الرياضيات المتناهية الأمثلة

$6 x - 2 y = 32$ , $5 x - 7 y = - 8$

خطوة 1

Write the system as a matrix.

$[\begin{array}{c} 6 & - 2 & 32 \\ 5 & - 7 & - 8 \end{array}]$

خطوة 2

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 2.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

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خطوة 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{6}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{6}{6} & \frac{- 2}{6} & \frac{32}{6} \\ 5 & - 7 & - 8 \end{array}]$

خطوة 2.1.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{16}{3} \\ 5 & - 7 & - 8 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{16}{3} \\ 5 - 5 \cdot 1 & - 7 - 5 (- \frac{1}{3}) & - 8 - 5 (\frac{16}{3}) \end{array}]$

خطوة 2.2.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{16}{3} \\ 0 & - \frac{16}{3} & - \frac{104}{3} \end{array}]$

خطوة 2.3

Multiply each element of $R_{2}$ by $- \frac{3}{16}$ to make the entry at $2, 2$ a $1$ .

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خطوة 2.3.1

Multiply each element of $R_{2}$ by $- \frac{3}{16}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{16}{3} \\ - \frac{3}{16} \cdot 0 & - \frac{3}{16} (- \frac{16}{3}) & - \frac{3}{16} (- \frac{104}{3}) \end{array}]$

خطوة 2.3.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & \frac{16}{3} \\ 0 & 1 & \frac{13}{2} \end{array}]$

خطوة 2.4

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

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خطوة 2.4.1

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

$[\begin{array}{c} 1 + \frac{1}{3} \cdot 0 & - \frac{1}{3} + \frac{1}{3} \cdot 1 & \frac{16}{3} + \frac{1}{3} \cdot \frac{13}{2} \\ 0 & 1 & \frac{13}{2} \end{array}]$

خطوة 2.4.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{15}{2} \\ 0 & 1 & \frac{13}{2} \end{array}]$

خطوة 3

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

$x = \frac{15}{2}$

$y = \frac{13}{2}$

خطوة 4

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

$(\frac{15}{2}, \frac{13}{2})$