الرياضيات المتناهية الأمثلة

$2 x + 6 x + 15 x = - 12$ , $4 x + 7 x + 13 x = - 10$ , $3 x + 6 x + 12 x = - 9$

خطوة 1

Move variables to the left and constant terms to the right.

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خطوة 1.1

أضف $2 x$ و $6 x$ .

$8 x + 15 x = - 12$

$4 x + 7 x + 13 x = - 10$

$3 x + 6 x + 12 x = - 9$

خطوة 1.2

أضف $8 x$ و $15 x$ .

$23 x = - 12$

$4 x + 7 x + 13 x = - 10$

$3 x + 6 x + 12 x = - 9$

خطوة 1.3

أضف $4 x$ و $7 x$ .

$23 x = - 12$

$11 x + 13 x = - 10$

$3 x + 6 x + 12 x = - 9$

خطوة 1.4

أضف $11 x$ و $13 x$ .

$23 x = - 12$

$24 x = - 10$

$3 x + 6 x + 12 x = - 9$

خطوة 1.5

أضف $3 x$ و $6 x$ .

$23 x = - 12$

$24 x = - 10$

$9 x + 12 x = - 9$

خطوة 1.6

أضف $9 x$ و $12 x$ .

$23 x = - 12$

$24 x = - 10$

$21 x = - 9$

$23 x = - 12$

$24 x = - 10$

$21 x = - 9$

خطوة 2

Write the system as a matrix.

$[\begin{array}{c} 23 & - 12 \\ 24 & - 10 \\ 21 & - 9 \end{array}]$

خطوة 3

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 3.1

Multiply each element of $R_{1}$ by $\frac{1}{23}$ to make the entry at $1, 1$ a $1$ .

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خطوة 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{23}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{23}{23} & \frac{- 12}{23} \\ 24 & - 10 \\ 21 & - 9 \end{array}]$

خطوة 3.1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{12}{23} \\ 24 & - 10 \\ 21 & - 9 \end{array}]$

خطوة 3.2

Perform the row operation $R_{2} = R_{2} - 24 R_{1}$ to make the entry at $2, 1$ a $0$ .

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خطوة 3.2.1

Perform the row operation $R_{2} = R_{2} - 24 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{12}{23} \\ 24 - 24 \cdot 1 & - 10 - 24 (- \frac{12}{23}) \\ 21 & - 9 \end{array}]$

خطوة 3.2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{12}{23} \\ 0 & \frac{58}{23} \\ 21 & - 9 \end{array}]$

خطوة 3.3

Perform the row operation $R_{3} = R_{3} - 21 R_{1}$ to make the entry at $3, 1$ a $0$ .

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خطوة 3.3.1

Perform the row operation $R_{3} = R_{3} - 21 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{12}{23} \\ 0 & \frac{58}{23} \\ 21 - 21 \cdot 1 & - 9 - 21 (- \frac{12}{23}) \end{array}]$

خطوة 3.3.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{12}{23} \\ 0 & \frac{58}{23} \\ 0 & \frac{45}{23} \end{array}]$

خطوة 3.4

Multiply each element of $R_{2}$ by $\frac{23}{58}$ to make the entry at $2, 2$ a $1$ .

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خطوة 3.4.1

Multiply each element of $R_{2}$ by $\frac{23}{58}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{12}{23} \\ \frac{23}{58} \cdot 0 & \frac{23}{58} \cdot \frac{58}{23} \\ 0 & \frac{45}{23} \end{array}]$

خطوة 3.4.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{12}{23} \\ 0 & 1 \\ 0 & \frac{45}{23} \end{array}]$

خطوة 3.5

Perform the row operation $R_{3} = R_{3} - \frac{45}{23} R_{2}$ to make the entry at $3, 2$ a $0$ .

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خطوة 3.5.1

Perform the row operation $R_{3} = R_{3} - \frac{45}{23} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{12}{23} \\ 0 & 1 \\ 0 - \frac{45}{23} \cdot 0 & \frac{45}{23} - \frac{45}{23} \cdot 1 \end{array}]$

خطوة 3.5.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{12}{23} \\ 0 & 1 \\ 0 & 0 \end{array}]$

خطوة 3.6

Perform the row operation $R_{1} = R_{1} + \frac{12}{23} R_{2}$ to make the entry at $1, 2$ a $0$ .

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خطوة 3.6.1

Perform the row operation $R_{1} = R_{1} + \frac{12}{23} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{12}{23} \cdot 0 & - \frac{12}{23} + \frac{12}{23} \cdot 1 \\ 0 & 1 \\ 0 & 0 \end{array}]$

خطوة 3.6.2

بسّط $R_{1}$ .

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

خطوة 4

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

$x = 0$

$0 = 1$

$0 = 0$

خطوة 5

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

$(error = error)$