Matemática discreta Ejemplos

$25 x + 50 y + 75 z + 300 = 1350$ , $40 x + 40 y + 40 z + 240 = 920$ , $40 x + 40 y + 60 z + 200 = 1080$

Paso 1

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

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Paso 1.1

Mueve todos los términos que no contengan una variable al lado derecho de la ecuación.

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Paso 1.1.1

Resta $300$ de ambos lados de la ecuación.

$25 x + 50 y + 75 z = 1350 - 300$

$40 x + 40 y + 40 z + 240 = 920$

$40 x + 40 y + 60 z + 200 = 1080$

Paso 1.1.2

Resta $300$ de $1350$ .

$25 x + 50 y + 75 z = 1050$

$40 x + 40 y + 40 z + 240 = 920$

$40 x + 40 y + 60 z + 200 = 1080$

$25 x + 50 y + 75 z = 1050$

$40 x + 40 y + 40 z + 240 = 920$

$40 x + 40 y + 60 z + 200 = 1080$

Paso 1.2

Mueve todos los términos que no contengan una variable al lado derecho de la ecuación.

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Paso 1.2.1

Resta $240$ de ambos lados de la ecuación.

$25 x + 50 y + 75 z = 1050$

$40 x + 40 y + 40 z = 920 - 240$

$40 x + 40 y + 60 z + 200 = 1080$

Paso 1.2.2

Resta $240$ de $920$ .

$25 x + 50 y + 75 z = 1050$

$40 x + 40 y + 40 z = 680$

$40 x + 40 y + 60 z + 200 = 1080$

$25 x + 50 y + 75 z = 1050$

$40 x + 40 y + 40 z = 680$

$40 x + 40 y + 60 z + 200 = 1080$

Paso 1.3

Mueve todos los términos que no contengan una variable al lado derecho de la ecuación.

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Paso 1.3.1

Resta $200$ de ambos lados de la ecuación.

$25 x + 50 y + 75 z = 1050$

$40 x + 40 y + 40 z = 680$

$40 x + 40 y + 60 z = 1080 - 200$

Paso 1.3.2

Resta $200$ de $1080$ .

$25 x + 50 y + 75 z = 1050$

$40 x + 40 y + 40 z = 680$

$40 x + 40 y + 60 z = 880$

$25 x + 50 y + 75 z = 1050$

$40 x + 40 y + 40 z = 680$

$40 x + 40 y + 60 z = 880$

$25 x + 50 y + 75 z = 1050$

$40 x + 40 y + 40 z = 680$

$40 x + 40 y + 60 z = 880$

Paso 2

Write the system as a matrix.

$[\begin{array}{c} 25 & 50 & 75 & 1050 \\ 40 & 40 & 40 & 680 \\ 40 & 40 & 60 & 880 \end{array}]$

Paso 3

Obtén la forma escalonada reducida por filas.

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Paso 3.1

Multiply each element of $R_{1}$ by $\frac{1}{25}$ to make the entry at $1, 1$ a $1$ .

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Paso 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{25}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{25}{25} & \frac{50}{25} & \frac{75}{25} & \frac{1050}{25} \\ 40 & 40 & 40 & 680 \\ 40 & 40 & 60 & 880 \end{array}]$

Paso 3.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 40 & 40 & 40 & 680 \\ 40 & 40 & 60 & 880 \end{array}]$

Paso 3.2

Perform the row operation $R_{2} = R_{2} - 40 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Paso 3.2.1

Perform the row operation $R_{2} = R_{2} - 40 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 40 - 40 \cdot 1 & 40 - 40 \cdot 2 & 40 - 40 \cdot 3 & 680 - 40 \cdot 42 \\ 40 & 40 & 60 & 880 \end{array}]$

Paso 3.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 0 & - 40 & - 80 & - 1000 \\ 40 & 40 & 60 & 880 \end{array}]$

Paso 3.3

Perform the row operation $R_{3} = R_{3} - 40 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Paso 3.3.1

Perform the row operation $R_{3} = R_{3} - 40 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 0 & - 40 & - 80 & - 1000 \\ 40 - 40 \cdot 1 & 40 - 40 \cdot 2 & 60 - 40 \cdot 3 & 880 - 40 \cdot 42 \end{array}]$

Paso 3.3.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 0 & - 40 & - 80 & - 1000 \\ 0 & - 40 & - 60 & - 800 \end{array}]$

Paso 3.4

Multiply each element of $R_{2}$ by $- \frac{1}{40}$ to make the entry at $2, 2$ a $1$ .

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Paso 3.4.1

Multiply each element of $R_{2}$ by $- \frac{1}{40}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ - \frac{1}{40} \cdot 0 & - \frac{1}{40} \cdot - 40 & - \frac{1}{40} \cdot - 80 & - \frac{1}{40} \cdot - 1000 \\ 0 & - 40 & - 60 & - 800 \end{array}]$

Paso 3.4.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 0 & 1 & 2 & 25 \\ 0 & - 40 & - 60 & - 800 \end{array}]$

Paso 3.5

Perform the row operation $R_{3} = R_{3} + 40 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Paso 3.5.1

Perform the row operation $R_{3} = R_{3} + 40 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 0 & 1 & 2 & 25 \\ 0 + 40 \cdot 0 & - 40 + 40 \cdot 1 & - 60 + 40 \cdot 2 & - 800 + 40 \cdot 25 \end{array}]$

Paso 3.5.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 0 & 1 & 2 & 25 \\ 0 & 0 & 20 & 200 \end{array}]$

Paso 3.6

Multiply each element of $R_{3}$ by $\frac{1}{20}$ to make the entry at $3, 3$ a $1$ .

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Paso 3.6.1

Multiply each element of $R_{3}$ by $\frac{1}{20}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 0 & 1 & 2 & 25 \\ \frac{0}{20} & \frac{0}{20} & \frac{20}{20} & \frac{200}{20} \end{array}]$

Paso 3.6.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 0 & 1 & 2 & 25 \\ 0 & 0 & 1 & 10 \end{array}]$

Paso 3.7

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

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Paso 3.7.1

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

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 25 - 2 \cdot 10 \\ 0 & 0 & 1 & 10 \end{array}]$

Paso 3.7.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 42 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 10 \end{array}]$

Paso 3.8

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

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Paso 3.8.1

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

$[\begin{array}{c} 1 - 3 \cdot 0 & 2 - 3 \cdot 0 & 3 - 3 \cdot 1 & 42 - 3 \cdot 10 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 10 \end{array}]$

Paso 3.8.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 2 & 0 & 12 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 10 \end{array}]$

Paso 3.9

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

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Paso 3.9.1

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

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 & 12 - 2 \cdot 5 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 10 \end{array}]$

Paso 3.9.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 5 \\ 0 & 0 & 1 & 10 \end{array}]$

Paso 4

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

$x = 2$

$y = 5$

$z = 10$

Paso 5

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

$(2, 5, 10)$