Precálculo Ejemplos

$3 x + 2 y + 4 z = 25$ , $2 x - y + z = 3$ , $6 x + 5 y - z = 17$

Paso 1

Write the system as a matrix.

$[\begin{array}{c} 3 & 2 & 4 & 25 \\ 2 & - 1 & 1 & 3 \\ 6 & 5 & - 1 & 17 \end{array}]$

Paso 2

Obtén la forma escalonada reducida por filas.

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Paso 2.1

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

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Paso 2.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 2 & - 1 & 1 & 3 \\ 6 & 5 & - 1 & 17 \end{array}]$

Paso 2.1.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 2 & - 1 & 1 & 3 \\ 6 & 5 & - 1 & 17 \end{array}]$

Paso 2.2

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

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Paso 2.2.1

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

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 2 - 2 \cdot 1 & - 1 - 2 (\frac{2}{3}) & 1 - 2 (\frac{4}{3}) & 3 - 2 (\frac{25}{3}) \\ 6 & 5 & - 1 & 17 \end{array}]$

Paso 2.2.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 0 & - \frac{7}{3} & - \frac{5}{3} & - \frac{41}{3} \\ 6 & 5 & - 1 & 17 \end{array}]$

Paso 2.3

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 2.3.1

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

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 0 & - \frac{7}{3} & - \frac{5}{3} & - \frac{41}{3} \\ 6 - 6 \cdot 1 & 5 - 6 (\frac{2}{3}) & - 1 - 6 (\frac{4}{3}) & 17 - 6 (\frac{25}{3}) \end{array}]$

Paso 2.3.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 0 & - \frac{7}{3} & - \frac{5}{3} & - \frac{41}{3} \\ 0 & 1 & - 9 & - 33 \end{array}]$

Paso 2.4

Multiply each element of $R_{2}$ by $- \frac{3}{7}$ to make the entry at $2, 2$ a $1$ .

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Paso 2.4.1

Multiply each element of $R_{2}$ by $- \frac{3}{7}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ - \frac{3}{7} \cdot 0 & - \frac{3}{7} (- \frac{7}{3}) & - \frac{3}{7} (- \frac{5}{3}) & - \frac{3}{7} (- \frac{41}{3}) \\ 0 & 1 & - 9 & - 33 \end{array}]$

Paso 2.4.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 0 & 1 & \frac{5}{7} & \frac{41}{7} \\ 0 & 1 & - 9 & - 33 \end{array}]$

Paso 2.5

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

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Paso 2.5.1

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

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 0 & 1 & \frac{5}{7} & \frac{41}{7} \\ 0 - 0 & 1 - 1 & - 9 - \frac{5}{7} & - 33 - \frac{41}{7} \end{array}]$

Paso 2.5.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 0 & 1 & \frac{5}{7} & \frac{41}{7} \\ 0 & 0 & - \frac{68}{7} & - \frac{272}{7} \end{array}]$

Paso 2.6

Multiply each element of $R_{3}$ by $- \frac{7}{68}$ to make the entry at $3, 3$ a $1$ .

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Paso 2.6.1

Multiply each element of $R_{3}$ by $- \frac{7}{68}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 0 & 1 & \frac{5}{7} & \frac{41}{7} \\ - \frac{7}{68} \cdot 0 & - \frac{7}{68} \cdot 0 & - \frac{7}{68} (- \frac{68}{7}) & - \frac{7}{68} (- \frac{272}{7}) \end{array}]$

Paso 2.6.2

Simplifica $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 0 & 1 & \frac{5}{7} & \frac{41}{7} \\ 0 & 0 & 1 & 4 \end{array}]$

Paso 2.7

Perform the row operation $R_{2} = R_{2} - \frac{5}{7} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Paso 2.7.1

Perform the row operation $R_{2} = R_{2} - \frac{5}{7} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 0 - \frac{5}{7} \cdot 0 & 1 - \frac{5}{7} \cdot 0 & \frac{5}{7} - \frac{5}{7} \cdot 1 & \frac{41}{7} - \frac{5}{7} \cdot 4 \\ 0 & 0 & 1 & 4 \end{array}]$

Paso 2.7.2

Simplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{4}{3} & \frac{25}{3} \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 4 \end{array}]$

Paso 2.8

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

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Paso 2.8.1

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

$[\begin{array}{c} 1 - \frac{4}{3} \cdot 0 & \frac{2}{3} - \frac{4}{3} \cdot 0 & \frac{4}{3} - \frac{4}{3} \cdot 1 & \frac{25}{3} - \frac{4}{3} \cdot 4 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 4 \end{array}]$

Paso 2.8.2

Simplifica $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & 0 & 3 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 4 \end{array}]$

Paso 2.9

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

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Paso 2.9.1

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

$[\begin{array}{c} 1 - \frac{2}{3} \cdot 0 & \frac{2}{3} - \frac{2}{3} \cdot 1 & 0 - \frac{2}{3} \cdot 0 & 3 - \frac{2}{3} \cdot 3 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 4 \end{array}]$

Paso 2.9.2

Simplifica $R_{1}$ .

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

Paso 3

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

$x = 1$

$y = 3$

$z = 4$

Paso 4

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

$(1, 3, 4)$