Matematica discreta Esempi

$3 x - 2 y + z = 11$ , $- x + y + 2 z = - 11$ , $x - y - 4 z = 17$

Passaggio 1

Write the system as a matrix.

$[\begin{array}{c} 3 & - 2 & 1 & 11 \\ - 1 & 1 & 2 & - 11 \\ 1 & - 1 & - 4 & 17 \end{array}]$

Passaggio 2

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 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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Passaggio 2.1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{- 2}{3} & \frac{1}{3} & \frac{11}{3} \\ - 1 & 1 & 2 & - 11 \\ 1 & - 1 & - 4 & 17 \end{array}]$

Passaggio 2.1.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ - 1 & 1 & 2 & - 11 \\ 1 & - 1 & - 4 & 17 \end{array}]$

Passaggio 2.2

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

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Passaggio 2.2.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ - 1 + 1 \cdot 1 & 1 - \frac{2}{3} & 2 + \frac{1}{3} & - 11 + \frac{11}{3} \\ 1 & - 1 & - 4 & 17 \end{array}]$

Passaggio 2.2.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ 0 & \frac{1}{3} & \frac{7}{3} & - \frac{22}{3} \\ 1 & - 1 & - 4 & 17 \end{array}]$

Passaggio 2.3

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

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Passaggio 2.3.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ 0 & \frac{1}{3} & \frac{7}{3} & - \frac{22}{3} \\ 1 - 1 & - 1 + \frac{2}{3} & - 4 - \frac{1}{3} & 17 - \frac{11}{3} \end{array}]$

Passaggio 2.3.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ 0 & \frac{1}{3} & \frac{7}{3} & - \frac{22}{3} \\ 0 & - \frac{1}{3} & - \frac{13}{3} & \frac{40}{3} \end{array}]$

Passaggio 2.4

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

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Passaggio 2.4.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ 3 \cdot 0 & 3 (\frac{1}{3}) & 3 (\frac{7}{3}) & 3 (- \frac{22}{3}) \\ 0 & - \frac{1}{3} & - \frac{13}{3} & \frac{40}{3} \end{array}]$

Passaggio 2.4.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ 0 & 1 & 7 & - 22 \\ 0 & - \frac{1}{3} & - \frac{13}{3} & \frac{40}{3} \end{array}]$

Passaggio 2.5

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

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Passaggio 2.5.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ 0 & 1 & 7 & - 22 \\ 0 + \frac{1}{3} \cdot 0 & - \frac{1}{3} + \frac{1}{3} \cdot 1 & - \frac{13}{3} + \frac{1}{3} \cdot 7 & \frac{40}{3} + \frac{1}{3} \cdot - 22 \end{array}]$

Passaggio 2.5.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ 0 & 1 & 7 & - 22 \\ 0 & 0 & - 2 & 6 \end{array}]$

Passaggio 2.6

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

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Passaggio 2.6.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ 0 & 1 & 7 & - 22 \\ - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 6 \end{array}]$

Passaggio 2.6.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ 0 & 1 & 7 & - 22 \\ 0 & 0 & 1 & - 3 \end{array}]$

Passaggio 2.7

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

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Passaggio 2.7.1

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

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ 0 - 7 \cdot 0 & 1 - 7 \cdot 0 & 7 - 7 \cdot 1 & - 22 - 7 \cdot - 3 \\ 0 & 0 & 1 & - 3 \end{array}]$

Passaggio 2.7.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{1}{3} & \frac{11}{3} \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & 1 & - 3 \end{array}]$

Passaggio 2.8

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

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Passaggio 2.8.1

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

$[\begin{array}{c} 1 - \frac{1}{3} \cdot 0 & - \frac{2}{3} - \frac{1}{3} \cdot 0 & \frac{1}{3} - \frac{1}{3} \cdot 1 & \frac{11}{3} - \frac{1}{3} \cdot - 3 \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & 1 & - 3 \end{array}]$

Passaggio 2.8.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & 0 & \frac{14}{3} \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & 1 & - 3 \end{array}]$

Passaggio 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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Passaggio 2.9.1

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

$[\begin{array}{c} 1 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & 0 + \frac{2}{3} \cdot 0 & \frac{14}{3} + \frac{2}{3} \cdot - 1 \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & 1 & - 3 \end{array}]$

Passaggio 2.9.2

Semplifica $R_{1}$ .

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

Passaggio 3

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

$x = 4$

$y = - 1$

$z = - 3$

Passaggio 4

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

$(4, - 1, - 3)$