Matemática discreta Ejemplos

$4 x - 3 y + z + 1 = 0$ , $5 x + 7 y + 2 z + 2 = 0$ , $3 x - 5 y - z - 1 = 0$

Paso 1

Move variables to the left and constant terms to the right.

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Paso 1.1

Resta $1$ de ambos lados de la ecuación.

$4 x - 3 y + z = - 1$

$5 x + 7 y + 2 z + 2 = 0$

$3 x - 5 y - z - 1 = 0$

Paso 1.2

Resta $2$ de ambos lados de la ecuación.

$4 x - 3 y + z = - 1$

$5 x + 7 y + 2 z = - 2$

$3 x - 5 y - z - 1 = 0$

Paso 1.3

Suma $1$ a ambos lados de la ecuación.

$4 x - 3 y + z = - 1$

$5 x + 7 y + 2 z = - 2$

$3 x - 5 y - z = 1$

$4 x - 3 y + z = - 1$

$5 x + 7 y + 2 z = - 2$

$3 x - 5 y - z = 1$

Paso 2

Write the system as a matrix.

$[\begin{array}{c} 4 & - 3 & 1 & - 1 \\ 5 & 7 & 2 & - 2 \\ 3 & - 5 & - 1 & 1 \end{array}]$

Paso 3

Obtén la forma escalonada reducida por filas.

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Paso 3.1

Multiply each element of $R_{1}$ by $\frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

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Paso 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{4}{4} & \frac{- 3}{4} & \frac{1}{4} & \frac{- 1}{4} \\ 5 & 7 & 2 & - 2 \\ 3 & - 5 & - 1 & 1 \end{array}]$

Paso 3.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 5 & 7 & 2 & - 2 \\ 3 & - 5 & - 1 & 1 \end{array}]$

Paso 3.2

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Paso 3.2.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 5 - 5 \cdot 1 & 7 - 5 (- \frac{3}{4}) & 2 - 5 (\frac{1}{4}) & - 2 - 5 (- \frac{1}{4}) \\ 3 & - 5 & - 1 & 1 \end{array}]$

Paso 3.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 0 & \frac{43}{4} & \frac{3}{4} & - \frac{3}{4} \\ 3 & - 5 & - 1 & 1 \end{array}]$

Paso 3.3

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Paso 3.3.1

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 0 & \frac{43}{4} & \frac{3}{4} & - \frac{3}{4} \\ 3 - 3 \cdot 1 & - 5 - 3 (- \frac{3}{4}) & - 1 - 3 (\frac{1}{4}) & 1 - 3 (- \frac{1}{4}) \end{array}]$

Paso 3.3.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 0 & \frac{43}{4} & \frac{3}{4} & - \frac{3}{4} \\ 0 & - \frac{11}{4} & - \frac{7}{4} & \frac{7}{4} \end{array}]$

Paso 3.4

Multiply each element of $R_{2}$ by $\frac{4}{43}$ to make the entry at $2, 2$ a $1$ .

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Paso 3.4.1

Multiply each element of $R_{2}$ by $\frac{4}{43}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ \frac{4}{43} \cdot 0 & \frac{4}{43} \cdot \frac{43}{4} & \frac{4}{43} \cdot \frac{3}{4} & \frac{4}{43} (- \frac{3}{4}) \\ 0 & - \frac{11}{4} & - \frac{7}{4} & \frac{7}{4} \end{array}]$

Paso 3.4.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 0 & 1 & \frac{3}{43} & - \frac{3}{43} \\ 0 & - \frac{11}{4} & - \frac{7}{4} & \frac{7}{4} \end{array}]$

Paso 3.5

Perform the row operation $R_{3} = R_{3} + \frac{11}{4} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Paso 3.5.1

Perform the row operation $R_{3} = R_{3} + \frac{11}{4} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 0 & 1 & \frac{3}{43} & - \frac{3}{43} \\ 0 + \frac{11}{4} \cdot 0 & - \frac{11}{4} + \frac{11}{4} \cdot 1 & - \frac{7}{4} + \frac{11}{4} \cdot \frac{3}{43} & \frac{7}{4} + \frac{11}{4} (- \frac{3}{43}) \end{array}]$

Paso 3.5.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 0 & 1 & \frac{3}{43} & - \frac{3}{43} \\ 0 & 0 & - \frac{67}{43} & \frac{67}{43} \end{array}]$

Paso 3.6

Multiply each element of $R_{3}$ by $- \frac{43}{67}$ to make the entry at $3, 3$ a $1$ .

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Paso 3.6.1

Multiply each element of $R_{3}$ by $- \frac{43}{67}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 0 & 1 & \frac{3}{43} & - \frac{3}{43} \\ - \frac{43}{67} \cdot 0 & - \frac{43}{67} \cdot 0 & - \frac{43}{67} (- \frac{67}{43}) & - \frac{43}{67} \cdot \frac{67}{43} \end{array}]$

Paso 3.6.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 0 & 1 & \frac{3}{43} & - \frac{3}{43} \\ 0 & 0 & 1 & - 1 \end{array}]$

Paso 3.7

Perform the row operation $R_{2} = R_{2} - \frac{3}{43} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Paso 3.7.1

Perform the row operation $R_{2} = R_{2} - \frac{3}{43} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 0 - \frac{3}{43} \cdot 0 & 1 - \frac{3}{43} \cdot 0 & \frac{3}{43} - \frac{3}{43} \cdot 1 & - \frac{3}{43} - \frac{3}{43} \cdot - 1 \\ 0 & 0 & 1 & - 1 \end{array}]$

Paso 3.7.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & - \frac{1}{4} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & - 1 \end{array}]$

Paso 3.8

Perform the row operation $R_{1} = R_{1} - \frac{1}{4} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Paso 3.8.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{4} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{4} \cdot 0 & - \frac{3}{4} - \frac{1}{4} \cdot 0 & \frac{1}{4} - \frac{1}{4} \cdot 1 & - \frac{1}{4} - \frac{1}{4} \cdot - 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & - 1 \end{array}]$

Paso 3.8.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & - 1 \end{array}]$

Paso 3.9

Perform the row operation $R_{1} = R_{1} + \frac{3}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Paso 3.9.1

Perform the row operation $R_{1} = R_{1} + \frac{3}{4} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{3}{4} \cdot 0 & - \frac{3}{4} + \frac{3}{4} \cdot 1 & 0 + \frac{3}{4} \cdot 0 & 0 + \frac{3}{4} \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & - 1 \end{array}]$

Paso 3.9.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & - 1 \end{array}]$

Paso 4

Use the result matrix to declare the final solution to the system of equations.

$x = 0$

$y = 0$

$z = - 1$

Paso 5

The solution is the set of ordered pairs that make the system true.

$(0, 0, - 1)$