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$6 x + 4 y + 5 z = 43$ , $4 x + 4 y - 3 z = 7$ , $3 x - 3 y - 4 z = 12$

Bước 1

Write the system as a matrix.

$[\begin{array}{c} 6 & 4 & 5 & 43 \\ 4 & 4 & - 3 & 7 \\ 3 & - 3 & - 4 & 12 \end{array}]$

Bước 2

Tìm dạng ma trận hàng bậc thang rút gọn.

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Bước 2.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

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Bước 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{6}{6} & \frac{4}{6} & \frac{5}{6} & \frac{43}{6} \\ 4 & 4 & - 3 & 7 \\ 3 & - 3 & - 4 & 12 \end{array}]$

Bước 2.1.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 4 & 4 & - 3 & 7 \\ 3 & - 3 & - 4 & 12 \end{array}]$

Bước 2.2

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

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Bước 2.2.1

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

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 4 - 4 \cdot 1 & 4 - 4 (\frac{2}{3}) & - 3 - 4 (\frac{5}{6}) & 7 - 4 (\frac{43}{6}) \\ 3 & - 3 & - 4 & 12 \end{array}]$

Bước 2.2.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 0 & \frac{4}{3} & - \frac{19}{3} & - \frac{65}{3} \\ 3 & - 3 & - 4 & 12 \end{array}]$

Bước 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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Bước 2.3.1

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

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 0 & \frac{4}{3} & - \frac{19}{3} & - \frac{65}{3} \\ 3 - 3 \cdot 1 & - 3 - 3 (\frac{2}{3}) & - 4 - 3 (\frac{5}{6}) & 12 - 3 (\frac{43}{6}) \end{array}]$

Bước 2.3.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 0 & \frac{4}{3} & - \frac{19}{3} & - \frac{65}{3} \\ 0 & - 5 & - \frac{13}{2} & - \frac{19}{2} \end{array}]$

Bước 2.4

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

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Bước 2.4.1

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

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ \frac{3}{4} \cdot 0 & \frac{3}{4} \cdot \frac{4}{3} & \frac{3}{4} (- \frac{19}{3}) & \frac{3}{4} (- \frac{65}{3}) \\ 0 & - 5 & - \frac{13}{2} & - \frac{19}{2} \end{array}]$

Bước 2.4.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 0 & 1 & - \frac{19}{4} & - \frac{65}{4} \\ 0 & - 5 & - \frac{13}{2} & - \frac{19}{2} \end{array}]$

Bước 2.5

Perform the row operation $R_{3} = R_{3} + 5 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Bước 2.5.1

Perform the row operation $R_{3} = R_{3} + 5 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 0 & 1 & - \frac{19}{4} & - \frac{65}{4} \\ 0 + 5 \cdot 0 & - 5 + 5 \cdot 1 & - \frac{13}{2} + 5 (- \frac{19}{4}) & - \frac{19}{2} + 5 (- \frac{65}{4}) \end{array}]$

Bước 2.5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 0 & 1 & - \frac{19}{4} & - \frac{65}{4} \\ 0 & 0 & - \frac{121}{4} & - \frac{363}{4} \end{array}]$

Bước 2.6

Multiply each element of $R_{3}$ by $- \frac{4}{121}$ to make the entry at $3, 3$ a $1$ .

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Bước 2.6.1

Multiply each element of $R_{3}$ by $- \frac{4}{121}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 0 & 1 & - \frac{19}{4} & - \frac{65}{4} \\ - \frac{4}{121} \cdot 0 & - \frac{4}{121} \cdot 0 & - \frac{4}{121} (- \frac{121}{4}) & - \frac{4}{121} (- \frac{363}{4}) \end{array}]$

Bước 2.6.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 0 & 1 & - \frac{19}{4} & - \frac{65}{4} \\ 0 & 0 & 1 & 3 \end{array}]$

Bước 2.7

Perform the row operation $R_{2} = R_{2} + \frac{19}{4} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Bước 2.7.1

Perform the row operation $R_{2} = R_{2} + \frac{19}{4} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 0 + \frac{19}{4} \cdot 0 & 1 + \frac{19}{4} \cdot 0 & - \frac{19}{4} + \frac{19}{4} \cdot 1 & - \frac{65}{4} + \frac{19}{4} \cdot 3 \\ 0 & 0 & 1 & 3 \end{array}]$

Bước 2.7.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & \frac{5}{6} & \frac{43}{6} \\ 0 & 1 & 0 & - 2 \\ 0 & 0 & 1 & 3 \end{array}]$

Bước 2.8

Perform the row operation $R_{1} = R_{1} - \frac{5}{6} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Bước 2.8.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{6} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{5}{6} \cdot 0 & \frac{2}{3} - \frac{5}{6} \cdot 0 & \frac{5}{6} - \frac{5}{6} \cdot 1 & \frac{43}{6} - \frac{5}{6} \cdot 3 \\ 0 & 1 & 0 & - 2 \\ 0 & 0 & 1 & 3 \end{array}]$

Bước 2.8.2

Rút gọn $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{3} & 0 & \frac{14}{3} \\ 0 & 1 & 0 & - 2 \\ 0 & 0 & 1 & 3 \end{array}]$

Bước 2.9

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

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Bước 2.9.1

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

$[\begin{array}{c} 1 - \frac{2}{3} \cdot 0 & \frac{2}{3} - \frac{2}{3} \cdot 1 & 0 - \frac{2}{3} \cdot 0 & \frac{14}{3} - \frac{2}{3} \cdot - 2 \\ 0 & 1 & 0 & - 2 \\ 0 & 0 & 1 & 3 \end{array}]$

Bước 2.9.2

Rút gọn $R_{1}$ .

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

Bước 3

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

$x = 6$

$y = - 2$

$z = 3$

Bước 4

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

$(6, - 2, 3)$