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$x - y = - 9$ , $2 x + y = - 6$ , $3 x - y = - 19$ , $4 x + y = - 16$

Bước 1

Write the system as a matrix.

$[\begin{array}{c} 1 & - 1 & - 9 \\ 2 & 1 & - 6 \\ 3 & - 1 & - 19 \\ 4 & 1 & - 16 \end{array}]$

Bước 2

Tìm dạng ma trận hàng bậc thang rút gọn.

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Bước 2.1

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

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Bước 2.1.1

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

$[\begin{array}{c} 1 & - 1 & - 9 \\ 2 - 2 \cdot 1 & 1 - 2 \cdot - 1 & - 6 - 2 \cdot - 9 \\ 3 & - 1 & - 19 \\ 4 & 1 & - 16 \end{array}]$

Bước 2.1.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 9 \\ 0 & 3 & 12 \\ 3 & - 1 & - 19 \\ 4 & 1 & - 16 \end{array}]$

Bước 2.2

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 2.2.1

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

$[\begin{array}{c} 1 & - 1 & - 9 \\ 0 & 3 & 12 \\ 3 - 3 \cdot 1 & - 1 - 3 \cdot - 1 & - 19 - 3 \cdot - 9 \\ 4 & 1 & - 16 \end{array}]$

Bước 2.2.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 9 \\ 0 & 3 & 12 \\ 0 & 2 & 8 \\ 4 & 1 & - 16 \end{array}]$

Bước 2.3

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

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Bước 2.3.1

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

$[\begin{array}{c} 1 & - 1 & - 9 \\ 0 & 3 & 12 \\ 0 & 2 & 8 \\ 4 - 4 \cdot 1 & 1 - 4 \cdot - 1 & - 16 - 4 \cdot - 9 \end{array}]$

Bước 2.3.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 9 \\ 0 & 3 & 12 \\ 0 & 2 & 8 \\ 0 & 5 & 20 \end{array}]$

Bước 2.4

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

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Bước 2.4.1

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

$[\begin{array}{c} 1 & - 1 & - 9 \\ \frac{0}{3} & \frac{3}{3} & \frac{12}{3} \\ 0 & 2 & 8 \\ 0 & 5 & 20 \end{array}]$

Bước 2.4.2

Rút gọn $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & - 9 \\ 0 & 1 & 4 \\ 0 & 2 & 8 \\ 0 & 5 & 20 \end{array}]$

Bước 2.5

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

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Bước 2.5.1

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

$[\begin{array}{c} 1 & - 1 & - 9 \\ 0 & 1 & 4 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & 8 - 2 \cdot 4 \\ 0 & 5 & 20 \end{array}]$

Bước 2.5.2

Rút gọn $R_{3}$ .

$[\begin{array}{c} 1 & - 1 & - 9 \\ 0 & 1 & 4 \\ 0 & 0 & 0 \\ 0 & 5 & 20 \end{array}]$

Bước 2.6

Perform the row operation $R_{4} = R_{4} - 5 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Bước 2.6.1

Perform the row operation $R_{4} = R_{4} - 5 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - 1 & - 9 \\ 0 & 1 & 4 \\ 0 & 0 & 0 \\ 0 - 5 \cdot 0 & 5 - 5 \cdot 1 & 20 - 5 \cdot 4 \end{array}]$

Bước 2.6.2

Rút gọn $R_{4}$ .

$[\begin{array}{c} 1 & - 1 & - 9 \\ 0 & 1 & 4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]$

Bước 2.7

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

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Bước 2.7.1

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

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & - 9 + 1 \cdot 4 \\ 0 & 1 & 4 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]$

Bước 2.7.2

Rút gọn $R_{1}$ .

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

Bước 3

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

$x = - 5$

$y = 4$

$0 = 0$

Bước 4

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

$(- 5, 4)$