tex]\[ y = -4 \][/tex]
### 2. Sistema de ecuaciones:
[tex]\(\left\{\begin{array}{l}-x+y=-1 \quad (3) \\ 4 x-2 y=5 \quad (4)\end{array}\right.\)[/tex]
Paso 1: Despejamos [tex]\(x\)[/tex] en la ecuación (3):
[tex]\[ -x + y = -1 \implies -x = -1 - y \implies x = 1 + y \][/tex]
Paso 2: Sustituimos [tex]\(x = 1 + y\)[/tex] en la ecuación (4):
[tex]\[ 4(1 + y) - 2y = 5 \][/tex]
[tex]\[ 4 + 4y - 2y = 5 \][/tex]
[tex]\[ 4 + 2y = 5 \][/tex]
[tex]\[ 2y = 1 \][/tex]
[tex]\[ y = \frac{1}{2} \][/tex]
Paso 3: Sustituimos el valor de [tex]\(y\)[/tex] en [tex]\(x = 1 + y\)[/tex]:
[tex]\[ x = 1 + \frac{1}{2} \][/tex]
[tex]\[ x = \frac{3}{2} \][/tex]
Solución del segundo sistema:
[tex]\[ x = \frac{3}{2} \][/tex]
[tex]\[ y = \frac{1}{2} \][/tex]
### 3. Sistema de ecuaciones:
[tex]\(\left\{\begin{array}{c}4 x+5 y=-3 \quad (5) \\ -7 x+3 y=-13 \quad (6)\end{array}\right.\)[/tex]
Paso 1: Despejamos [tex]\(x\)[/tex] en la ecuación (5):
[tex]\[ 4x + 5y = -3 \implies 4x = -3 - 5y \implies x = \frac{-3 - 5y}{4} \][/tex]
Paso 2: Sustituimos [tex]\(x = \frac{-3 - 5y}{4}\)[/tex] en la ecuación (6):
[tex]\[ -7\left(\frac{-3 - 5y}{4}\right) + 3y = -13 \][/tex]
[tex]\[ \frac{21 + 35y}{4} + 3y = -13 \][/tex]
Multiplicamos todo por 4 para eliminar el denominador:
[tex]\[ 21 + 35y + 12y = -52 \][/tex]
[tex]\[ 21 + 47y = -52 \][/tex]
[tex]\[ 47y = -52 - 21 \][/tex]
[tex]\[ 47y = -73 \][/tex]
[tex]\[ y = \frac{-73}{47} \][/tex]
Paso 3: Sustituimos el valor de [tex]\(y\)[/tex] en [tex]\(x = \frac{-3 - 5y}{4}\)[/tex]:
[tex]\[ x = \frac{-3 - 5\left(\frac{-73}{47}\right)}{4} \][/tex]
[tex]\[ x = \frac{-3 + \frac{365}{47}}{4} \][/tex]
[tex]\[ x = \frac{-3 \cdot \frac{47}{47} + \frac{365}{47}}{4} \][/tex]
[tex]\[ x = \frac{\frac{-141 + 365}{47}}{4} \][/tex]
[tex]\[ x = \frac{\frac{224}{47}}{4} \][/tex]
[tex]\[ x = \frac{224}{4 \cdot 47} \][/tex]
[tex]\[ x = \frac{224}{188} \][/tex]
[tex]\[ x = \frac{56}{47} \][/tex]
Solución del tercer sistema:
[tex]\[ x = \frac{56}{47} \][/tex]
[tex]\[ y = \frac{-73}{47} \][/tex]
Por lo tanto, las soluciones de los sistemas de ecuaciones son:
1. [tex]\( x = 3, y = -4 \)[/tex]
2. [tex]\( x = \frac{3}{2}, y = \frac{1}{2} \)[/tex]
3. [tex]\( x = \frac{56}{47}, y = \frac{-73}{47} \)[/tex]
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