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$6 x - 3 y = 3$ , $7 x - y = 3$

Bước 1

Write the system as a matrix.

$[\begin{array}{c} 6 & - 3 & 3 \\ 7 & - 1 & 3 \end{array}]$

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 $1$ .

$[\begin{array}{c} \frac{6}{6} & \frac{- 3}{6} & \frac{3}{6} \\ 7 & - 1 & 3 \end{array}]$

Bước 2.1.2

Rút gọn $R_{1}$ .

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

Bước 2.2

Perform the row operation $R_{2} = R_{2} - 7 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Bước 2.2.1

Perform the row operation $R_{2} = R_{2} - 7 R_{1}$ to make the entry at $2, 1$ a $0$ .

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

Bước 2.2.2

Rút gọn $R_{2}$ .

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

Bước 2.3

Multiply each element of $R_{2}$ by $\frac{2}{5}$ to make the entry at $2, 2$ a $1$ .

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Bước 2.3.1

Multiply each element of $R_{2}$ by $\frac{2}{5}$ to make the entry at $2, 2$ a $1$ .

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

Bước 2.3.2

Rút gọn $R_{2}$ .

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

Bước 2.4

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

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Bước 2.4.1

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

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

Bước 2.4.2

Rút gọn $R_{1}$ .

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

Bước 3

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

$x = \frac{2}{5}$

$y = - \frac{1}{5}$

Bước 4

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

$(\frac{2}{5}, - \frac{1}{5})$