الرياضيات المتناهية الأمثلة

$7 x + 5 y = 32$ , $- 3 x + y = - 20$

خطوة 1

Write the system as a matrix.

$[\begin{array}{c} 7 & 5 & 32 \\ - 3 & 1 & - 20 \end{array}]$

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 $1$ .

$[\begin{array}{c} \frac{7}{7} & \frac{5}{7} & \frac{32}{7} \\ - 3 & 1 & - 20 \end{array}]$

خطوة 2.1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{32}{7} \\ - 3 & 1 & - 20 \end{array}]$

خطوة 2.2

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

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خطوة 2.2.1

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

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{32}{7} \\ - 3 + 3 \cdot 1 & 1 + 3 (\frac{5}{7}) & - 20 + 3 (\frac{32}{7}) \end{array}]$

خطوة 2.2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{32}{7} \\ 0 & \frac{22}{7} & - \frac{44}{7} \end{array}]$

خطوة 2.3

Multiply each element of $R_{2}$ by $\frac{7}{22}$ to make the entry at $2, 2$ a $1$ .

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خطوة 2.3.1

Multiply each element of $R_{2}$ by $\frac{7}{22}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{32}{7} \\ \frac{7}{22} \cdot 0 & \frac{7}{22} \cdot \frac{22}{7} & \frac{7}{22} (- \frac{44}{7}) \end{array}]$

خطوة 2.3.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{5}{7} & \frac{32}{7} \\ 0 & 1 & - 2 \end{array}]$

خطوة 2.4

Perform the row operation $R_{1} = R_{1} - \frac{5}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

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خطوة 2.4.1

Perform the row operation $R_{1} = R_{1} - \frac{5}{7} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{5}{7} \cdot 0 & \frac{5}{7} - \frac{5}{7} \cdot 1 & \frac{32}{7} - \frac{5}{7} \cdot - 2 \\ 0 & 1 & - 2 \end{array}]$

خطوة 2.4.2

بسّط $R_{1}$ .

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

خطوة 3

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

$x = 6$

$y = - 2$

خطوة 4

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

$(6, - 2)$