Matematika Berhingga Contoh

$- 3 x - 9 y + 3 z = - 36$ , $- 6 x - 18 y + 6 z = - 72$ , $3 x + 9 y - 3 z = 36$

Langkah 1

Write the system as a matrix.

$[\begin{array}{c} - 3 & - 9 & 3 & - 36 \\ - 6 & - 18 & 6 & - 72 \\ 3 & 9 & - 3 & 36 \end{array}]$

Langkah 2

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 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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Langkah 2.1.1

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

$[\begin{array}{c} - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 9 & - \frac{1}{3} \cdot 3 & - \frac{1}{3} \cdot - 36 \\ - 6 & - 18 & 6 & - 72 \\ 3 & 9 & - 3 & 36 \end{array}]$

Langkah 2.1.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 3 & - 1 & 12 \\ - 6 & - 18 & 6 & - 72 \\ 3 & 9 & - 3 & 36 \end{array}]$

Langkah 2.2

Perform the row operation $R_{2} = R_{2} + 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Langkah 2.2.1

Perform the row operation $R_{2} = R_{2} + 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 3 & - 1 & 12 \\ - 6 + 6 \cdot 1 & - 18 + 6 \cdot 3 & 6 + 6 \cdot - 1 & - 72 + 6 \cdot 12 \\ 3 & 9 & - 3 & 36 \end{array}]$

Langkah 2.2.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & 3 & - 1 & 12 \\ 0 & 0 & 0 & 0 \\ 3 & 9 & - 3 & 36 \end{array}]$

Langkah 2.3

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

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Langkah 2.3.1

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

$[\begin{array}{c} 1 & 3 & - 1 & 12 \\ 0 & 0 & 0 & 0 \\ 3 - 3 \cdot 1 & 9 - 3 \cdot 3 & - 3 - 3 \cdot - 1 & 36 - 3 \cdot 12 \end{array}]$

Langkah 2.3.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & 3 & - 1 & 12 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 3

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

$x + 3 y - z = 12$

$0 = 0$

Langkah 4

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

$(12 - 3 y + z, y, z)$