Matematika Berhingga Contoh

$2 x - 4 y + 5 z = 1$ , $x - 3 z = 2$ , $5 x - 8 y + 7 z = 6$ , $3 x - 4 y + 2 z = 3$

Langkah 1

Write the system as a matrix.

$[\begin{array}{c} 2 & - 4 & 5 & 1 \\ 1 & 0 & - 3 & 2 \\ 5 & - 8 & 7 & 6 \\ 3 & - 4 & 2 & 3 \end{array}]$

Langkah 2

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 2.1

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

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Langkah 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{- 4}{2} & \frac{5}{2} & \frac{1}{2} \\ 1 & 0 & - 3 & 2 \\ 5 & - 8 & 7 & 6 \\ 3 & - 4 & 2 & 3 \end{array}]$

Langkah 2.1.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 1 & 0 & - 3 & 2 \\ 5 & - 8 & 7 & 6 \\ 3 & - 4 & 2 & 3 \end{array}]$

Langkah 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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Langkah 2.2.1

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

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 1 - 1 & 0 + 2 & - 3 - \frac{5}{2} & 2 - \frac{1}{2} \\ 5 & - 8 & 7 & 6 \\ 3 & - 4 & 2 & 3 \end{array}]$

Langkah 2.2.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{3}{2} \\ 5 & - 8 & 7 & 6 \\ 3 & - 4 & 2 & 3 \end{array}]$

Langkah 2.3

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Langkah 2.3.1

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{3}{2} \\ 5 - 5 \cdot 1 & - 8 - 5 \cdot - 2 & 7 - 5 (\frac{5}{2}) & 6 - 5 (\frac{1}{2}) \\ 3 & - 4 & 2 & 3 \end{array}]$

Langkah 2.3.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{3}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{7}{2} \\ 3 & - 4 & 2 & 3 \end{array}]$

Langkah 2.4

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

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Langkah 2.4.1

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

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{3}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{7}{2} \\ 3 - 3 \cdot 1 & - 4 - 3 \cdot - 2 & 2 - 3 (\frac{5}{2}) & 3 - 3 (\frac{1}{2}) \end{array}]$

Langkah 2.4.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{3}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{7}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{3}{2} \end{array}]$

Langkah 2.5

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

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Langkah 2.5.1

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

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ \frac{0}{2} & \frac{2}{2} & \frac{- \frac{11}{2}}{2} & \frac{\frac{3}{2}}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{7}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{3}{2} \end{array}]$

Langkah 2.5.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 1 & - \frac{11}{4} & \frac{3}{4} \\ 0 & 2 & - \frac{11}{2} & \frac{7}{2} \\ 0 & 2 & - \frac{11}{2} & \frac{3}{2} \end{array}]$

Langkah 2.6

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

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Langkah 2.6.1

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

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 1 & - \frac{11}{4} & \frac{3}{4} \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - \frac{11}{2} - 2 (- \frac{11}{4}) & \frac{7}{2} - 2 (\frac{3}{4}) \\ 0 & 2 & - \frac{11}{2} & \frac{3}{2} \end{array}]$

Langkah 2.6.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 1 & - \frac{11}{4} & \frac{3}{4} \\ 0 & 0 & 0 & 2 \\ 0 & 2 & - \frac{11}{2} & \frac{3}{2} \end{array}]$

Langkah 2.7

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

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Langkah 2.7.1

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

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 1 & - \frac{11}{4} & \frac{3}{4} \\ 0 & 0 & 0 & 2 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - \frac{11}{2} - 2 (- \frac{11}{4}) & \frac{3}{2} - 2 (\frac{3}{4}) \end{array}]$

Langkah 2.7.2

Sederhanakan $R_{4}$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 1 & - \frac{11}{4} & \frac{3}{4} \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 2.8

Multiply each element of $R_{3}$ by $\frac{1}{2}$ to make the entry at $3, 4$ a $1$ .

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Langkah 2.8.1

Multiply each element of $R_{3}$ by $\frac{1}{2}$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 1 & - \frac{11}{4} & \frac{3}{4} \\ \frac{0}{2} & \frac{0}{2} & \frac{0}{2} & \frac{2}{2} \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 2.8.2

Sederhanakan $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 1 & - \frac{11}{4} & \frac{3}{4} \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 2.9

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

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Langkah 2.9.1

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

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 - \frac{3}{4} \cdot 0 & 1 - \frac{3}{4} \cdot 0 & - \frac{11}{4} - \frac{3}{4} \cdot 0 & \frac{3}{4} - \frac{3}{4} \cdot 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 2.9.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & \frac{1}{2} \\ 0 & 1 & - \frac{11}{4} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 2.10

Perform the row operation $R_{1} = R_{1} - \frac{1}{2} R_{3}$ to make the entry at $1, 4$ a $0$ .

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Langkah 2.10.1

Perform the row operation $R_{1} = R_{1} - \frac{1}{2} R_{3}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & - 2 - \frac{1}{2} \cdot 0 & \frac{5}{2} - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 \\ 0 & 1 & - \frac{11}{4} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 2.10.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & \frac{5}{2} & 0 \\ 0 & 1 & - \frac{11}{4} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 2.11

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

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Langkah 2.11.1

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

$[\begin{array}{c} 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & \frac{5}{2} + 2 (- \frac{11}{4}) & 0 + 2 \cdot 0 \\ 0 & 1 & - \frac{11}{4} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}]$

Langkah 2.11.2

Sederhanakan $R_{1}$ .

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

Langkah 3

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

$x - 3 z = 0$

$y - \frac{11}{4} z = 0$

$0 = 1$

$0 = 0$

Langkah 4

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

$(3 z, \frac{11 z}{4}, z)$