Matematika Berhingga Contoh

$2 x - 6 y = 7$ , $5 x + 2 y = 10$

Langkah 1

Write the system as a matrix.

$[\begin{array}{c} 2 & - 6 & 7 \\ 5 & 2 & 10 \end{array}]$

Langkah 2

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 2.1

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

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Langkah 2.1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{- 6}{2} & \frac{7}{2} \\ 5 & 2 & 10 \end{array}]$

Langkah 2.1.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - 3 & \frac{7}{2} \\ 5 & 2 & 10 \end{array}]$

Langkah 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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Langkah 2.2.1

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

$[\begin{array}{c} 1 & - 3 & \frac{7}{2} \\ 5 - 5 \cdot 1 & 2 - 5 \cdot - 3 & 10 - 5 (\frac{7}{2}) \end{array}]$

Langkah 2.2.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & \frac{7}{2} \\ 0 & 17 & - \frac{15}{2} \end{array}]$

Langkah 2.3

Multiply each element of $R_{2}$ by $\frac{1}{17}$ to make the entry at $2, 2$ a $1$ .

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Langkah 2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{17}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 3 & \frac{7}{2} \\ \frac{0}{17} & \frac{17}{17} & \frac{- \frac{15}{2}}{17} \end{array}]$

Langkah 2.3.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & \frac{7}{2} \\ 0 & 1 & - \frac{15}{34} \end{array}]$

Langkah 2.4

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

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Langkah 2.4.1

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

$[\begin{array}{c} 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & \frac{7}{2} + 3 (- \frac{15}{34}) \\ 0 & 1 & - \frac{15}{34} \end{array}]$

Langkah 2.4.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{37}{17} \\ 0 & 1 & - \frac{15}{34} \end{array}]$

Langkah 3

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

$x = \frac{37}{17}$

$y = - \frac{15}{34}$

Langkah 4

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

$(\frac{37}{17}, - \frac{15}{34})$