الرياضيات المتناهية الأمثلة

$7 x + 12 y = - 1$ , $5 x - 3 y = 7$

خطوة 1

Write the system as a matrix.

$[\begin{array}{c} 7 & 12 & - 1 \\ 5 & - 3 & 7 \end{array}]$

خطوة 2

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 2.1

Multiply each element of $R_{1}$ by $\frac{1}{7}$ to make the entry at $1, 1$ a $1$ .

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خطوة 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{7}$ to make the entry at $1, 1$ a $1$ .

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

خطوة 2.1.2

بسّط $R_{1}$ .

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

خطوة 2.2

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

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خطوة 2.2.1

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

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

خطوة 2.2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{12}{7} & - \frac{1}{7} \\ 0 & - \frac{81}{7} & \frac{54}{7} \end{array}]$

خطوة 2.3

Multiply each element of $R_{2}$ by $- \frac{7}{81}$ to make the entry at $2, 2$ a $1$ .

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خطوة 2.3.1

Multiply each element of $R_{2}$ by $- \frac{7}{81}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{12}{7} & - \frac{1}{7} \\ - \frac{7}{81} \cdot 0 & - \frac{7}{81} (- \frac{81}{7}) & - \frac{7}{81} \cdot \frac{54}{7} \end{array}]$

خطوة 2.3.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{12}{7} & - \frac{1}{7} \\ 0 & 1 & - \frac{2}{3} \end{array}]$

خطوة 2.4

Perform the row operation $R_{1} = R_{1} - \frac{12}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

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خطوة 2.4.1

Perform the row operation $R_{1} = R_{1} - \frac{12}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{12}{7} \cdot 0 & \frac{12}{7} - \frac{12}{7} \cdot 1 & - \frac{1}{7} - \frac{12}{7} (- \frac{2}{3}) \\ 0 & 1 & - \frac{2}{3} \end{array}]$

خطوة 2.4.2

بسّط $R_{1}$ .

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

خطوة 3

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

$x = 1$

$y = - \frac{2}{3}$

خطوة 4

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

$(1, - \frac{2}{3})$