Matemática discreta Ejemplos

$3 x \cdot 1 + 9 x \cdot 2 - 21 x \cdot 3 = 15$ , $2 x \cdot 1 + x \cdot 2 - 4 x \cdot 3 = 0$ , $x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1

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

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Paso 1.1

Simplifica cada término.

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Paso 1.1.1

Multiplica $3$ por $1$ .

$3 x + 9 x \cdot 2 - 21 x \cdot 3 = 15$

$2 x \cdot 1 + x \cdot 2 - 4 x \cdot 3 = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1.1.2

Multiplica $2$ por $9$ .

$3 x + 18 x - 21 x \cdot 3 = 15$

$2 x \cdot 1 + x \cdot 2 - 4 x \cdot 3 = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1.1.3

Multiplica $3$ por $- 21$ .

$3 x + 18 x - 63 x = 15$

$2 x \cdot 1 + x \cdot 2 - 4 x \cdot 3 = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

$3 x + 18 x - 63 x = 15$

$2 x \cdot 1 + x \cdot 2 - 4 x \cdot 3 = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1.2

Suma $3 x$ y $18 x$ .

$21 x - 63 x = 15$

$2 x \cdot 1 + x \cdot 2 - 4 x \cdot 3 = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1.3

Resta $63 x$ de $21 x$ .

$- 42 x = 15$

$2 x \cdot 1 + x \cdot 2 - 4 x \cdot 3 = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1.4

Simplifica cada término.

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Paso 1.4.1

Multiplica $2$ por $1$ .

$- 42 x = 15$

$2 x + x \cdot 2 - 4 x \cdot 3 = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1.4.2

Mueve $2$ a la izquierda de $x$ .

$- 42 x = 15$

$2 x + 2 \cdot x - 4 x \cdot 3 = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1.4.3

Multiplica $3$ por $- 4$ .

$- 42 x = 15$

$2 x + 2 x - 12 x = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

$- 42 x = 15$

$2 x + 2 x - 12 x = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1.5

Suma $2 x$ y $2 x$ .

$- 42 x = 15$

$4 x - 12 x = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1.6

Resta $12 x$ de $4 x$ .

$- 42 x = 15$

$- 8 x = 0$

$x \cdot 1 + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1.7

Simplifica cada término.

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Paso 1.7.1

Multiplica $x$ por $1$ .

$- 42 x = 15$

$- 8 x = 0$

$x + x \cdot 2 - 3 x \cdot 3 = 1$

Paso 1.7.2

Mueve $2$ a la izquierda de $x$ .

$- 42 x = 15$

$- 8 x = 0$

$x + 2 \cdot x - 3 x \cdot 3 = 1$

Paso 1.7.3

Multiplica $3$ por $- 3$ .

$- 42 x = 15$

$- 8 x = 0$

$x + 2 x - 9 x = 1$

$- 42 x = 15$

$- 8 x = 0$

$x + 2 x - 9 x = 1$

Paso 1.8

Suma $x$ y $2 x$ .

$- 42 x = 15$

$- 8 x = 0$

$3 x - 9 x = 1$

Paso 1.9

Resta $9 x$ de $3 x$ .

$- 42 x = 15$

$- 8 x = 0$

$- 6 x = 1$

$- 42 x = 15$

$- 8 x = 0$

$- 6 x = 1$

Paso 2

Write the system as a matrix.

$[\begin{array}{c} - 42 & 15 \\ - 8 & 0 \\ - 6 & 1 \end{array}]$

Paso 3

Obtén la forma escalonada reducida por filas.

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Paso 3.1

Multiply each element of $R_{1}$ by $- \frac{1}{42}$ to make the entry at $1, 1$ a $1$ .

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Paso 3.1.1

Multiply each element of $R_{1}$ by $- \frac{1}{42}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{42} \cdot - 42 & - \frac{1}{42} \cdot 15 \\ - 8 & 0 \\ - 6 & 1 \end{array}]$

Paso 3.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{5}{14} \\ - 8 & 0 \\ - 6 & 1 \end{array}]$

Paso 3.2

Perform the row operation $R_{2} = R_{2} + 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Paso 3.2.1

Perform the row operation $R_{2} = R_{2} + 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{14} \\ - 8 + 8 \cdot 1 & 0 + 8 (- \frac{5}{14}) \\ - 6 & 1 \end{array}]$

Paso 3.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{14} \\ 0 & - \frac{20}{7} \\ - 6 & 1 \end{array}]$

Paso 3.3

Perform the row operation $R_{3} = R_{3} + 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Paso 3.3.1

Perform the row operation $R_{3} = R_{3} + 6 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{14} \\ 0 & - \frac{20}{7} \\ - 6 + 6 \cdot 1 & 1 + 6 (- \frac{5}{14}) \end{array}]$

Paso 3.3.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{5}{14} \\ 0 & - \frac{20}{7} \\ 0 & - \frac{8}{7} \end{array}]$

Paso 3.4

Multiply each element of $R_{2}$ by $- \frac{7}{20}$ to make the entry at $2, 2$ a $1$ .

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Paso 3.4.1

Multiply each element of $R_{2}$ by $- \frac{7}{20}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{5}{14} \\ - \frac{7}{20} \cdot 0 & - \frac{7}{20} (- \frac{20}{7}) \\ 0 & - \frac{8}{7} \end{array}]$

Paso 3.4.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{5}{14} \\ 0 & 1 \\ 0 & - \frac{8}{7} \end{array}]$

Paso 3.5

Perform the row operation $R_{3} = R_{3} + \frac{8}{7} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Paso 3.5.1

Perform the row operation $R_{3} = R_{3} + \frac{8}{7} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{5}{14} \\ 0 & 1 \\ 0 + \frac{8}{7} \cdot 0 & - \frac{8}{7} + \frac{8}{7} \cdot 1 \end{array}]$

Paso 3.5.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{5}{14} \\ 0 & 1 \\ 0 & 0 \end{array}]$

Paso 3.6

Perform the row operation $R_{1} = R_{1} + \frac{5}{14} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Paso 3.6.1

Perform the row operation $R_{1} = R_{1} + \frac{5}{14} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{5}{14} \cdot 0 & - \frac{5}{14} + \frac{5}{14} \cdot 1 \\ 0 & 1 \\ 0 & 0 \end{array}]$

Paso 3.6.2

Simplifica $R_{1}$ .

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

Paso 4

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

$x = 0$

$0 = 1$

$0 = 0$

Paso 5

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

$(error = error)$