Matematica discreta Esempi

$2 x + 5 y + 7 z = 8$ , $7 x + 4 y + 7 z = 36$ , $2 x + 9 y + 7 z = 16$

Passaggio 1

Write the system as a matrix.

$[\begin{array}{c} 2 & 5 & 7 & 8 \\ 7 & 4 & 7 & 36 \\ 2 & 9 & 7 & 16 \end{array}]$

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 $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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Passaggio 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{5}{2} & \frac{7}{2} & \frac{8}{2} \\ 7 & 4 & 7 & 36 \\ 2 & 9 & 7 & 16 \end{array}]$

Passaggio 2.1.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ 7 & 4 & 7 & 36 \\ 2 & 9 & 7 & 16 \end{array}]$

Passaggio 2.2

Perform the row operation $R_{2} = R_{2} - 7 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Passaggio 2.2.1

Perform the row operation $R_{2} = R_{2} - 7 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ 7 - 7 \cdot 1 & 4 - 7 (\frac{5}{2}) & 7 - 7 (\frac{7}{2}) & 36 - 7 \cdot 4 \\ 2 & 9 & 7 & 16 \end{array}]$

Passaggio 2.2.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ 0 & - \frac{27}{2} & - \frac{35}{2} & 8 \\ 2 & 9 & 7 & 16 \end{array}]$

Passaggio 2.3

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

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Passaggio 2.3.1

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

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ 0 & - \frac{27}{2} & - \frac{35}{2} & 8 \\ 2 - 2 \cdot 1 & 9 - 2 (\frac{5}{2}) & 7 - 2 (\frac{7}{2}) & 16 - 2 \cdot 4 \end{array}]$

Passaggio 2.3.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ 0 & - \frac{27}{2} & - \frac{35}{2} & 8 \\ 0 & 4 & 0 & 8 \end{array}]$

Passaggio 2.4

Multiply each element of $R_{2}$ by $- \frac{2}{27}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 2.4.1

Multiply each element of $R_{2}$ by $- \frac{2}{27}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ - \frac{2}{27} \cdot 0 & - \frac{2}{27} (- \frac{27}{2}) & - \frac{2}{27} (- \frac{35}{2}) & - \frac{2}{27} \cdot 8 \\ 0 & 4 & 0 & 8 \end{array}]$

Passaggio 2.4.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ 0 & 1 & \frac{35}{27} & - \frac{16}{27} \\ 0 & 4 & 0 & 8 \end{array}]$

Passaggio 2.5

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

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Passaggio 2.5.1

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

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ 0 & 1 & \frac{35}{27} & - \frac{16}{27} \\ 0 - 4 \cdot 0 & 4 - 4 \cdot 1 & 0 - 4 (\frac{35}{27}) & 8 - 4 (- \frac{16}{27}) \end{array}]$

Passaggio 2.5.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ 0 & 1 & \frac{35}{27} & - \frac{16}{27} \\ 0 & 0 & - \frac{140}{27} & \frac{280}{27} \end{array}]$

Passaggio 2.6

Multiply each element of $R_{3}$ by $- \frac{27}{140}$ to make the entry at $3, 3$ a $1$ .

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Passaggio 2.6.1

Multiply each element of $R_{3}$ by $- \frac{27}{140}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ 0 & 1 & \frac{35}{27} & - \frac{16}{27} \\ - \frac{27}{140} \cdot 0 & - \frac{27}{140} \cdot 0 & - \frac{27}{140} (- \frac{140}{27}) & - \frac{27}{140} \cdot \frac{280}{27} \end{array}]$

Passaggio 2.6.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ 0 & 1 & \frac{35}{27} & - \frac{16}{27} \\ 0 & 0 & 1 & - 2 \end{array}]$

Passaggio 2.7

Perform the row operation $R_{2} = R_{2} - \frac{35}{27} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Passaggio 2.7.1

Perform the row operation $R_{2} = R_{2} - \frac{35}{27} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{5}{2} & \frac{7}{2} & 4 \\ 0 - \frac{35}{27} \cdot 0 & 1 - \frac{35}{27} \cdot 0 & \frac{35}{27} - \frac{35}{27} \cdot 1 & - \frac{16}{27} - \frac{35}{27} \cdot - 2 \\ 0 & 0 & 1 & - 2 \end{array}]$

Passaggio 2.7.2

Semplifica $R_{2}$ .

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

Passaggio 2.8

Perform the row operation $R_{1} = R_{1} - \frac{7}{2} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Passaggio 2.8.1

Perform the row operation $R_{1} = R_{1} - \frac{7}{2} R_{3}$ to make the entry at $1, 3$ a $0$ .

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

Passaggio 2.8.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{2} & 0 & 11 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & - 2 \end{array}]$

Passaggio 2.9

Perform the row operation $R_{1} = R_{1} - \frac{5}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Passaggio 2.9.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{5}{2} \cdot 0 & \frac{5}{2} - \frac{5}{2} \cdot 1 & 0 - \frac{5}{2} \cdot 0 & 11 - \frac{5}{2} \cdot 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & - 2 \end{array}]$

Passaggio 2.9.2

Semplifica $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 6 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & - 2 \end{array}]$

Passaggio 3

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

$x = 6$

$y = 2$

$z = - 2$

Passaggio 4

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

$(6, 2, - 2)$