Matemática discreta Ejemplos

$x - 2 y = 0$ , $x + y + z = 310$ , $45000 x + 40000 y + 80000 z = 16000000$

Paso 1

Representa el sistema de ecuaciones en el formato de la matriz.

$[\begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$

Paso 2

Find the determinant of the coefficient matrix $[\begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array}]$ .

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Paso 2.1

Write $[\begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array}]$ in determinant notation.

$| \begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 1 \\ 45000 & 40000 & 80000 \end{array} |$

Paso 2.2

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Paso 2.2.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Paso 2.2.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Paso 2.2.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

Paso 2.2.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

Paso 2.2.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

Paso 2.2.6

Multiply element $a_{12}$ by its cofactor.

$2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

Paso 2.2.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Paso 2.2.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Paso 2.2.9

Add the terms together.

$1 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} | + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Paso 2.3

Multiplica $0$ por $| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$ .

$1 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} | + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

Paso 2.4

Evalúa $| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$ .

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Paso 2.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (1 \cdot 80000 - 40000 \cdot 1) + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

Paso 2.4.2

Simplifica el determinante.

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Paso 2.4.2.1

Simplifica cada término.

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Paso 2.4.2.1.1

Multiplica $80000$ por $1$ .

$1 (80000 - 40000 \cdot 1) + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

Paso 2.4.2.1.2

Multiplica $- 40000$ por $1$ .

$1 (80000 - 40000) + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

Paso 2.4.2.2

Resta $40000$ de $80000$ .

$1 \cdot 40000 + 2 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0$

Paso 2.5

Evalúa $| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$ .

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Paso 2.5.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot 40000 + 2 (1 \cdot 80000 - 45000 \cdot 1) + 0$

Paso 2.5.2

Simplifica el determinante.

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Paso 2.5.2.1

Simplifica cada término.

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Paso 2.5.2.1.1

Multiplica $80000$ por $1$ .

$1 \cdot 40000 + 2 (80000 - 45000 \cdot 1) + 0$

Paso 2.5.2.1.2

Multiplica $- 45000$ por $1$ .

$1 \cdot 40000 + 2 (80000 - 45000) + 0$

Paso 2.5.2.2

Resta $45000$ de $80000$ .

$1 \cdot 40000 + 2 \cdot 35000 + 0$

Paso 2.6

Simplifica el determinante.

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Paso 2.6.1

Simplifica cada término.

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Paso 2.6.1.1

Multiplica $40000$ por $1$ .

$40000 + 2 \cdot 35000 + 0$

Paso 2.6.1.2

Multiplica $2$ por $35000$ .

$40000 + 70000 + 0$

Paso 2.6.2

Suma $40000$ y $70000$ .

$110000 + 0$

Paso 2.6.3

Suma $110000$ y $0$ .

$110000$

$D = 110000$

Paso 3

Since the determinant is not $0$ , the system can be solved using Cramer's Rule.

Paso 4

Find the value of $x$ by Cramer's Rule, which states that $x = \frac{D_{x}}{D}$ .

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Paso 4.1

Replace column $1$ of the coefficient matrix that corresponds to the $x$ -coefficients of the system with $[\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$ .

$| \begin{array}{c} 0 & - 2 & 0 \\ 310 & 1 & 1 \\ 16000000 & 40000 & 80000 \end{array} |$

Paso 4.2

Find the determinant.

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Paso 4.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Paso 4.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Paso 4.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Paso 4.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

Paso 4.2.1.4

Multiply element $a_{11}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$

Paso 4.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

Paso 4.2.1.6

Multiply element $a_{12}$ by its cofactor.

$2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

Paso 4.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

Paso 4.2.1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

Paso 4.2.1.9

Add the terms together.

$0 | \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} | + 2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 | \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

Paso 4.2.2

Multiplica $0$ por $| \begin{array}{c} 1 & 1 \\ 40000 & 80000 \end{array} |$ .

$0 + 2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 | \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$

Paso 4.2.3

Multiplica $0$ por $| \begin{array}{c} 310 & 1 \\ 16000000 & 40000 \end{array} |$ .

$0 + 2 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0$

Paso 4.2.4

Evalúa $| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$ .

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Paso 4.2.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$0 + 2 (310 \cdot 80000 - 16000000 \cdot 1) + 0$

Paso 4.2.4.2

Simplifica el determinante.

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Paso 4.2.4.2.1

Simplifica cada término.

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Paso 4.2.4.2.1.1

Multiplica $310$ por $80000$ .

$0 + 2 (24800000 - 16000000 \cdot 1) + 0$

Paso 4.2.4.2.1.2

Multiplica $- 16000000$ por $1$ .

$0 + 2 (24800000 - 16000000) + 0$

Paso 4.2.4.2.2

Resta $16000000$ de $24800000$ .

$0 + 2 \cdot 8800000 + 0$

Paso 4.2.5

Simplifica el determinante.

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Paso 4.2.5.1

Multiplica $2$ por $8800000$ .

$0 + 17600000 + 0$

Paso 4.2.5.2

Suma $0$ y $17600000$ .

$17600000 + 0$

Paso 4.2.5.3

Suma $17600000$ y $0$ .

$17600000$

$D_{x} = 17600000$

Paso 4.3

Use the formula to solve for $x$ .

$x = \frac{D_{x}}{D}$

Paso 4.4

Substitute $110000$ for $D$ and $17600000$ for $D_{x}$ in the formula.

$x = \frac{17600000}{110000}$

Paso 4.5

Divide $17600000$ por $110000$ .

$x = 160$

Paso 5

Find the value of $y$ by Cramer's Rule, which states that $y = \frac{D_{y}}{D}$ .

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Paso 5.1

Replace column $2$ of the coefficient matrix that corresponds to the $y$ -coefficients of the system with $[\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$ .

$| \begin{array}{c} 1 & 0 & 0 \\ 1 & 310 & 1 \\ 45000 & 16000000 & 80000 \end{array} |$

Paso 5.2

Find the determinant.

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Paso 5.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Paso 5.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Paso 5.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Paso 5.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

Paso 5.2.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$

Paso 5.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

Paso 5.2.1.6

Multiply element $a_{12}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$

Paso 5.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Paso 5.2.1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Paso 5.2.1.9

Add the terms together.

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} | + 0 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Paso 5.2.2

Multiplica $0$ por $| \begin{array}{c} 1 & 1 \\ 45000 & 80000 \end{array} |$ .

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 + 0 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Paso 5.2.3

Multiplica $0$ por $| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$ .

$1 | \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} | + 0 + 0$

Paso 5.2.4

Evalúa $| \begin{array}{c} 310 & 1 \\ 16000000 & 80000 \end{array} |$ .

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Paso 5.2.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (310 \cdot 80000 - 16000000 \cdot 1) + 0 + 0$

Paso 5.2.4.2

Simplifica el determinante.

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Paso 5.2.4.2.1

Simplifica cada término.

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Paso 5.2.4.2.1.1

Multiplica $310$ por $80000$ .

$1 (24800000 - 16000000 \cdot 1) + 0 + 0$

Paso 5.2.4.2.1.2

Multiplica $- 16000000$ por $1$ .

$1 (24800000 - 16000000) + 0 + 0$

Paso 5.2.4.2.2

Resta $16000000$ de $24800000$ .

$1 \cdot 8800000 + 0 + 0$

Paso 5.2.5

Simplifica el determinante.

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Paso 5.2.5.1

Multiplica $8800000$ por $1$ .

$8800000 + 0 + 0$

Paso 5.2.5.2

Suma $8800000$ y $0$ .

$8800000 + 0$

Paso 5.2.5.3

Suma $8800000$ y $0$ .

$8800000$

$D_{y} = 8800000$

Paso 5.3

Use the formula to solve for $y$ .

$y = \frac{D_{y}}{D}$

Paso 5.4

Substitute $110000$ for $D$ and $8800000$ for $D_{y}$ in the formula.

$y = \frac{8800000}{110000}$

Paso 5.5

Divide $8800000$ por $110000$ .

$y = 80$

Paso 6

Find the value of $z$ by Cramer's Rule, which states that $z = \frac{D_{z}}{D}$ .

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Paso 6.1

Replace column $3$ of the coefficient matrix that corresponds to the $z$ -coefficients of the system with $[\begin{array}{c} 0 \\ 310 \\ 16000000 \end{array}]$ .

$| \begin{array}{c} 1 & - 2 & 0 \\ 1 & 1 & 310 \\ 45000 & 40000 & 16000000 \end{array} |$

Paso 6.2

Find the determinant.

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Paso 6.2.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Paso 6.2.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Paso 6.2.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Paso 6.2.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} |$

Paso 6.2.1.4

Multiply element $a_{11}$ by its cofactor.

$1 | \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} |$

Paso 6.2.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Paso 6.2.1.6

Multiply element $a_{12}$ by its cofactor.

$2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$

Paso 6.2.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Paso 6.2.1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Paso 6.2.1.9

Add the terms together.

$1 | \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} | + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0 | \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$

Paso 6.2.2

Multiplica $0$ por $| \begin{array}{c} 1 & 1 \\ 45000 & 40000 \end{array} |$ .

$1 | \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} | + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

Paso 6.2.3

Evalúa $| \begin{array}{c} 1 & 310 \\ 40000 & 16000000 \end{array} |$ .

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Paso 6.2.3.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 (1 \cdot 16000000 - 40000 \cdot 310) + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

Paso 6.2.3.2

Simplifica el determinante.

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Paso 6.2.3.2.1

Simplifica cada término.

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Paso 6.2.3.2.1.1

Multiplica $16000000$ por $1$ .

$1 (16000000 - 40000 \cdot 310) + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

Paso 6.2.3.2.1.2

Multiplica $- 40000$ por $310$ .

$1 (16000000 - 12400000) + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

Paso 6.2.3.2.2

Resta $12400000$ de $16000000$ .

$1 \cdot 3600000 + 2 | \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} | + 0$

Paso 6.2.4

Evalúa $| \begin{array}{c} 1 & 310 \\ 45000 & 16000000 \end{array} |$ .

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Paso 6.2.4.1

El determinante de una matriz $2 \times 2$ puede obtenerse usando la fórmula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot 3600000 + 2 (1 \cdot 16000000 - 45000 \cdot 310) + 0$

Paso 6.2.4.2

Simplifica el determinante.

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Paso 6.2.4.2.1

Simplifica cada término.

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Paso 6.2.4.2.1.1

Multiplica $16000000$ por $1$ .

$1 \cdot 3600000 + 2 (16000000 - 45000 \cdot 310) + 0$

Paso 6.2.4.2.1.2

Multiplica $- 45000$ por $310$ .

$1 \cdot 3600000 + 2 (16000000 - 13950000) + 0$

Paso 6.2.4.2.2

Resta $13950000$ de $16000000$ .

$1 \cdot 3600000 + 2 \cdot 2050000 + 0$

Paso 6.2.5

Simplifica el determinante.

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Paso 6.2.5.1

Simplifica cada término.

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Paso 6.2.5.1.1

Multiplica $3600000$ por $1$ .

$3600000 + 2 \cdot 2050000 + 0$

Paso 6.2.5.1.2

Multiplica $2$ por $2050000$ .

$3600000 + 4100000 + 0$

Paso 6.2.5.2

Suma $3600000$ y $4100000$ .

$7700000 + 0$

Paso 6.2.5.3

Suma $7700000$ y $0$ .

$7700000$

$D_{z} = 7700000$

Paso 6.3

Use the formula to solve for $z$ .

$z = \frac{D_{z}}{D}$

Paso 6.4

Substitute $110000$ for $D$ and $7700000$ for $D_{z}$ in the formula.

$z = \frac{7700000}{110000}$

Paso 6.5

Divide $7700000$ por $110000$ .

$z = 70$

Paso 7

Enumera la solución del sistema de ecuaciones.

$x = 160$

$y = 80$

$z = 70$