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$[\begin{array}{c} 125 a & 25 b & 5 c & 49.8 \\ 1000 a & 100 b & 10 c & 50.7 \\ 3375 a & 225 b & 15 c & 51.3 \end{array}]$

Bước 1

Multiply each element of $R_{1}$ by $\frac{1}{125 a}$ to make the entry at $1, 1$ a $1$ .

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Bước 1.1

Multiply each element of $R_{1}$ by $\frac{1}{125 a}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{125 a}{125 a} & \frac{25 b}{125 a} & \frac{5 c}{125 a} & \frac{49.8}{125 a} \\ 1000 a & 100 b & 10 c & 50.7 \\ 3375 a & 225 b & 15 c & 51.3 \end{array}]$

Bước 1.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 1000 a & 100 b & 10 c & 50.7 \\ 3375 a & 225 b & 15 c & 51.3 \end{array}]$

Bước 2

Perform the row operation $R_{2} = R_{2} - 1000 a R_{1}$ to make the entry at $2, 1$ a $0$ .

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Bước 2.1

Perform the row operation $R_{2} = R_{2} - 1000 a R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 1000 a - 1000 a \cdot 1 & 100 b - 1000 a \frac{b}{5 a} & 10 c - 1000 a \frac{c}{25 a} & 50.7 - 1000 a \frac{0.3984}{a} \\ 3375 a & 225 b & 15 c & 51.3 \end{array}]$

Bước 2.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 0 & - 100 b & - 30 c & - 347.7 \\ 3375 a & 225 b & 15 c & 51.3 \end{array}]$

Bước 3

Perform the row operation $R_{3} = R_{3} - 3375 a R_{1}$ to make the entry at $3, 1$ a $0$ .

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Bước 3.1

Perform the row operation $R_{3} = R_{3} - 3375 a R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 0 & - 100 b & - 30 c & - 347.7 \\ 3375 a - 3375 a \cdot 1 & 225 b - 3375 a \frac{b}{5 a} & 15 c - 3375 a \frac{c}{25 a} & 51.3 - 3375 a \frac{0.3984}{a} \end{array}]$

Bước 3.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 0 & - 100 b & - 30 c & - 347.7 \\ 0 & - 450 b & - 120 c & - 1293.3 \end{array}]$

Bước 4

Multiply each element of $R_{2}$ by $\frac{1}{- 100 b}$ to make the entry at $2, 2$ a $1$ .

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Bước 4.1

Multiply each element of $R_{2}$ by $\frac{1}{- 100 b}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ \frac{0}{- 100 b} & \frac{- 100 b}{- 100 b} & \frac{- 30 c}{- 100 b} & \frac{- 347.7}{- 100 b} \\ 0 & - 450 b & - 120 c & - 1293.3 \end{array}]$

Bước 4.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 0 & 1 & \frac{3 c}{10 b} & \frac{3.477}{b} \\ 0 & - 450 b & - 120 c & - 1293.3 \end{array}]$

Bước 5

Perform the row operation $R_{3} = R_{3} + 450 b R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 5.1

Perform the row operation $R_{3} = R_{3} + 450 b R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 0 & 1 & \frac{3 c}{10 b} & \frac{3.477}{b} \\ 0 + 450 b \cdot 0 & - 450 b + 450 b \cdot 1 & - 120 c + 450 b \frac{3 c}{10 b} & - 1293.3 + 450 b \frac{3.477}{b} \end{array}]$

Bước 5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 0 & 1 & \frac{3 c}{10 b} & \frac{3.477}{b} \\ 0 & 0 & 15 c & 271.35 \end{array}]$

Bước 6

Multiply each element of $R_{3}$ by $\frac{1}{15 c}$ to make the entry at $3, 3$ a $1$ .

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Bước 6.1

Multiply each element of $R_{3}$ by $\frac{1}{15 c}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 0 & 1 & \frac{3 c}{10 b} & \frac{3.477}{b} \\ \frac{0}{15 c} & \frac{0}{15 c} & \frac{15 c}{15 c} & \frac{271.35}{15 c} \end{array}]$

Bước 6.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 0 & 1 & \frac{3 c}{10 b} & \frac{3.477}{b} \\ 0 & 0 & 1 & \frac{18.09}{c} \end{array}]$

Bước 7

Perform the row operation $R_{2} = R_{2} - \frac{3 c}{10 b} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Bước 7.1

Perform the row operation $R_{2} = R_{2} - \frac{3 c}{10 b} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 0 - \frac{3 c}{10 b} \cdot 0 & 1 - \frac{3 c}{10 b} \cdot 0 & \frac{3 c}{10 b} - \frac{3 c}{10 b} \cdot 1 & \frac{3.477}{b} - \frac{3 c}{10 b} \cdot \frac{18.09}{c} \\ 0 & 0 & 1 & \frac{18.09}{c} \end{array}]$

Bước 7.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & \frac{c}{25 a} & \frac{0.3984}{a} \\ 0 & 1 & 0 & - \frac{1.95}{b} \\ 0 & 0 & 1 & \frac{18.09}{c} \end{array}]$

Bước 8

Perform the row operation $R_{1} = R_{1} - \frac{c}{25 a} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Bước 8.1

Perform the row operation $R_{1} = R_{1} - \frac{c}{25 a} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{c}{25 a} \cdot 0 & \frac{b}{5 a} - \frac{c}{25 a} \cdot 0 & \frac{c}{25 a} - \frac{c}{25 a} \cdot 1 & \frac{0.3984}{a} - \frac{c}{25 a} \cdot \frac{18.09}{c} \\ 0 & 1 & 0 & - \frac{1.95}{b} \\ 0 & 0 & 1 & \frac{18.09}{c} \end{array}]$

Bước 8.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & \frac{b}{5 a} & 0 & - \frac{0.3252}{a} \\ 0 & 1 & 0 & - \frac{1.95}{b} \\ 0 & 0 & 1 & \frac{18.09}{c} \end{array}]$

Bước 9

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

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Bước 9.1

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

$[\begin{array}{c} 1 - \frac{b}{5 a} \cdot 0 & \frac{b}{5 a} - \frac{b}{5 a} \cdot 1 & 0 - \frac{b}{5 a} \cdot 0 & - \frac{0.3252}{a} - \frac{b}{5 a} (- \frac{1.95}{b}) \\ 0 & 1 & 0 & - \frac{1.95}{b} \\ 0 & 0 & 1 & \frac{18.09}{c} \end{array}]$

Bước 9.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{0.0648}{a} \\ 0 & 1 & 0 & - \frac{1.95}{b} \\ 0 & 0 & 1 & \frac{18.09}{c} \end{array}]$