Algebra lineare Esempi

$x + y - z = 2$ , $x + 2 y + 3 z = 7$ , $x + 3 y - z = 4$

Passaggio 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 1 & - 1 & 2 \\ 1 & 2 & 3 & 7 \\ 1 & 3 & - 1 & 4 \end{array}]$

Passaggio 2

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 2.1

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

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Passaggio 2.1.1

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

$[\begin{array}{c} 1 & 1 & - 1 & 2 \\ 1 - 1 & 2 - 1 & 3 + 1 & 7 - 2 \\ 1 & 3 & - 1 & 4 \end{array}]$

Passaggio 2.1.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 1 & - 1 & 2 \\ 0 & 1 & 4 & 5 \\ 1 & 3 & - 1 & 4 \end{array}]$

Passaggio 2.2

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

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Passaggio 2.2.1

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

$[\begin{array}{c} 1 & 1 & - 1 & 2 \\ 0 & 1 & 4 & 5 \\ 1 - 1 & 3 - 1 & - 1 + 1 & 4 - 2 \end{array}]$

Passaggio 2.2.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 1 & 2 \\ 0 & 1 & 4 & 5 \\ 0 & 2 & 0 & 2 \end{array}]$

Passaggio 2.3

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

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Passaggio 2.3.1

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

$[\begin{array}{c} 1 & 1 & - 1 & 2 \\ 0 & 1 & 4 & 5 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 4 & 2 - 2 \cdot 5 \end{array}]$

Passaggio 2.3.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 1 & 2 \\ 0 & 1 & 4 & 5 \\ 0 & 0 & - 8 & - 8 \end{array}]$

Passaggio 2.4

Multiply each element of $R_{3}$ by $- \frac{1}{8}$ to make the entry at $3, 3$ a $1$ .

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Passaggio 2.4.1

Multiply each element of $R_{3}$ by $- \frac{1}{8}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & - 1 & 2 \\ 0 & 1 & 4 & 5 \\ - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot 0 & - \frac{1}{8} \cdot - 8 & - \frac{1}{8} \cdot - 8 \end{array}]$

Passaggio 2.4.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 1 & - 1 & 2 \\ 0 & 1 & 4 & 5 \\ 0 & 0 & 1 & 1 \end{array}]$

Passaggio 2.5

Perform the row operation $R_{2} = R_{2} - 4 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Passaggio 2.5.1

Perform the row operation $R_{2} = R_{2} - 4 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & - 1 & 2 \\ 0 - 4 \cdot 0 & 1 - 4 \cdot 0 & 4 - 4 \cdot 1 & 5 - 4 \cdot 1 \\ 0 & 0 & 1 & 1 \end{array}]$

Passaggio 2.5.2

Semplifica $R_{2}$ .

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

Passaggio 2.6

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

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Passaggio 2.6.1

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

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

Passaggio 2.6.2

Semplifica $R_{1}$ .

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

Passaggio 2.7

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

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Passaggio 2.7.1

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

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

Passaggio 2.7.2

Semplifica $R_{1}$ .

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

Passaggio 3

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

$x = 2$

$y = 1$

$z = 1$

Passaggio 4

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

$(2, 1, 1)$