Matematika Berhingga Contoh

$8 x - 7 y = - 2$ , $16 x + 14 y = 2$

Langkah 1

Write the system as a matrix.

$[\begin{array}{c} 8 & - 7 & - 2 \\ 16 & 14 & 2 \end{array}]$

Langkah 2

Tentukan bentuk eselon baris yang dikurangi.

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Langkah 2.1

Multiply each element of $R_{1}$ by $\frac{1}{8}$ to make the entry at $1, 1$ a $1$ .

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Langkah 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{8}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{8}{8} & \frac{- 7}{8} & \frac{- 2}{8} \\ 16 & 14 & 2 \end{array}]$

Langkah 2.1.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{7}{8} & - \frac{1}{4} \\ 16 & 14 & 2 \end{array}]$

Langkah 2.2

Perform the row operation $R_{2} = R_{2} - 16 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Langkah 2.2.1

Perform the row operation $R_{2} = R_{2} - 16 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{7}{8} & - \frac{1}{4} \\ 16 - 16 \cdot 1 & 14 - 16 (- \frac{7}{8}) & 2 - 16 (- \frac{1}{4}) \end{array}]$

Langkah 2.2.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{7}{8} & - \frac{1}{4} \\ 0 & 28 & 6 \end{array}]$

Langkah 2.3

Multiply each element of $R_{2}$ by $\frac{1}{28}$ to make the entry at $2, 2$ a $1$ .

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Langkah 2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{28}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{7}{8} & - \frac{1}{4} \\ \frac{0}{28} & \frac{28}{28} & \frac{6}{28} \end{array}]$

Langkah 2.3.2

Sederhanakan $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{7}{8} & - \frac{1}{4} \\ 0 & 1 & \frac{3}{14} \end{array}]$

Langkah 2.4

Perform the row operation $R_{1} = R_{1} + \frac{7}{8} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Langkah 2.4.1

Perform the row operation $R_{1} = R_{1} + \frac{7}{8} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{7}{8} \cdot 0 & - \frac{7}{8} + \frac{7}{8} \cdot 1 & - \frac{1}{4} + \frac{7}{8} \cdot \frac{3}{14} \\ 0 & 1 & \frac{3}{14} \end{array}]$

Langkah 2.4.2

Sederhanakan $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{1}{16} \\ 0 & 1 & \frac{3}{14} \end{array}]$

Langkah 3

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

$x = - \frac{1}{16}$

$y = \frac{3}{14}$

Langkah 4

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

$(- \frac{1}{16}, \frac{3}{14})$