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x e^{2 t} [\begin{array}{c} 1 \\ 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$x e^{2 t} [\begin{array}{c} 1 \\ 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1

بسّط.

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خطوة 1.1

بسّط كل حد.

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خطوة 1.1.1

اضرب

x e^{2 t}

$x e^{2 t}$ في كل عنصر من عناصر المصفوفة.

[\begin{array}{c} x e^{2 t} \cdot 1 \\ x e^{2 t} \cdot 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \cdot 1 \\ x e^{2 t} \cdot 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.1.2

بسّط كل عنصر في المصفوفة.

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خطوة 1.1.2.1

اضرب

x

$x$ في

1

$1$ .

[\begin{array}{c} x e^{2 t} \\ x e^{2 t} \cdot 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ x e^{2 t} \cdot 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.1.2.2

اضرب

x e^{2 t} \cdot 0

$x e^{2 t} \cdot 0$ .

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خطوة 1.1.2.2.1

اضرب

0

$0$ في

x

$x$ .

[\begin{array}{c} x e^{2 t} \\ 0 e^{2 t} \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 e^{2 t} \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.1.2.2.2

اضرب

0

$0$ في

e^{2 t}

$e^{2 t}$ .

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + y e^{2 t} [\begin{array}{c} 0 \\ - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.1.3

اضرب

y e^{2 t}

$y e^{2 t}$ في كل عنصر من عناصر المصفوفة.

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} y e^{2 t} \cdot 0 \\ y e^{2 t} \cdot - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} y e^{2 t} \cdot 0 \\ y e^{2 t} \cdot - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.1.4

بسّط كل عنصر في المصفوفة.

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خطوة 1.1.4.1

اضرب

y e^{2 t} \cdot 0

$y e^{2 t} \cdot 0$ .

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خطوة 1.1.4.1.1

اضرب

0

$0$ في

y

$y$ .

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 e^{2 t} \\ y e^{2 t} \cdot - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 e^{2 t} \\ y e^{2 t} \cdot - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.1.4.1.2

اضرب

0

$0$ في

e^{2 t}

$e^{2 t}$ .

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ y e^{2 t} \cdot - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ y e^{2 t} \cdot - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ y e^{2 t} \cdot - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ y e^{2 t} \cdot - 1 \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.1.4.2

انقُل

- 1

$- 1$ إلى يسار

y e^{2 t}

$y e^{2 t}$ .

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - 1 \cdot (y e^{2 t}) \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - 1 \cdot (y e^{2 t}) \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.1.4.3

أعِد كتابة

- 1 (y e^{2 t})

$- 1 (y e^{2 t})$ بالصيغة

- (y e^{2 t})

$- (y e^{2 t})$ .

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + t [\begin{array}{c} 1 \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.1.5

اضرب

t

$t$ في كل عنصر من عناصر المصفوفة.

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \cdot 1 \\ t \cdot 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \cdot 1 \\ t \cdot 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.1.6

بسّط كل عنصر في المصفوفة.

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خطوة 1.1.6.1

اضرب

t

$t$ في

1

$1$ .

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ t \cdot 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ t \cdot 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.1.6.2

اضرب

t

$t$ في

0

$0$ .

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 \end{array}] + [\begin{array}{c} 0 \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.2

اجمع العناصر المتناظرة.

[\begin{array}{c} x e^{2 t} + 0 \\ 0 - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} + 0 \\ 0 - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.3

Simplify each element.

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خطوة 1.3.1

أضف

x e^{2 t}

$x e^{2 t}$ و

0

$0$ .

[\begin{array}{c} x e^{2 t} \\ 0 - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ 0 - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.3.2

اطرح

y e^{2 t}

$y e^{2 t}$ من

0

$0$ .

[\begin{array}{c} x e^{2 t} \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

[\begin{array}{c} x e^{2 t} \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} \\ - y e^{2 t} \end{array}] + [\begin{array}{c} t \\ 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.4

اجمع العناصر المتناظرة.

[\begin{array}{c} x e^{2 t} + t \\ - y e^{2 t} + 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} + t \\ - y e^{2 t} + 0 \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 1.5

أضف

- y e^{2 t}

$- y e^{2 t}$ و

0

$0$ .

[\begin{array}{c} x e^{2 t} + t \\ - y e^{2 t} \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} + t \\ - y e^{2 t} \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

[\begin{array}{c} x e^{2 t} + t \\ - y e^{2 t} \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]

$[\begin{array}{c} x e^{2 t} + t \\ - y e^{2 t} \end{array}] = [\begin{array}{c} 0 \\ 1 \end{array}]$

خطوة 2

يمكن كتابة معادلة المصفوفة في صورة مجموعة من المعادلات.

x e^{2 t} + t = 0

$x e^{2 t} + t = 0$

- y e^{2 t} = 1

$- y e^{2 t} = 1$