الرياضيات المتناهية الأمثلة

$4 x + 2 y = 20.94$ , $6 x + 2 y + 2 z = 42.9$ , $4 x + 1 z = 19.95$

خطوة 1

اضرب $z$ في $1$ .

$4 x + 2 y = 20.94$

$6 x + 2 y + 2 z = 42.9$

$4 x + z = 19.95$

خطوة 2

Write the system as a matrix.

$[\begin{array}{c} 4 & 2 & 0 & 20.94 \\ 6 & 2 & 2 & 42.9 \\ 4 & 0 & 1 & 19.95 \end{array}]$

خطوة 3

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 3.1

Multiply each element of $R_{1}$ by $\frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

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خطوة 3.1.1

Multiply each element of $R_{1}$ by $\frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{4}{4} & \frac{2}{4} & \frac{0}{4} & \frac{20.94}{4} \\ 6 & 2 & 2 & 42.9 \\ 4 & 0 & 1 & 19.95 \end{array}]$

خطوة 3.1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 6 & 2 & 2 & 42.9 \\ 4 & 0 & 1 & 19.95 \end{array}]$

خطوة 3.2

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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خطوة 3.2.1

Perform the row operation $R_{2} = R_{2} - 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 6 - 6 \cdot 1 & 2 - 6 (\frac{1}{2}) & 2 - 6 \cdot 0 & 42.9 - 6 \cdot 5.235 \\ 4 & 0 & 1 & 19.95 \end{array}]$

خطوة 3.2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 0 & - 1 & 2 & 11.49 \\ 4 & 0 & 1 & 19.95 \end{array}]$

خطوة 3.3

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

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خطوة 3.3.1

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 0 & - 1 & 2 & 11.49 \\ 4 - 4 \cdot 1 & 0 - 4 (\frac{1}{2}) & 1 - 4 \cdot 0 & 19.95 - 4 \cdot 5.235 \end{array}]$

خطوة 3.3.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 0 & - 1 & 2 & 11.49 \\ 0 & - 2 & 1 & - 0.99 \end{array}]$

خطوة 3.4

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

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خطوة 3.4.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ - 0 & - - 1 & - 1 \cdot 2 & - 1 \cdot 11.49 \\ 0 & - 2 & 1 & - 0.99 \end{array}]$

خطوة 3.4.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 0 & 1 & - 2 & - 11.49 \\ 0 & - 2 & 1 & - 0.99 \end{array}]$

خطوة 3.5

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

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خطوة 3.5.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 0 & 1 & - 2 & - 11.49 \\ 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 1 + 2 \cdot - 2 & - 0.99 + 2 \cdot - 11.49 \end{array}]$

خطوة 3.5.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 0 & 1 & - 2 & - 11.49 \\ 0 & 0 & - 3 & - 23.97 \end{array}]$

خطوة 3.6

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

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خطوة 3.6.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 0 & 1 & - 2 & - 11.49 \\ - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot - 23.97 \end{array}]$

خطوة 3.6.2

بسّط $R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 0 & 1 & - 2 & - 11.49 \\ 0 & 0 & 1 & 7.99 \end{array}]$

خطوة 3.7

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

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خطوة 3.7.1

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

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 0 + 2 \cdot 0 & 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 11.49 + 2 \cdot 7.99 \\ 0 & 0 & 1 & 7.99 \end{array}]$

خطوة 3.7.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & 0 & 5.235 \\ 0 & 1 & 0 & 4.49 \\ 0 & 0 & 1 & 7.99 \end{array}]$

خطوة 3.8

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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خطوة 3.8.1

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

$[\begin{array}{c} 1 - \frac{1}{2} \cdot 0 & \frac{1}{2} - \frac{1}{2} \cdot 1 & 0 - \frac{1}{2} \cdot 0 & 5.235 - \frac{1}{2} \cdot 4.49 \\ 0 & 1 & 0 & 4.49 \\ 0 & 0 & 1 & 7.99 \end{array}]$

خطوة 3.8.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 2.99 \\ 0 & 1 & 0 & 4.49 \\ 0 & 0 & 1 & 7.99 \end{array}]$

خطوة 4

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

$x = 2.99$

$y = 4.49$

$z = 7.99$

خطوة 5

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

$(2.99, 4.49, 7.99)$